Key Words to Understand China: The Fight Against COVID-19 for daily precautions, policy publicity, and information dissemination. Based on the translations and relevant data from Chinese and English media, this study makes a tentative analysis of the linguistic, ideological, and cultural features of English used in the translations. The findings are that China English is outstanding with respect to translating some Chinese-specific words and expressions in the translations, which generally results from the method of literal rendering. It is suggested that literal translation and transliteration can be used more frequently in English texts related to big events in order to strengthen the Chinese characteristics of China English as a major variety of world Englishes.During the COVID-19 outbreak, the China Academy of Translation translated and officially released five groups of  Although the origin of COVID-19 has not yet been confirmed, the virus caused by SARS-CoV-2 instead of viral escape from a laboratory “most likely arose in bats, and then spread to humans via an as-yet unidentified intermediary animal,” according to the World Health Organization (WHO).ril 2020 . In respanguages . The latanguages , 2020c. Some scholars have so far studied the characteristics of English as a first, second or foreign language used to report seminal global events, such as wars, PHEICs, terrorist attacks, and sports games. For example, It is of great importance to study English used in reports on big events, such as COVID-19, especially in countries and regions where English is used as a second or foreign language because such English has a great influence on English varieties, such as China English. As a major variety of world Englishes, China English, formerly labeled “Chinglish,” refers to “English with Chinese characteristics” , p. 1, aA Tale of Two Cities: “It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity. . ..”  at its WeChat public platform by the middle of April 2020, and the first three groups in more than a dozen foreign languages were officially approved and released by the CFLPA via various media and channels, such as the website of the Translators Association of China  Xinhuanet-English version; (4) The New York Times, The Guardian, US Today, The Australian, and New Scientist. There are 52,417 words/characters in the corpus of this study, the inclusion of the linguistic innovations into the sample is purely thematic, and the new verbal signs which occur once in the corpus are included into the sample. A few abbreviations are used in the case study for the convenience of discussion, including PY, WT, LT, and CT which, respectively, refer to Chinese Pinyin version, word-for-word translation, literal translation, and the CAT’s translation of Chinese examples. In the following sections, we analyze the linguistic, ideological, and cultural features of English used in the CAT’s translations from bilingual, translation, and bicultural perspectives.The KWUC-FAC is a Chinese-English bilingual version which consists of six sections, including “Decisions by the Central Leadership,” “Anti-Epidemic Guidelines and Arrangements,” “Effective Measures,” “About COVID-19,” “International Aid,” and “Brave Fighters” . In each section, the key words, phrases or sentences in the Chinese government documents are used as titles and followed by a detailed account of them. There are a total of 615 key words in the five groups in which 181 words in the first three groups had been authorized by the CFLPA. A comparative approach is taken in this study to compare Chinese originals with their English translations, and these translations are also compared with similar English expressions in Chinese and Western media in order to summarize the features of epidemic-related China English. Data for this project is taken from the websites or WeChat public platforms of the following sources: (1) China Academy of Translation; (2) 1    新增  确诊  病例xīnzēng  quèzhěn  bìnglìPY:  CT:  newly confirmed cases2  对  抗疫  医务  人员  保护、  关心、  爱护du  kàngyì  yīwù  rényuán  băohù    guānxīn  àihùPY:  CT:  to provide full protection and care to medical workers fighting against the epidemic3  打赢  疫情  防控   狙击战dăyíng  yìqíng  fángkòng   zŭjīzhànPY:  CT:  to fight and win the battle against the epidemicTranslation can be defined as “the replacement of textual material in one language (SL) by equivalent textual material in another language (TL)” , p. 20. dă, “fight”) in the set phrase “打赢”  seems unnecessary and thus can be omitted because “win the battle” presupposes “fight the battle.” In Chinese, omission of some sentence part gives rise to obscurity in meaning or semantic implicitness which is an important feature of Chinese 5  外防  输入、  内防   扩散wàifáng  shūrù,  nèifáng  kuòsàn  WT  outside prevent entering, inside prevent spreadingPY:  CT:  preventing the coronavirus from entering and spreading within a region6  应收  尽收、  应治  尽治yīngshōu  jìnshōu,  yīngzhì  jìnzhì  WT:  should admit all admitted, should treat all treatedPY:  CT:  admitting all suspected and confirmed cases for treatmentIn examples 1 and 2, the expressions “newly confirmed cases” and “fighting against the epidemic” are acceptable, but they are not as concise as “new infections” and “fight the coronavirus” used by Western media . In exam Chinese , p. 101,wŭhànshì) and the object is “four groups of people”  which include confirmed cases, suspected cases, febrile patients who cannot rule out the possibility of infection, and close contacts. Likewise, in examples 5 and 6, it is not clear what should be prevented from entering from outside and spreading inside, and who should be admitted and treated. The CAT’s translations illustrate that they are “coronavirus” and “suspected and confirmed cases.” Specifically, the context indicates that as for “外防输入、内防扩散,” the subject is “the community”  and the object is “imported cases”  and “the outbreak” ; as for “应收尽收、应治尽治,” the subject is “the hospitals”  and the object is “confirmed and suspected cases” . Therefore, the implied context-bound meaning should be clarified by adding proper words in translating in order to help the international community to understand China’s official guidelines on the epidemic control completely. Parataxis is another feature of Chinese in which phrases and clauses are often juxtaposed without the use of connectors, while English is a hypotactic language in which phrases and clauses are joined with conjunctions or prepositions In example 4, the Chinese original does not tell us who should carry out the screening work and what should be screened, and according to the context, the subject is “Wuhan City”  is used four times in the original. This lexical repetition is a striking feature of Chinese which can be viewed as a special form of hypotaxis . But the word “集中” is omitted in the translation, and its meaning is implied by “dedicated” and “all.” Let us look at one more example:8  早  发现、  早  报告、  早  隔离、  早  治疗zăo  fāxiàn,  zăo  bàogào,  zăo gélí,    zăo  zhìliáoPY:  WT:  early discovery, early report, early isolation, early treatmentCT:  early detection, reporting, isolation and treatmentIn example 7, the Chinese original is a typical paratactic sentence in which four clauses are put together without using a connector in between. However, in the CAT’s translation, the Chinese paratactic sentence is transformed into a hypotactic one in the English version via the use of such connectors as “in,” “by,” “from,” “and,” and “with.” Moreover, the word “集中” (zăo) is used four times in the original, while its literal rendering “early” is used only once in the English version because repetition should, in many cases, be avoided in English via ellipsis or substitution 10  对口  支援  duìkŏu  zhīyuánPY: CT:  pairing assistance In example 8, the word “早” , “奋战” , “作战” , “战斗” , “备战” , “战胜” , “打赢” , “阻击战” , “战斗力” , “攻坚战” , “总体战” , “人民战争” , “战线” , “前线” , “防线” , “军令状” , “堡垒” , and “牺牲” . The use of war metaphors in the Chinese discourse of battling the coronavirus reflects the militarism of the Chinese people 12  这  是  动员令,  也是  宣言书,  更是 zhè  shì  dòngyuánling,  yěshì  xuānyánshū,  gèngshì PY:   军令状。jūnlìngzhuàng.WT:   This is mobilization order, also is manifesto, moreover is military order.CT:   This is a mobilization and an order.In containing the coronavirus spread, great achievements had been achieved in China under the leadership of the CPC and the Central Government whose efforts were acknowledged by the international community. Just as the WHO pointed out, “China has rolled out perhaps the most ambitious, agile and aggressive disease containment effort in history” . The scie people , p. 127,rénmín zhànzhēng, “people’s war”), “总体战” , and “阻击战” , but none of them is retained in the translation. Instead, the general expression “the nation’s war” is used to cover and represent their meanings. Similarly, the powerful expressions of “动员令” , “宣言书” , and “军令状”  are not faithfully represented in the English version in example 12 because “令”  in “动员令,” “宣言书,” and “军令”  in “军令状” are not translated, resulting in the weakening of the expected effectiveness of publicity. Other types of metaphor are also seen in Xi’s talks. For example:13  生命    重于  泰山。  shēngmìng  zhòngyú  tàishān.PY: LT:   Life is weightier than Mount Tai.CT:   Saving lives is of paramount importance.14  疫情  就是  命令，  防控  就是  责任。yìqíng  jiùshì  mìngling,  fángkòng  jiùshì  zérèn.PY:   LT:   COVID-19 is the order; its prevention and control is the responsibility.CT:   Go where there is epidemic, fight it till it perishes.Three war-related expressions are used in the original in example 11, including “人民战争” In the Chinese sentences in examples 13 and 14, people’s life is compared to Mount Tai, and the epidemic to a military order, which emphasized on the significance of safeguarding the people’s lives and controlling the virus spread. The belief contained in example 13 that the life of all people is of equal importance constitutes a sharp contrast to some Western leaders’ philosophy of herd immunity and natural elimination which put elderly people at great risk . In the shèng, “win”), President Xi clearly pointed out the key to battling the virus by saying that China would eventually win the war of battling the epidemic if Wuhan and Hubei could successfully control the virus, and he called on the whole nation to provide assistance to them. This example shows that there are also good repetitions of words in English, just like the opening sentence in Charles Dickens’ A Tale of Two Cities. It should be noted that unique political language was often seen in the CPC’s guidelines, and some expressions excerpted from Xi’s talks had become slogans for all Party members and cadres in combating the coronavirus. For example:16  让  党旗  在  防控  疫情  斗争    第一线ràng  dăngqí  zài  fángkòng  yìqíng  dòuzhēng  dìyīxiànPY:   高高  飘扬。gāogāo piāoyáng.CT:   Let the Party flag fly high on the front line of the anti-epidemic war.17   一个  党员   就是  一面  旗帜,  一个  支部  就是  yīgè  dăngyuán   jiùshì  yīmiàn  qízhì,  yīgè  zhībù  jiùshìPY:   一座  堡垒。yīzuò  băolěi.CAT:  A Party member is like a flag and a Party branch is like a fortress.COVID-19 first broke out in Wuhan, Hubei’s capital city in January 2020, and Hubei was the hardest-hit province in China. Therefore, the central task of the whole country focused on the epidemic control in Wuhan and Hubei. The English version in example 15 is a literal rendering, and by repeating the word “胜” (dăngyuán gànbù xiàchén) during the outbreak, which means the deep community engagement of all Party members and cadres in the epidemic prevention and control . Finally, the shift in emotional coloring is sometimes seen in translating commendatory terms. For example, as a commendatory word, “殉职”  is rendered as the neutral word “die”; “崇高”  in “崇高的敬意”  is also a commendatory word but omitted, resulting in the weakening of the original commendatory coloring .The political slogans in examples 16 and 17 have special connotations in the context of the CPC leadership. Confronted with the epidemic crisis, President Xi asked all Party members and Party organizations at all levels to take the lead to bring into full play their abilities by going deep into grass-roots communities and villages to fight the virus. This was clearly illustrated in the catchphrase “党员干部下沉” 19  雷神山    医院léishénshān  yīyuànPY:   CT:   Leishenshan Hospital20  中西医    并用zhōngxīyī  bìngyòngPY:   CT:   combined use of TCM and Western medicineDue to the lockdown and self-quarantine, COVID-19 had a great impact on China’s politics, economy, culture, healthcare, education, travel, and daily life. This impact was reflected in the CAT’s translations. For example:方舱医院,” fāngcāng yīyuàn) were built to deal with suspected or mild cases , and it can be argued that the Chinese original “方舱”  in the name of hospitals of this category might allude to Noah’s ark which is translated into Chinese as “诺亚方舟” (nuòyà fāngzhōu). As the saying goes, God help those who help themselves. The self-reliance of the Chinese nation played a decisive role in controlling COVID-19. As seen in example 20, in the treatment of infected people, traditional Chinese medicine (TCM) and Western medicine were combined because they have their own advantages, and their combined use “had achieved good effects during China’s fight against SARS in 2003” . During the period of lockdown, hundreds of millions of people had been placed in some form of isolation in order to contain the virus spread 22    解决  好  生活   必需品供应    的  PY:   jiějué  hăo  shēnghuó   bìxūpĭn  gòngyīng  de“最后  一  公里”  问题“zuìhòu  yī  gōnglĭ”   wèntíCT:   to ensure the “last kilometer” delivery of daily necessities23    武汉  快递  小哥  汪勇：  组织  志愿者PY:   wǔhàn  kuàidì  xiăogē  wāng  yŏng:  zŭzhī  zhìyuànzhě接送  医护  人员jiēsòng yīhù rényuánCT:   Wang Yong: a courier volunteerWuhan was the epicenter of the epidemic in China and the first city in lockdown, with more than 50,000 infections. As the number of infected people skyrocketed, the hospitals available in Wuhan were unable to tackle confirmed cases. As a result, two big hospitals were built within 20 days, a “Chinese speed” in times of urgency, and they were named “Huoshenshan Hospital” and “Leishenshan Hospital.” As illustrated in examples 18 and 19, “Huoshenshan” and “Leishenshan” are the Chinese Pinyin versions or transliterations of “火神山” and “雷神山” whose literal meanings are “Fire God Mountain” and “Thunder God Mountain.” It was believed that the Fire God and the Thunder God could help the Chinese to stop the epidemic because they take control of fire and lightning which can kill the virus . Moreoves spread . The supkuàidì xiǎogē) who were praised by Western media because they took the risk of losing their lives to do so On behalf of the CPC Central Committee, Vice Premier Sun Chunlan, together with the Central Guiding Team, arrived in Wuhan on January 27, 2020, and found that daily necessities could not be sent to residents fast and smoothly . Therefore, it was proposed that the governors and city mayors should take full responsibility for food supplies, such as Chinese cabbage, pakchoi seedlings, and musang king durians , and ensure the “last kilometer” smooth delivery of food and other daily necessities. With the help of tens of thousands of community cadres and courier volunteers, such as Wang Yong, ample and timely daily supplies were guaranteed, as indicated in examples 21, 22, and 23. As for quarantined city dwellers who were unable to do the cooking by themselves, the meals were brought to them by thousands of “deliverymen”  The outbreak forced China’s Ministry of Education (MOE) to postpone the commencement of the spring semester, and launch the “Home Study Initiative”  . Based oChina Daily, 27 February 2020). As a form of gratitude, there appeared many English versions of the verse line by the Chinese, such as “Miles apart, but close at heart” in example 25 and “Rivers low, mountains high; the same moon in the sky” by Professor Zhao Yanchun, the translator of the Chinese classic Three Character Classic.The Tokyo-based Japan Youth Development Association donated 20,000 masks to Hubei, on the boxes of which was written the Chinese verse line “山川异域, 风月同天” , and this powerful poetic message had a very positive response on Chinese social media  as “China will do everything possible to combat it”  is not as good as the literal rendering of “Better be in vain in nine out of ten prevention efforts than lose the defense line in one out of ten thousand efforts.” Although a number of translation methods are employed in the CAT’s translations of the 25 examples above with a varying use frequency, such as annotation (2 times), free translation (2 times), addition (3 times), omission (8 times), and literal translation/transliteration (10 times), it can be argued that the methods of literal translation and transliteration can best keep Chinese characteristics in rendering Chinese-specific words and expressions, such as “fight and win” for “打赢,” “actively win” for “积极争取,” and “Huoshenshan Hospital” for “火神山医院.” It is not advisable to achieve clarity in meaning and brevity in expression at the cost of losing Chinese uniqueness in foreign publicity.Lexical repetition can be retained in translating in most cases in order to achieve semantic emphasis and represent Chinese characteristics. For instance, the CAT’s translation of “early detection, reporting, isolation and treatment” for “早发现, 早报告, 早隔离, 早治疗” can be modified as “early detection, early reporting, early isolation, early treatment,” which can better show China’s fast responses to the coronavirus. War metaphors as an important feature in the epidemic-related Chinese discourse can also be maintained, as illustrated in “people’s war, total warfare, tough fighting” for “人民战争, 总体战, 阻击战” which is better than the CAT’s generalization of “the nation’s war.” Similarly, the CAT’s generalization of “宁可十防九空, 不可失防万一” , a world peace and security-oriented concept proposed by China . The pandemic has proved that all of us live in one and the same world, and only when the whole world should be united as one, could the virus be wiped out. However, the ideological confrontation seems to continue to exist between China and the West. The fact that COVID-19 first broke out in China made some Westerners call it “Chinese virus” or “China virus,” as used in Western media’s headlines, such as “Schools isolate students in danger of China virus” . The WHO named the new virus “COVID-19” for the very purpose of preventing the use of such prejudiced terms which are related to “a geographical location, an animal, an individual or a group of people” , decided to go on a voyage to Japan; after six attempts and the loss of his eyesight, Jianzhen finally arrived, and made a significant contribution to the spread of Buddhism in Japan” . The story behind the verse line invoked both shared history and mutual appreciation for ancient poetry in Chinese. Many Chinese people were touched by the poetic expression of support from the Japanese during the COVID-19 outbreak.During the extended lockdown, millions of families were forced to live in confined spaces in Wuhan and other cities in Hubei. Many problems, such as boredom and physical abuse, took place in these families . MoreoveEnglish was used widely and frequently in Chinese media during the COVID-19 outbreak for daily precautions, policy publicity, and information dissemination, and the CAT’s translations provided the media with importance reference for the use of English. China English was outstanding with respect to translating some Chinese-specific words and expressions in the translations, and it generally resulted from the method of literal rendering. With this method, some of the CAT’s translations can be improved with more Chinese characteristics if we take linguistic acceptability into consideration at the same time. In some sense, translation, especially literal translation, is an important way to develop and enrich China English. As for ideological publicity, by virtue of lexical repetitions and use of war metaphors, the greatest importance was attached to people’s lives in President Xi’s talks which highlighted high efficiency and strategic rationality in combating the virus. Although there were some problems during the period of lockdown, such as boredom and physical abuse, poetic message and musical power brought comfort and warmth to quarantined people in cold China in the late winter and early spring of 2020.


The original version of this article unfortunately contained a mistake. The wrong Table 2 was published and in Table 5, document measures in the column “Range” were mistakenly listed as dates.The corrected Tables 3D−2D”In the section “Single linear regression analysis of the angle ρ and the ΔBoth equations should have a “minus” sign in the beginning :p < 0.0001, R2 = 0.0446, Fig. 5c). On the left, angle ρ showed a linear regression relationship with the difference of AV angles Δ3D−2D .. On the left, angle ρ showed a linear regression relationship with the difference of AV angles Δ3D−2D .”“which means that angle ρIt should be:Δ3D−2D on the right ”.“which means that angle ρ has a significant negative influence on


Experimental design applications for discriminating between models have been hampered by the assumption to know beforehand which model is the true one, which is counter to the very aim of the experiment. Previous approaches to alleviate this requirement were either symmetrizations of asymmetric techniques, or Bayesian, minimax, and sequential approaches. Here we present a genuinely symmetric criterion based on a linearized distance between mean‐value surfaces and the newly introduced tool of flexible nominal sets. We demonstrate the computational efficiency of the approach using the proposed criterion and provide a Monte‐Carlo evaluation of its discrimination performance on the basis of the likelihood ratio. An application for a pair of competing models in enzyme kinetics is given. The crucial problem is that one typically cannot construct an optimal model‐discrimination design without already knowing which model is the true one, and what are the true values of its parameters. In this respect, the situation is analogous to the problem of optimal experimental design for parameter estimation in nonlinear statistical models  linear, T‐optimality, albeit being feasible, are not a natural choice. We further suppose that we do not use the full prior distribution of the unknown parameters of the models, which rules out Bayesian approaches such as Felsenstein , int(Θ1), and ε1,…,εn are unobservable random errors. For both k=0,1 and any x∈X, we will assume that the functions ηk are differentiable on int(Θk); the gradient of ηk in θk∈int(Θk) will be denoted by ∇ηk. Our principal assumption is that one of the models is true but we do not know which, that is, for k=0 or for k=1 there exists θ¯k∈Θk such that yi=ηk+εi.Let N, where σ2∈. The assumption of the same variances of the errors for both models is plausible if, for instance, the errors are due to the measurement device and hence do not significantly depend on the value being measured. The situation with different error variances requires a more elaborate approach, compared with Fedorov and Pázman −logL(θ^1)<>0. Under the normality, homoskedasticity, and independence assumptions, this decision is equivalent to a decision based on the proximity of the vector (yi)i=1n of observations to the vectors of estimated mean values )i=1n and )i=1n.The choice of the best discrimination rule based on the observations is generally a nontrivial problem. However, it is natural to compute the maximum likelihood estimates m0≠m1 to counterbalance favoring models with greater number of parameters, Cox (L(θ^0)/L(θ^1)(em1/em0)n/n∼, which corresponds to the Bayesian information criterion; see Schwarz  be the design used for the collection of data prior to the decision, and assume that model η0 is true, with the corresponding parameter value θ¯0. Note that this comes without loss of generality and symmetry as we can equivalently assume model η1 to be true. Then, the probability of the correct decision based on the likelihood ratio is equal to(yi)i=1n follows the normal distribution with mean )i=1n and covariance σ2In.Let d, where Ej is the set of all possible mean values of the observations under the model j, j=0,1, and d denotes the infimum distance between all pairs of elements of two sets, is developed as follows.Clearly, probability , and denote y:=(yi)i=1n, ηj(θj):=)i=1n for j=0,1. Note that we can express P[d≤d]. Now, let R=∥ε∥, where ε=y−η0(θ¯0), be the norm of the vector of errors. Assuming R≤d/2 we obtaind/2≤d) and consequently[R≤d/2] implies the event [d≤d], that is, d, which depends on the underlying experimental design. Although this maximization is much simpler than maximizing d through linearization, as will be explained in the following section.Consider a fixed experimental design 1.1η0=θ0x and η1=eθ1x. Furthermore, for the moment we assume just two observations y1,y2 at fixed design points x1=−1 and x2=1, respectively. In this case evidently θ^0=y2−y12 and θ^1 is the solution of 2e−θ(y1−e−θ)−2eθ(y2−eθ)=0, which for −2≤y1≤2 is the log root of the polynomial γ4−γ3y2+γy1−1. Figure Let 2θ∼0∈int(Θ0) and θ∼1∈int(Θ1) be nominal parameter values, which satisfy the basic discriminability conditionη0≠η1 for some x∈X. Let us introduce regions Θ∼0⊆int(Θ0)⊆Rm and Θ∼1⊆int(Θ1)⊆Rm containing θ∼0 and θ∼1; we will consequently call Θ∼0 and Θ∼1flexible nominal sets. It is evident that optimal designs depend on the parameter spaces in the same way as on our flexible nominal sets  be a design. Let us perform the following particular linearization of Model ηk=0,1 in θ∼k:Fk(D) is the n×m matrix given byak(D) is the n‐dimensional vectorε=T is a vector of independent N errors.Let ak(D) plays an important role and, although it is known, we cannot subtract it from the vector of observations, as is usual when we linearize a single nonlinear regression model. However, if ηk corresponds to the standard linear model then ak(D)=0n for any D.Note that for the proposed method the vector 2.1θ0∈Θ∼0,θ1∈Θ∼1. The criterion δ can be viewed as an approximation of the nearest distance d of the mean‐value surfaces of the models, in the neighborhoods of the vectors )i=1n and )i=1n; see the illustrative Figure Consider the design criterionD= represented by the counting measure ξ on X defined as# means the size of a set. Let θ∼=T. For all x∈X letθ0∈Θ∼0, θ1∈Θ∼1 and θ=T we haveM in Equations i· is the ith row of the matrix , with parameter θ and independent, homoskedastic errors ε1,…,εn with mean 0; we will call response difference model.We will now express the δ‐criterion as a function of the design 2.2D, expression δ2(D|θ) is a quadratic function of θ=T. Moreover, both δ(D|θ) and δ2(D|θ) are convex because they are compositions of an affine function of θ and convex functions ∥.∥ and ∥.∥2, respectively. Clearly, if the flexible nominal sets are compact, convex, and polyhedral, optimization For a fixed design δ(D|θ) as follows. AsΘ∼:=Θ∼0×Θ∼1 in the response difference model with artificial observationsAlternatively, we can view the computation of Θ∼0=Θ∼1=Rm, the infimum in R package bvls; see Stark and Parker  be defined via formula δapp2 is linear on Ξ. Moreover, letδapp2 is positive homogeneous and concave on Ξ.For The following simple proposition collects the analytic properties of a natural analogue of δ defined on the linear vector space Ξ of all finite signed measures on δapp2 implies that an s‐fold replication of an exact design leads to an s‐fold increase of its δ2 value. Consequently, a natural and statistically interpretable definition of relative δ‐efficiency of two designs D1 and D2 is given by δ2(D1)/δ2(D2), provided that δ2(D2)>0.Positive homogeneity of D be the set of all n‐point designs. A design D∗∈D will be called δ‐optimal, ifΘ∼0={θ∼0} and Θ∼1={θ∼1}, then δ(D∗) is strictly positive. However, for larger flexible nominal sets it can happen that δ(D∗)=0.Let As the evaluation of the δ‐criterion is generally very rapid, the calculation of a δ‐optimal, or nearly δ‐optimal design is similar to that for standard design criteria. For instance, in small problems we can use complete‐enumeration and in larger problems we can employ an exchange heuristic, such as the KL exchange algorithm :=Rm, Θ∼1(∞):=Rm, such that r can be considered a tuning (set) parameter governing the size of the flexible nominal sets. In Θ∼0(1) and Θ∼1(1) are “unit” nondegenerate compact cuboids centered on respective nominal parameters. For any design D and r∈, we defineδr‐optimal values of the problem will be denoted byProposition 2.2D be a design. Functions δr2(D), δr(D), o2(r), o(r) are nonincreasing and convex in r on the entire interval . (b) There exists r∗<∞, such that for all r≥r∗: (i) o(r)=o(∞); (ii) Any δ∞‐optimal design is also a δr‐optimal design.(a) Let For simplicity, we will focus on cuboid flexible nominal sets centered at the nominal parameter values. This choice can be justified by the results of Sidak , in partD be an n‐point design and let 0≤r1≤r2∈.(a) Let δr12(D)≥δr22(D) follows from definitions o2(r1)≥o2(r2) follows from the fact that a maximum of nonincreasing functions is a nonincreasing function. Monotonicity of δr(D) and o(r) in r can be shown analogously.Inequality δr2(D) in r, let α∈ and let rα=αr1+(1−α)r2. For all r∈, let θ^r denote a minimizer of δr2(D|·) on Θ∼(r):=Θ∼0(r)×Θ∼1(r). Convexity of δ2(D|θ) in θ and a simple fact αθ^r1+(1−α)θ^r2∈Θ∼(rα) yieldδr2(D) is convex in r. The convexity of δr(D) in r can be shown analogously. The functions o2 and o, as pointwise maxima of a system of convex functions, are also convex.To prove the convexity of D of size n, the function δ∞2(D|·) is nonnegative and quadratic on R2m, therefore its minimum is attained in some θD∈R2m. There is only a finite number of exact designs of size n, and Θ∼(r)↑rR2m, which means that there exists r∗<∞ such that θD∈Θ∼(r∗) for all designs D of size n. Let r≥r∗. We haveD(∞) be any δ∞‐optimal n‐trial design. The equality (i) and the fact that δr(D(∞)) and o(r) are nonincreasing with respect to r gives□(b) For any design  of relevant set parameters; increasing the set parameter beyond r∗ leaves the optimal designs as well as the optimal value of the δ‐criterion unchanged. We will call any such r∗ a set upper bound.The second part of Proposition r∗. Our experience shows that it usually requires only a small number of recomputations of the δr‐optimal design, even if rini is small and q is close to 1, resulting in a good set upper bound r∗ , as well as to obtain δr‐optimal designs in steps 2 and 9 of the algorithm itself.Algorithm Algorithm 12.4X={1.00,1.01,…,2.00}, θ∼0=e, and θ∼1=1. Note that these nominal values satisfy η0=η1. Moreover, let us set Θ∼0(1)= and Θ∼1(1)=, and let the required size of the experiment be n=6. First, we computed the value o2(∞)≈0.02614. Next, we used Algorithm rini=0.3 and q=1+10−6, which returned a set upper bound r∗≈0.6787 after as few as seven computations of δr‐optimal designs. Informed by r∗, we computed δr‐optimal designs for r=0.01,0.1,0.2,…,0.7. The resulting δr‐optimal designs are displayed in Figure Θ∼(r)s are very narrow, the δr‐optimal design is concentrated in the design point x=2, effectively maximizing the difference between η0 and η1. For larger values of r, the δr‐optimal design has a 2‐point and ultimately a 3‐point support.Consider the models from the motivating example. Let r∗, beyond which the values of δr are constantly 0 for all designs. These cases can be identified by solving a linear programming (LP) problem, as we show next.Proposition 2.3D¯ be the design that performs exactly one trial in each point of X. Consider the following LP problem with variables r∈R, θ0∈Rm, θ1∈Rm:T. Then, r∗ is a finite set upper bound. Moreover, o(r)=0 for all r∈.Let For some pairs of competing models there exists a set upper bound D and its nonreplication version Dnr we have δr(Dnr)=0 implies δr(D)=0. Moreover, if D2⪰D1 in the sense that D2 is an augmentation of D1 then δr(D2)=0 implies δr(D1)=0. Now let T be a solution of r≥r∗ and let D be any design. Definition of δr and the form of δr(D¯)=0. From D¯⪰Dnr we see that then δr(Dnr)=0, hence δr(D)=0. The proposition follows.□From the expression r∗ obtained using Proposition n, that is, it is a set upper bound simultaneously valid for all design sizes. The basic discriminability condition implies that r∗≠0.Note that a0(D¯) and a1(D¯) are zero. Therefore, T for any r≥0 such that both Θ∼0(r) and Θ∼1(r) cover 0m. That is, for the case of linear models, there is a finite set upper bound r∗ beyond which the δr‐values of all designs vanish. However, the same holds for specific nonlinear models, including the ones from Section Proposition 2.4r∗ such that o(r)=0 for all r∈.Assume that both competing regression models are linear provided that we consider a proper subset of their parameters as known constants. Then  has a fiIf the competing models are linear, vectors erefore,  has a fek0<m components of θ0 converts Model 0 to a linear model. More precisely, let θ01,…,θ0m denote the components of θ0 and assume thatγj(0), j=k0+1,…,m. Choose θ^0 such that θ^0j=θ∼0j for j=1,…,k0, and θ^0j=0 for j=k0+1,…,m. Make an analogous assumption for Model 1 and also define θ^1 analogously. It is then straightforward to verify that for the design D¯ from Proposition Fk(D¯)θ^k+ak(D¯)=0d, where d=#X, for both k=0,1. Therefore, any T such that θ^0∈Θ∼0(r) and θ^1∈Θ∼1(r) is a solution of □Without loss of generality, assume that fixing the first In the following, we numerically demonstrate that the δ design criterion leads to designs which yield a high probability of correct discrimination.3y is alternatively modeled asx1 denotes the concentration of the substrate and x2 the concentration of an inhibitor. The data used in Bogacka et al. , θ0,2,θ1,2∈ designs for λ and employ them for model discrimination. Note that also this method is not fully symmetric as it requires a nominal value for λ for linearization of In Atkinson , the twoθ∼01=θ∼11=θ∼21=10, θ∼02=θ∼12=θ∼22=4.36, θ∼03=2.58, θ∼13=5.16, and θ∼23=3.096. However, note that particularly for model X=X1×X2=×.The nominal values used in Atkinson  obviouslT−optimal designs assuming λ=0 (A1) and λ=1 (A4), a compound T‐optimal design (A3), and a Ds‐optimum (A2) for the encompassing model . We will compare our δ‐optimal designs against exact versions of these designs, properly rounded by the method of Pukelsheim and Rieder . We then simulated the data‐generating process under each model for N=10000 times and calculated the total percentages of correct discrimination (hit rates) when using the likelihood ratio as decision rule.As we are assuming equal variances for the two models we are using the estimate for the error standard deviation δ1,δ2, and δ3, which represent a range of different nominal intervals. Specifically we chose Θ∼k=[θ∼k1±rσ∼k1]×[θ∼k2±rσ∼k2]×[θ∼k3±rσ∼k3]k=0,1, where we chose θ∼kj=θ^kj and σ∼kj=σ^kj for k=0,1 and j=1,2,3. The tuning parameter r was set to three levels: r=1 (which is close to the lower bound of still providing a regular design), r=5 and r=15 , respectively. To make the latter more precise: the models in considerations are such that if we fix the last two out of the three parameters, then they become one‐parameter linear models. Therefore, using Proposition r∗. Solving r∗≈64.02. Note that the same bound is valid for all design sizes n. Designs A1–A4 and δ1 all contain four support points, while δ2 has six and δ3 has five, respectively. A graphical depiction of the designs is given in Figure We are comparing the designs A1–A4 to three specific δ designs Robustness study: As we would like to avoid comparing designs only when the data are generated from the nominal values , we perturbed the data‐generating process by drawing parameters from uniform distributions drawn at θ∼±c×σ∼θ, where c then acts as a perturbation parameter. Under these settings all these designs fare pretty well as can be seen from Table A4 and δ2 seem to outperform the other competing designs by usually narrow margins except perhaps for A1, which is consistently doing worst. Note that in a real situation the true competitors of δ‐optimal designs are just A2 and A3 as it is unknown beforehand which model is true.3.2n=60 is nearly perfect, we are required to inflate the error variance. However, using additive normal errors in the data‐generating process and inflating the variance by a large enough factor, would generate a large number of negative observations, which renders likelihood estimation invalid. So, the data‐generating process was adapted to use multiplicative log‐normal errors. The observations were then rescaled to match the means from the original process. This way we are at liberty to inflate the error variance by any factor without producing faulty observations. Note that now the data‐generating process does not fully match the assumptions under which the designs were generated, but this can just be considered an extended robustness study as it holds for all compared designs equally. We could of course also have calculated the designs under the same data‐generating process, but as the fit of the model to the original data is not greatly improved and models As the discriminatory power of all the designs for 5×σ^ (and c=0). The respective designs δ1–3 were qualitatively similar to those given in Figure n=60 observations from these designs a thousand times.Perturbation of the parameters here did not exhibit a discernible effect, while the error inflation still does. For brevity, we here report only again the results for using 1 (remember it being the T‐optimum design assuming η0 true), while δ1 and δ2 come close (and beat the true competitors A2 and A3) with A1 again being clearly the worst.The corresponding boxplots of the correct classification rates are given in Figure 4T‐optimality, can be undertaken with efficient routines of quadratic optimization that in general enhance the speed of computations by an order of magnitude. An optimal exact design problem is a problem of discrete optimization, and the efficiency of its solution critically depends on the speed of evaluation of the design criterion. By a series of approximations, we substituted the theoretically ideal but numerically infeasible computation of the probability of correct discrimination with a simple convex optimization, which can be solved rapidly and reliably. Combined with the proposed methodology of flexible nominal sets, we can construct an entire sequence of exact experimental designs efficient for discrimination between models. Also it was shown in an example that resulting designs are competitive in their actual discriminatory abilities.We have presented a novel design criterion for symmetric model discrimination. Its main advantage is that design computations, unlike for The notion of flexible nominal sets may have independent merit. Note again the distinction between parametric spaces and flexible nominal sets . Parametric spaces usually encompass all theoretically possible values of the parameters, while flexible nominal sets can contain the unknown parameters with very high likelihood, and still be significantly smaller than the original parameter spaces. In this paper, we do not completely specify the process of constructing the flexible nominal sets, but if we perform a two stage experiment, with a second, discriminatory phase, the potential specification through confidence intervals is an important problem.As the approach suggested offers a fundamentally new way of constructing discriminatory designs, many properties are yet unexplored. A nonexhaustive list of questions follows.Sequential procedure. The proposed method lends itself naturally to a two‐stage procedure, where parameter estimates and confidence intervals are employed as nominal values in the second stage. Even sequential generation of design points can be straightforwardly implemented.Approximate designs. Proposition δapp2 is concave on the set of all approximate designs. Therefore, it is possible to work out a minimax‐type equivalence theorem for δ‐optimal approximate designs, and use specific convex optimization methods to find a δ‐optimal approximate designs numerically. For instance, it would be possible to employ methods analogous to Burclová and Pázman , a natural criterion closely related to δr‐optimality can be defined asδ∼r requires a multivariate nonconvex optimization for the evaluation in each design D, which entails possible numerical difficulties and a long time to compute an optimal design. However, the δr‐optimal design, which can be computed rapidly and reliably, can serve as efficient initial design for the optimization of δ∼r. Note that if Θ∼0 is a singleton containing only the nominal parameter value for Model 0, the δr‐optimal designs could potentially be used as efficient initial designs for computing the exact version of the criterion of T‐optimality.Selection of the best design from a finite set of possible candidates. As most proposals for the construction of optimal experimental designs, the method depends on the choice of some tuning parameters or even on entire prior distributions (in the Bayesian approach), which always results in a set of possible designs. It would be interesting to develop a comprehensive Monte‐Carlo methodology for the choice of the best design out of this pre‐selected small set of candidate designs. A useful generalization of the rule would take into account possibly unequal losses for the wrong classification.Noncuboid sets. The methodology could certainly be extended to other types of flexible nominal sets, particularly when we are interested in functional relations among the parameters. However, then the particularly efficient box constrained quadratic programming algorithm could not be utilized.Higher order approximations. As a referee remarked it is possible to employ tighter approximations of the sets of mean values of responses than the one that we suggest. For instance, it would be possible to use the local curvature of the mean‐value function. However, this may also lead to the loss of numerical efficiency of the method.More than two rival models. Another referee remark leads us to point out the natural extension to investigate a weighted sum or the minimum δ over all paired comparisons. The implications of these suggestions, however, requires deeper investigations.Different error variances. Yet another referee requested a clarification for how to proceed in case of unequal error variances for the two models. In case the functional form of these variances are known simple standardizations of the models will suffice. All other cases, including dependencies of the errors, will require more elaborate strategies.Combination with other criteria. The proposed method can produce poor or even singular designs for estimating model parameters. Because of this problem, already mentioned in Atkinson and Fedorov (DT‐optimality. The same approach is possible for δ‐optimality. However, our numerical experience suggests that for a large enough size of the flexible nominal set, the δ‐optimal designs tend to be supported on a set that is large enough for estimability of the parameters, without any combination with an auxiliary criterion. A detailed analysis goes beyond the scope of this paper. Fedorov , Atkinso Fedorov  used a cThe authors declare that there is no conflict of interest.This article has earned an Open Data badge for making publicly available the digitally‐shareable data necessary to reproduce the reported results. The data is available in the Reproducible Research” for making publicly available the code necessary to reproduce the reported results. The results reported in this article could fully be reproduced.This article has earned an open data badge “SUPPORTING INFORMATIONClick here for additional data file.


This article has been withdrawn by the authors. The Journal statesthat the immunoblots corresponding to ‘Rac1’ in figure 4B and 4H were duplicated.The immunoblots corresponding to ‘beta-Tubulin’ in figure 4B and ‘Pak2’ in figure 5Ewere duplicated. Possible micrograph reuse for ‘RhoA ASO3 + Thrombin’ panel infigure 4C and ‘Control ASO’ panel in figure 10D. Additional potential reuse in ‘Pak2ASO2 + Thrombin’ panel in figure 5C and ‘G-alpha-qASO2’ panel in figure 10D.Possible reuse in immunoblot panels for ‘GTP-RhoA’ in figure 4E and ‘GTP-Rac1’ infigure 10J. Additional potential reuse between micrograph panels ‘SCH79797+Thrombin’in figure 8A and ‘Control ASO’ in figure 2C. There was possible image reuse betweenimmunoblots ‘beta-Tubulin’ panel in figure 7A and ‘Pak2’ panel in figure 7F.Finally, there was possible image reuse for ‘PY20’ immunoblot panel in figures 7Dand 9E, which represent different experimental conditions. Due to these immunoblotand image duplications, the authors wish to withdraw this article.


These maps can be realized as attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure‐preserving maps f the background Oxtoby–Ulam measures induced by Lebesgue measure for f on the interval are physical on the disc and in addition there is a dense set of maps f defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; that is, the parametrized family is statistically stable. We also find an arc in the generic Lebesgue measure‐preserving set of maps and construct a family of disk homeomorphisms parametrized by this arc which induces a continuously varying family of pseudo‐arc attractors with prime ends rotation numbers varying continuously in . It follows that there are uncountably many dynamically non‐equivalent embeddings of the pseudo‐arc in this family of attractors.The main goal of this paper is to study topological and measure‐theoretic properties of an intriguing family of strange planar attractors. Building toward these results, we first show that any generic Lebesgue measure‐preserving map  We provide a parametrized family of strange attractors where all the attracting sets are homeomorphic but nevertheless, as we shall see later, they exhibit a variety of rich dynamical behavior and have good measure‐theoretic and statistical properties. The attracting sets of this family are all homeomorphic to the one‐dimensional space of much interest in Continuum Theory and beyond, called the pseudo‐arc. A continuum is a non‐empty compact connected metric space. The pseudo‐arc may be regarded as the most intriguing planar continuum not separating the plane. On the one hand its structure is quite complicated, since it does not contain any arc. On the other hand it reflects much regularity in its shape, since it is homeomorphic to any of its proper subcontinua. For the history of the pseudo‐arc and numerous results connecting it to other mathematical fields we refer the reader to the introduction in residual iated in  that theTheorem 1.1Gδ set T⊂Cλ(I) such that if f∈T then for every δ>0 there exists a positive integer n so that fn is δ‐crooked.There is a dense Here we prove another topological property of Lebesgue measure‐preserving maps, which might be the most surprising of the properties yet; namely we prove:δ‐crookedness is not an easy‐to‐state property (see Definition δ‐crookedness in the sense of Theorem Corollary 1.2Cλ(I)‐generic map as a single bonding map is the pseudo‐arc.The inverse limit with any The One should note that all interesting “global” dynamics in interval maps can be reflected in Lebesgue measure‐preserving maps  and let f^:I^→I^ be the natural extension off (or the shift homeomorphism). A natural projectionπ0:I^→I defined by π0(x)=x0semi‐conjugatesf^ to f.At this point, let us mention that one can view inverse limits with single bonding maps as the simplest invertible dynamical extensions of the dynamics given by the bonding map. Let us state this fact more precisely. Denote by Y let g:Y→Y be an invertible dynamical system and let p:Y→I factor g to f. Then p factors through π0: that is, f^ is the minimal invertible system which extends f ⊂C(I) the class of interval maps with the dense set of periodic points and by CDP(I)¯ its closure. Building on the well‐known properties of interval maps mentioned earlier (see Remark Corollary 1.3CDP(I)¯‐generic map as a single bonding map is the pseudo‐arc.The inverse limit with any Denote by C(I) (where C(I) denotes the class of all continuous interval maps). On the other hand, Bing 27]). ThiC ) the set of all continuous mappings from a metric space X to a metric space Y . We equip the space C with the metric of uniform convergence ρ. Let D⊂R2 denote a closed topological disk. We say that a compact set K⊂D⊂R2 is the  attractor of h:D→D in D if for every x∈D∖∂D, the omega limit setωh(x)⊂K and for some z∈K we have that ωh(z)=K.Denote by K⊂D∖∂D⊂R2 we can associate the circle of prime ends as the compactification of D∖K. If h∼:R2→R2 preserves orientation and h∼(K)=K, h∼(D)=D then h∼ induces an orientation‐preserving homeomorphism of the prime ends circle, and therefore it gives a natural prime ends rotation number. The prime ends rotation number allows one to study boundary dynamics of underlying global attractors and distinguish their embeddings from dynamical point of view, see Definition Theorem 1.5T be a dense Gδ subset of Cλ(I) from Theorem {ft}t∈⊂T⊂Cλ(I) and a parametrized family of homeomorphisms {Φt}t∈⊂H varying continuously with t having Φt‐invariant pseudo‐arc attractors Λt⊂D for every t∈ so that(a)Φt|Λt is topologically conjugate to f^t:I^f→I^f.(b){Λt}t∈ vary continuously in the Hausdorff metric.The attractors (c)Φt vary continuously with t in the interval .Prime ends rotation numbers of homeomorphisms (d){Λt}t∈.There are uncountably many dynamically non‐equivalent planar embeddings of the pseudo‐arc in the family Let To a non‐degenerate and non‐separating continuum Cλ(I) and it would be interesting to know if this set itself is pathwise connected. Third, let us mention that the homeomorphism group of the pseudo‐arc contains no non‐degenerate continua  should be continuous. There is also no immediate argument why homeomorphisms ht∘Φt|Λt∘ht−1 are different. Since in this work we are interested only in embeddings from dynamical perspective we leave the following question about the topological nature of embeddings open.Question 1t≠t′∈ the attractors Λt and Λt′  non‐equivalently embedded?Are for every This result is interesting also from several other aspects. First, it is the first example in the literature (to our knowledge) of a parametrized family of strange attractors where the attractors are proven to be homeomorphic, yet the boundary dynamics on the attractor is very rich. This result underlines the fact that pseudo‐arc is among one‐dimensional continua a special object with respect to its flexibility to permit a variety of different dynamical behavior. Furthermore, let us also note that Theorem continua , so Theosults of  in state equivalent if there is a homeomorphism h:R2→R2 such that h(Λt)=Λt′. Note that h does not have to intertwine the dynamics, so Theorem Recall that two planar embeddings are called Cλ(I), and we take Theorem M is called Oxtoby–Ulam (OU) or good if it is non‐atomic, positive on open sets, and assigns zero measure to the boundary of manifold M (if it exists) ⊂R denote the unit interval. Let diam(A) denote the diameter of A⊂I. Let λ denote the Lebesgue measure on the underlying Euclidean space. By C(I) we denote the family of all continuous interval maps. Furthermore, let Cλ(I)⊂C(I) denote the family of all continuous Lebesgue measure‐preserving functions of I. We equip both C(I) and Cλ(I) with the metric of uniform convergenceρ: we shall use B for the open ball of radius ξ centered at x∈X and for a set U⊂X we shall denoted to denote the Euclidean distance on the underlying Euclidean space. We say that a map f is piecewise linear (or piecewise affine) if it has finitely many critical points  and is linear on every interval of monotonicity . We say that an interval map f is leo if for every open interval J⊂I there exists a non‐negative integer n so that fn(J)=I. This property is also sometimes referred in the literature as topological exactness.Let 2.2Definition 2.1f∈C(I), let a,b∈I and let δ>0. We say that f is δ‐crooked betweena and b if for every two points c,d∈I such that f(c)=a and f(d)=b, there is a point c′ between c and d and there is a point d′ between c′ and d such that |b−f(c′)|<δ and |a−f(d′)|<δ. We will say that f is δ‐crooked if it is δ‐crooked between every pair of points.Let For an easier visualization of the concept defined in the following definition we refer the reader to Figure Definition 2.2(cr[n])n=1∞⊂N be the sequence defined in the following way: cr[1]:=1, cr[2]:=2 and cr[n]:=2cr[n−1]+cr[n−2] for each n⩾3.Let The following two definitions and the first part of the third definition were introduced in . We willDefinition 2.3g1 and g2 be two maps of I into itself such that g1(0)=g2(0)=0 and g1(1)=g2(1)=1. Suppose m⩾3 is an integer and s is a real number such that 0<s<1/2. Then ϕ is the function of I into itself defined by the formula:Let Definition 2.4n⩾3 denoten∈N, let simplen‐crooked mapσn:I→I be defined in the following way :=1−σn(t) for each t∈I. Let σnL:=σn| where σn|:→, σnR:=σn| where σn|:→. Similarly as above let σ−nL and σ−nR denote the reflections of σnL and σnR, respectively.For each integer Definition 2.5n⩾3 the maximal number of intervals of monotonicity of σn of the same length is denoted by #σn.For every integer Definition 2.6f:→ where ,⊂R and a<b,a′<b′ is an odd function around the point∈R2 if the graph of f| equals the rotation for angle π of the graph of f| around .We say that a function Observation 2.7n⩾3(1)σn is a piecewise linear continuous function.(2)#σn=cr[n]. In particular, for even n, #σn is even and it is odd for odd n.it holds that (3)σn has uniform slope being ±cr[n]n.(4)σn is an odd function around the point , that is, it holds that σ−nL(t)=σnR(t+1/2) and σ−nR(t+1/2)=σnL(t) for all t∈.For every integer ε=3/n, however let us note that this estimate is far from optimal for n small.Remark 2.8ε>0 and n is sufficiently large to ensure 2/n<ε, the map σn is ε‐crooked.By Proposition 3.5 in , if ε>0 We will often use the following remark with Remark 2.9ε‐crooked with small ε>0 it cannot be a small perturbation of the identity map. To work with small perturbations of identity cr[n]+cr[n−1]n.a continuous and piecewise linear function with the uniform slope (2).an odd function around the point For each odd integer For each odd integer formulaλ^n,k(t)Lemma 2.11n⩾7 and every integer k⩾1 it holds that if t,s∈ are such that |λ^n,k(t)−λ^n,k(s)|<n−1n then λ^n,k is 3n‐crooked between λ^n,k(t) and λ^n,k(s).For every odd integer λ^n,k is generated using rescaled σ−(n−1)R, σn, and σ−(n−1)L that are properly shifted vertically; to simplify the notation in this proof we will refer to the three parts of the definition of λ^n,k simply by σ−(n−1)R, σn, and σ−(n−1)L while remembering about the rescaling and shift. Therefore, λ^n,k consists of blocks of the formSi for i∈{1,…,2(n+k)} and which gives a well‐defined order on them.First note that the function a,b∈, so that |a−b|<n−1n. If points c,d∈, where f(c)=a and f(d)=b, are contained in some Si for i∈{1,…,2(n+k)}, the claim follows directly from Remark S1 or S2(n+k) note that the images are restrictions of σ−(n−1) and thus we can again use Remark Fix c,d∈ are contained in two adjacent blocks Sj<Sj+1, λ^n,k|Sj=σ−(n−1) and λ^n,k|Sj+1=σn for some j∈{1,…,2(n+k)−1}. Assume that c∈Sj and d∈Sj+1 . Then, since λ^n,k(Sj)⊂λ^n,k(Sj+1) it follows that there always exist c′∈Sj+1 such that c<c′<d and so that f(c′)=a. Thus we obtain the claim using Remark Now assume that points c,d∈ are contained in two non‐adjacent blocks, c∈Sj and d∈Sj′ for |j−j′|⩾2. If a∈λ^n,k(Sj′) or b∈λ^n,k(Sj) then we can find two adjacent blocks between Sj and Sj′ and use the arguments from the preceding paragraph as we assumed that |a−b|<n−1n. Note that a∈λ^n,k(Sj′) or b∈λ^n,k(Sj) holds always except if λ^n,k|Sj=λ^n,k|Sj′=σn, |j−j′|=2, and n−2n⩽|a−b|⩽n−1n. But in this case observe that 3n‐crookedness of σn assures that we can find a point between c and d in either Sj or Sj′ with the required image value.□Now let us assume that Definition 2.12n,k∈N define the flip mapFor all Definition 2.13n⩾7 and every k∈N define the map λn,k:I→I byFor every odd integer Now we have all the ingredients to define the final map with which we will work in this section.t∈I.for all λn,k for n=7 and note that the properly scaled “middle” building blocks of λn,k are as in Figure Observation 2.14n⩾7 and each integer k⩾1 the map λn,k is(1)±(cr[n]+cr[n−1]),a continuous piecewise linear function with the uniform slope being (2),an odd function around the point (3)λn,k(jn+k−1)=jn+k−1 for all j∈{0,…,n+k−1}.such that For each odd integer See Figure λn,k preserves Lebesgue measure for all odd n⩾7 and k⩾1.Definition 2.15j∈{1,…,n+k−1} and some fixed integer k⩾1 and odd integer n⩾7 denote by Ij:=1n+k−1⊂I and letjth vertical strip andjth horizontal strip. Let p1:I×I→I (p2:I×I→I) be the natural projection onto the first (second) coordinate. Furthermore, define the maximal number of intervals of monotonicity of λn,k of the diameter exactly 1n+k−1 in Vj  byFor every Now we will turn to the proof that the function Observation 2.16n⩾7 and each integer k⩾1 it holds that(1)λn,k|Ij is an odd function around the point (2j−12(n+k−1),2j−12(n+k−1)) for any j∈{n+12,…,k+n−12}.the function (2)#(Vj)=cr[n]+cr[n−1] for any j∈{1,…,n+k−1} (since function Fl does not change #(Vj)).(3)diam)=nn+k−1 if j∈{n+12,…,k+n−12}.(4)diam)=n+2j−12(n+k−1) if j∈{1,…,n−12}.For each odd integer λn,k preserves Lebesgue measure we will also implicitly use the following simple observation.Observation 2.17E differentiable interval map f preserves Lebesgue measure if and only iff∈Cλ(I) and any non‐degenerate interval J⊂I,Comp(f−1(J)) denotes the set of all connected components of f−1(J).A piecewise monotone and up to a finite points To check whether Proposition 2.18n⩾7 and each integer k⩾1 the map λn,k preserves Lebesgue measure on I.For each odd integer #(Vj)=cr[n]+cr[n−1] for all j∈{1,…,n+k−1} and since by Observation λn,k has uniform slope  we only need to show that #(Hj)=cr[n]+cr[n−1] since diam(p1(Vj))=diam(p2(Hj′)) for any j,j′∈{1,…,n+k−1}. We will consider boxesBj,l:=Vj∩Hl and the number of injective branches of λn,k of Bj,l denoted by # for all j,l∈{1,…,n+k−1}, see Figure From Observation j∈{n+12,…,k+n−12}. Then #(Vj)=#+⋯+#+⋯+#=#+2#+2#+⋯+2#, due to Observation n is odd. But note that =# for all m∈{j−n−12,…,j+n−12} and thus #(Hj)=#+⋯+#+⋯+#=#+⋯+#+⋯+#=#(Vj) which finishes this part of the proof.First assume that j∈{1,…,n−12}∪{k+n+12,…,n+k−1}. By Observation j∈{1,…,n+12}. By Definition #(Hj)=#+#⋯+#+⋯+#=#+2#+2#+⋯+2#=#(Vj) .□Now assume that Equation  and therλn,k fits in the context of Proposition 5 of  defined for every odd integer n⩾7 and every integer k⩾1 byλn,k from λ∼n,k applying the flip function from Definition a,b∈ so that |a−b|<ε we have that the map λ∼n,k is 3γ‐crooked between a and b. However, since one obtains λn,k from λ∼n,k and the flip function by the definition at most decreases the distances between the function values, the claim follows immediately.Now let us prove that for every (iii)A⊂I is a subinterval so that diam(A)<2(cr[n]+cr[n−1])(n+k−1), then by Observation diam)>(cr[n]+cr[n−1])diam(A)2>diam(A). If a subinterval A⊂I is such that γ>diam(A)⩾2(cr[n]+cr[n−1])(n+k−1), then it follows that diam)⩾γ>diam(A) because A contains at least one full interval of monotonicity of the diameter of image being γ. Now assume that γ⩽diam(A)⩽2γ. Then there are x,y∈A such that λn,k(y)=maxλn,k(Ij) and λn,k(x)=minλn,k(Ij′) where |j−j′|⩽1 and A∩Ij≠∅, A∩Ij′≠∅. Note that A is contained in at most three different intervals Ii and λn,k(Ii) covers itself and two neighboring intervals on each side, provided it is not the interval containing endpoints 0 or 1 ,λn,k(y)]⊂λn,k(A), in particular diam)⩾diam(A). If A contains 0 or 1 it follows from the construction of λn,k that A⊂λn,k(A) and thus subsequently diam)⩾diam(A); even more, diam)⩾n+12(n+k−1). When diam(A)>2γ, then there are j<j′ such that Ij∪…∪Ij′⊂A and diam(Ij∪…∪Ij′) is maximal possible for the interval Ij∪…∪Ij′ under inclusion . But then clearly A⊂ ε/2=n−12(n+k−1). Namely, |λ∼n,k(x)−λ∼n,k(y)|⩾n+1n+k−1 where x and y are as in the previous paragraph. After applying the flip function we obtain |λn,k(x)−λn,k(y)|⩾n+12(n+k−1), see Figure This part follows from the arguments in the last paragraph if one notes that (c)Ij the first maximal value of λn,k lies γ apart for all j∈{1,…,k+n−12} and it is exactly γ greater from the maximal value of λn,k(Ij−1) for all j∈{2,…,k+n−12} and we will use this fact heavily. Furthermore, if we denote α:=γcr[n]+cr[n−1] that is α is the length of the smallest interval of monotonicity, then λn,k(xi)=maxλn,k for each i∈{1,…,k+n−12−1}, where xi are as in Observation γ⩽diam(A)<2γ the conclusion follows from Observation For this part Observation A⊂I be an interval such that diam(A)⩾2γ, say A:=⊂I. Then there is maximal j such that Ij⊂A. If j⩾k+n−12 then maxλn,k(A)=1 and so λn,k=λn,k where b+r⩽1 for some non‐negative real number r. So assume that j<k+n−12. Next, let j′ be the maximal number such that xj′∈, where xj′ is as in Observation j′=k+n−12 then λn,k(xj′)=1 and clearly xj′−xj<r+γ so λn,k⊂B,r+γ). For the last case, fix any x∈. Then, by the repetative structure of building blocks of the graph λn,k ⩽λn,k(y)+sγ. But this shows that λn,k⊂B,r+γ), which concludes the proof.□Let piecewise linear maps that preserve Lebesgue measureλ by PLλ and piecewise linear maps that are leo and preserveλ by PLλ(leo). If additionally they satisfy Markov property, which means there is a partition 0=a0<a1<…<an=1 such that for each i the map f is monotone and there are s<t such that f=, then we call them Markov piecewise linear leo maps that preserveλ. The set of all such maps is denoted PLMλ(leo).Definition 2.21f∈C(I) is called admissible, if |f′(t)|⩾4 for every t∈I for which f′(t) exists and f is leo.A piecewise linear map We denote the set of λn,k from Lemma Lemma 2.22f:I→I be an admissible map. Let η and δ be two positive real numbers. Then there is an admissible map F:I→I and there is a positive integer n such that Fn is δ‐crooked and ρ<η. Moreover, if f∈Cλ(I), such F can be also chosen to be in Cλ(I).Let Having the appropriate Lebesgue measure‐preserving perturbations F=f∘λn,k and proceed with the proof as in ⊂I→I we say that they are λ‐equivalent if for each Borel set A∈B(I) (where B(I) from now on denotes Borelσ‐algebra on I) it holds that,f∈Cλ(I) and ⊂I we denote by C the set of all continuous maps λ‐equivalent to f|. We defineFor maps In what follows we will also need perturbations of maps that preserve Lebesgue measure, similarly as in .DefinitDefinition 2.26f∈Cλ(I) and ⊂I. For any fixed m∈N, let us define the map h=h⟨f;,m⟩:→I by :h⟨f;,m⟩∈C for each m and h⟨f;,m⟩∈C∗ for each m odd. In particular, if h=h⟨f;,m⟩, m odd, we will speak of regularm‐fold window perturbationh of f .Let The following definition is illustrated by Figure Lemma 2.27ε>0 and every leo map f∈Cλ(I) there exists F∈Cλ(I) such that F is admissible and ρ<ε.For every For more details on the perturbations from the previous definition we refer the reader to .Lemma 2Cλ(I), so let us start with such map g with ρ<ε/2. Let us choose a Markov partition 0=a0<a1<…<an=1 for g. Periodic points are dense for a leo map, so including points from periodic orbits as points in the partition, we may also require that |ai+1−ai|<δ for any fixed δ>0. In particular, we have that g is monotone on each interval , diam)<ε/2 and g= for some indices k<k′. Now, repeating construction in Lemma 5 of  by its regular m‐fold window perturbation, with odd and sufficiently large m. This way F is admissible with ρ⩽diam)<ε/2. Window perturbations are invariant for Cλ(I), hence F∈Cλ(I). Clearly also g=F=. Therefore, for each i there is n∈N such that Fn=I. But then, if we fix any open set U⊂I, then since slope on intervals of monotonicity of F is at least 4, there is M∈N such that FM(U) contains three consecutive intervals of monotonicity, and therefore FM+1(U)⊃.□By Lemma mma 5 of  we constLemma 2.28 be two maps so that ρ<ε. If f is δ‐crooked, then F is (δ+2ε)‐crooked.The following lemma is an essential ingredient in the construction of pseudo‐arc using inverse limits in .Lemma 2Now we are ready to prove the main theorem of this section.Proof of Theorem 1.1k⩾1 let the set Ak⊂Cλ(I) be contained in the set of maps f such that fn is (1/k−δ)‐crooked for some n and some sufficiently small δ>0. First observe that Ak is dense. Namely, by Lemma Cλ(I). If we start with such a map g then first applying Lemma g to a map f∈Ak by an arbitrarily small perturbation. But if f∈Ak and n,δ are constants from the definition of Ak, then by Lemma B⊂Ak. This shows that Ak contains an open dense set. But then the setGδ and clearly each element f∈T satisfies the conclusion of the theorem.□For any 2.3CDP(I)⊂C(I) the family of interval maps with a dense set of periodic points. First note that CDP(I) is not a closed space. However, since CDP(I)¯ is closed in C(I) it is thus a complete space as well. Now we state a useful remark that is given and explained in the introduction of  let f∼t be defined by f∼t(27)=f∼t(47)=f∼t(1721)=f∼t(1)=0 and f∼t(37)=f∼t(57)=f∼t(1921)=1 and piecewise linear between these points on the interval . Furthermore on the interval x∈ let:Proposition 4.1t∈, the map f∼t∈Cλ(I).For every For what follows we refer the reader to Figure d by f∼t2=f∼t(47)=f∼0,f∼1∈Cλ(I). For x∈ it holds that s0:=|f∼0′(x)|=7 and s1:=|f∼1′(x)|=212; thus s1/s0=3/2. Note that for any t∈ it holds that for x∈ and y∈I either there exist 3 points of f∼t−1 in  where f∼t has slope 212 or 2 points where ft∼ has slope 7. Therefore, invoking Observation f∼t∈Cλ(I) for all t∈.□Applying Observation Observation 4.2{f∼t}t∈ is a family of continuous piecewise linear maps varying continuously with t∈. Furthermore, f∼0 is an 8‐fold map with f∼0(0)=0 and f∼1 is a 9‐fold map so that f∼1(0)=1 and f∼1(1)=0. Moreover, since it holds that λn,k(0)=0 and λn,k(1)=1 for any odd n and k∈N it holds for g∼t:=f∼t∘λn1,k1∘…∘λnm,km for any k1,…,km⩾1 and odd n1,…,nm⩾1 that g∼0(0)=0, g∼1(0)=1, and g∼1(1)=0.Observation 4.3t∈ and for all points x∈I where f∼t′ is defined it holds that 212⩾|f∼t′(x)|⩾7.For every Observation 4.4t∈ and any subinterval A⊂I which does not contain two critical points it holds that diam(f∼t(A))>3diam(A).For every Due to the previous observation we obtain the following.Proposition 4.5t∈, the map f∼t is leo.For every ⊂I and any t∈. By Observation N∈N such that f∼tN contains two critical points. But then, by the definition of f∼t, we obtain that f∼tN+1=I. Indeed f∼t is leo for all t∈.□Fix any non‐degenerate interval Corollary 4.6t∈, the map f∼t is admissible.For every Combining Observation {f∼t}t∈ with the help of Lemma {ft}t∈⊂Cλ(I) of continuous maps varying continuously with t as well.Lemma 4.7β>0. Let g∼t:=f∼t∘λn1,k1∘…∘λnm,km for some k1,…,km⩾1 and odd n1,…,nm⩾1. There is an integer N⩾0 so that for every 0⩽a⩽b⩽1 with b−a>β and all t∈ it holds that g∼tN=I.Let Now we will perturb the maps N so that 3Nβ>1. By Observation λn,k does not shrink intervals  contains two critical points of f∼t. By the definition of f∼t we have g∼tj+1=I.□Take any Lemma 4.8η and δ be two positive real numbers fixed for the whole family {g∼t}t∈ where g∼t:=f∼t∘λn1,k1∘…∘λnm,km for some k1,…,km⩾1 and odd n1,…,nm⩾1. Then there is a positive integer N such that for every t∈ there exists an admissible map G∼t:I→I so that G∼tN is δ‐crooked and ρ<η. Moreover, G∼t∈Cλ(I) and G∼t=g∼t∘λnm+1,km+1 for some km+1⩾1 and odd nm+1⩾1.Let f in the Lemma from  and . All the maps g∼t have slopes bounded from the above by the same constant, call it s, since slopes of all f∼t are uniformly bounded from the above and all the maps in the composition are piecewise linear. Also, Lemma N for all g∼t which plays the role of Proposition 6 in  we use Lemma {F∼t}t∈∈A1. But if {F∼t}t∈∈A1 and m,δ are constants from the definition of A1, then by Lemma B⊂A1. For the second step we take the family {F∼t}t∈. Proceeding as in the rest of the proof of Theorem Ak the family {ft}t∈⊂T of continuous maps varying with t.□The procedure we take is the same as in the proof of Theorem {ft}t∈, see →D which decomposes D into a continuously varying family of arcs {α}x∈∂D⊂C, so that α(I) are pairwise disjoint except perhaps at the endpoints α, where one requires that α∈I. We can then associate a retractionr:D→I defined by r)=α for every x∈∂D corresponding to the given decomposition. The map is boundary retract, but we need to maintain the disc, so we will collapse only the “inner half” of it (see definition of R below). Recall also, that a continuous map between two compact metric spaces is called a near‐homeomorphism, if it is a uniform limit of homeomorphisms.Now we will briefly describe standard parametrized BBM construction for the family ,1], see  for moresmashR:D→D as a near‐homeomorphism so that:Having the above decomposition in arcs we define unwrapping of {ft}t∈⊂T as a continuously varying family f¯t:D→D of orientation‐preserving homeomorphisms so that for all t:(i)suppf¯t⊂{α;s⩾1/2},(ii)R∘f¯t|I=ft, For the purpose of easier discussion afterward let us fix the unwrappings to be the “rotated graphs” of the corresponding functions, following . Moreover, it follows from the choice of unwrapping that every point from the interior of D^t is attracted to I^t, therefore, I^t is a global attractor for H^t and thus Λt is a global attractor for Φt as well. By Theorem 3.1 from  vary continuously with t and the attractors {Λt}t∈ vary continuously in Hausdorff metric.As a consequence of (i) we obtain that for all  theorem , D^t:=liows from  that Φt|3.1 from  {Φt}t∈ is a family of orientation‐preserving homeomorphisms on a topological disk D⊂R2 continuously varying with t. For every t∈ let Kt⊂intD be a non‐degenerate sphere non‐separating continuum, invariant under Ψt, and assume that {Kt}t∈ vary continuously with t in the Hausdorff metric. Then the prime ends rotation numbers vary continuously with t∈.To a non‐degenerate and non‐separating continuum by Barge .Lemma 4Definition 4.11X and Y be metric spaces. Suppose that F:X→X and G:Y→Y are homeomorphisms and E:X→Y is an embedding. If E∘F=G∘E we say that the embedding E is a dynamical embedding of  into . If E, respectively, E′, are dynamical embeddings of  resp.  into , respectively, , and there is a homeomorphism H:Y→Y′ so that H(E(X))=E′(X′) which conjugates G|E(X) with G′|E′(X′) we say that the embeddings E and E′ are dynamically equivalent.Let Finally, let us define how we distinguish the embeddings from the dynamical perspective. In what follows we generalize the definition from  of equivRemark 4.12Y=Y′=R2 and X,X′ are pseudo‐arcs (in particular plane non‐separating continua). Thus, the dynamical equivalence from Definition H conjugates G with G′ on all R2.In our case Definition 4.13x∈K⊂R2 is accessible if there exists an arc A⊂R2 such that A∩K={x} and A∖{x}⊂R2∖K.We say that a point We will also use the following definition.Now let us prove the main theorem of this section.Proof of Theorem 1.5Items (a) and (b) follow directly from Theorem 3.1 of .Λ0=h(I^0) has an accessible point h0) fixed under Φ0. We choose a horizontal radial arc Q0⊂D that has an endpoint in 0∈I⊂D. Note that by the definition of H0 it holds that H0(Q0)=Q0 and H0|Q0 is a near‐homeomorphism. Thus, J0:=lim← is an arc by the result of Brown  looks like precisely , however we do not delve in that aspect of research in this work.While the embeddings from Theorem 5Cλ(I) is a complete space in C(I) with the supremum metric. However, the space Cλ(I) is not equicontinuous and thus by Arzelá–Ascoli theorem Cλ(I) is not compact. Therefore, we cannot apply the parametrized BBM construction from [Cλ(I) but a priori only from the topological perspective). Thus this section can be viewed as a generalization of the preceding section with the additional measure‐theoretic ingredients.Note that the set ion from  directlyion from  for some5.1X is a compact metric space and that f:X→X is continuous and onto and recall that we denote by X^:=lim← and by πn:X^→I the coordinate projections maps. Recall also that B(X) denotes the σ‐algebra of Borel sets in X. First we will need the following standard result.Theorem 5.1 Suppose ) is a separable Borel space and that f:X→X is onto and B(X)‐measurable. Let B(X^) be the smallest σ‐algebra on X^ such that all the projection maps πi are measurable. If {μn}n∈N0 is a sequence of probability measures on B(X) such that μn(A)=μn+1(f−1(A)) for all A∈B(X), then there exists a unique probability measure μ^ on B(X^) such that μ^(πn−1(A))=μn(A) for all A∈B(X) and each n∈N0.In this subsection we give some measure‐theoretic results that are required later in the construction. Suppose M(X) denote the set of all invariant probability measures on the Borelσ‐algebra B(X). For any μ∈M(X) a continuous function f:X→X induces a map f∗:M(X)→M(X) given by∈lim←(M(X),f∗) can be uniquely extended to a probability measure on X^, that is we have a function:Another result that we use is from . Let M Suppose f:X→X is a continuous function on a compact metric space. Let B(X^) be the smallest σ‐algebra such that all the projection maps πi are measurable and let μ^=G). Then μ^ is f^‐invariant and σ‐invariant.Theorem 6 in  shows thDefinition 5.3μ be an f‐invariant measure on X. Set Bμ is a basin ofμ for f if for all g∈C(X) and x∈Bμ:μphysical forf if there exists a basin Bμ of μ for f and a measurable set B so that B⊂Bμ and λ(B)>0.Let ν^ for the natural extension f^:X^→X^ is called inverse limit physical measure if ν^ has a basin B^ν^ so that λ(π0(B^ν^))>0.An invariant measure Theorem 5.4(Theorems 11 and 12 from ) If μ is a physical measure for f:X→X where X is an Euclidean space, then the induced measure μ^ on X^ is an inverse limit physical measure for the natural extension f^. In particular, there is a basin B^μ^:=π0−1(B) of μ^ for f^ with λ(B)>0.M(I) be the space of Borel probability measures on I equipped with the Prokhorov metricD defined byμ,ν∈M(I). The following (asymmetric) formulaD coincides with the weak∗topology for measures, in particular M(I) equipped with the metric D is a compact metric space (for more details on Prokhorov metric and weak* topology the reader is referred to [Let erred to ).5.2M is called Oxtoby–Ulam (OU) or good if it is non‐atomic, positive on open sets, and assigns zero measure to the boundary of manifold M (if it exists) [λ^f on the inverse limit lim← using Lebesgue measure λ on D, where the map f is a near‐homeomorphism of D and identity on ∂D. Then we will find a homeomorphism Θf:lim←→D and define a push‐forward measure λf=(Θf)∗λ^f. By this construction it is clear that λf is an OU‐measure.In what follows, we will adjust Oxtoby–Ulam technique of full Lebesgue measure transformation  to the c exists) , 46. In Θf, we need the following definitions.Definition 5.5lim← be an inverse limit where {Xi}i⩾0 are continua and {fi:Xi+1→Xi}i⩾0 a collection of continuous maps. A sequence (ai)i⩾0 of positive real numbers is a Lebesgue sequence for lim← if there is a sequence (bi)i⩾0 of positive real numbers such that:(1)∑i=0∞bi<∞,(2)xi,yi∈Xj and any i<j, if |xi−yi|<aj, then |fi+1∘…∘fj(xi)−fi+1∘…∘fj(yi)|<bj.for any  A sequence (ci)i⩾0 of positive real numbers is a measure sequence for lim← if:(1)∑i=n+1∞ci<cn/2 for any n⩾0,(2)x^≠y^∈lim← there exists a non‐negative integer N so that |xN+1−yN+1|>cN.for any two points Let To provide a parametrized version of Brown's theorem and in particular to construct a continuously varying family of homeomorphisms Now we are ready to prove the main theorem of this section.Proof of Theorem 1.6Cλ(I) be the family of Lebesgue measure‐preserving maps and let Q=∩n=0∞Qn⊂Cλ(I) be the intersection of open dense sets Qn satisfying Theorem λ), and Theorem 3 from [{fi}i=0∞⊂Q that are dense in Q. By assumptions we know that each of these maps is leo, Lebesgue measure is ergodic measure for each fi (it is even weakly mixing), has the shadowing property and by Theorem δ>0 there exists N∈N so that fiN is δ‐crooked. Let S⊂I be a set of full Lebesgue measure such that any x∈S is generic point of all fi.Let m 3 from   and ρ<ξin/4<bn/8 where the sequence ξin will be specified later (we will need its faster convergence). Note that ρ<ξin/4+ξjn+1/4<ξin/2 provided that ξin>ξjn+1. All the lower indices will be specified later.We also fix a sequence n there exist indices {jin}i∈N and δin<2−n such that if we denote Ain:=B then An:=∪iAin is dense (and open by the definition) in Q, {fi}i=0∞⊂An and Ain∩Ajn=∅ for i≠j. Simply, for any i the set {ρ:j≠i} is countable, and so we can choose δin outside this set, making construction of consecutive Ain possible by induction . We can also make each δin arbitrarily small, in particular for m>n and any i,j we may require that if Aim∩Ajn≠∅, then Aim⊂Ajn. We can also require that each Ain⊂Qn, since each Qn is open and contains all functions fi.Note that for each δin, because we will need them to be sufficiently small as will be specified later.In our construction we will implement additional requirements on values of D⊂R2, let I⊂int(D) be the unit interval on which the BBM construction will take place, let I⊂int(D1)⊂D1⊂int(D2)⊂D2⊂int(D) where D1 and D2 are two closed discs and let R:D→D be a near‐homeomorphism, such that R(D2)=I, R|D∖D2 is one‐to‐one and R is identity on the boundary of D. We also require that the smash R is done along radial lines. It is not hard to provide an analytic formula defining R. These maps and discs are fixed throughout the whole construction.For a closed disk f∈Cλ(I) we construct an unwrapping f¯:D→D in the following way:(1)f¯(I)⊂intD1 and as usually in BBMs f¯|I is a rotated graph of f,(2)f¯ is identity on D∖D1,(3)R∘f¯|I=f and every point in intD is attracted to I under iteration of R∘f¯ where I is identified with I in a standard way. We also denote f∼=f¯|I. One of the main features of the construction will be to ensure that unwrappings within the family that we construct vary continuously with f.Now let us briefly recall how BBM construction is performed in general, for more details see Section U⊂D2∖D1 of positive Lebesgue measure in D is transformed onto set R∘f¯(U)⊂I of positive Lebesgue measure on I. It is a consequence of Fubini's theorem, since the smash R is performed along radial lines, and so the base of integration needs to have positive Lebesgue measure.One important property to notice here is that any set {Ci}i=1∞⊂intD1 be a collection of Cantor sets such that Ci∩I=∅, Ci∩Cj=∅ and λ(D1)=∑i=1∞λ(Ci). In other words, these Cantor sets fill densely intD1∖I and carry full Lebesgue measure. Such family of Cantor sets can be chosen using standard arguments. We may require that ∪iR(Ci)⊂S, because the union of radial lines over S has the full Lebesgue measure.Let δi0. When fji0 is fixed, we also have f¯ji0 and so the images f¯ji0(Ck) are explicitly determined as well. Therefore, we may require thatNow it is a good moment to set the first restriction on ai0>0 be such that if d<ai0 for x,y∈D then d(hi0(x),hi0(y))<b0/2. We also require that 16δi0<b0. This implies that if map H:D→D satisfies ρ<4δi0 and d<ai0 thenLet k,l such that Ak1⊂Al0. Then fjk1∈B. Construct a homeomorphism G:D→D such that G(x)=x for all x∉B(hl0(I),2δl0) and for x∈I we haveρ<2δl0, because we move the graph along horizontal lines. Similarly, we construct a map H:D→D such that H(x)=x for x∉B(f¯jl0(C1),δl0),x∈B(f¯jl0(C1),δl0).Now let us explain how the first step of the construction will be made, the reader is also referred to Figure ⋃i=1∞Ci and sufficiently well approximate small portions covering Cantor set H(f¯jl0(C1)), then define maps between these small portions and selected Cantor sets, and then extend the map to a homeomorphism on sufficiently small neighborhoods where these translations of small portions take place. This is possible, because our Cantor sets are in the plane, so are tamely embeddedρ<4δl0, hk1|I=f∼jk1 and hk1(C1)⊂⋃i=1∞Ci. Similarly as above, we decrease δk1 so thathk1(I) and Cantor set hk1(C2) so we need the neighborhoods of sets disjoint ρ<2ρ<ξln/2,(2)hkn+1|I=f∼jkn+1,(3)hln|Cr=hkn+1|Cr for r=1,…,n,(4)hkn+1(Cr)⊂∪i=1∞Ci for r=1,…,n+1,(5)δkn+1<δln/2 and ξkn+1<ξln/2.This construction can be extended by induction in the following way. If δin and ξin so that we are able to repeat arguments from [Now we will perform an additional adjustment of the constants nts from . That isAjnn is already constructed for some index jn∈N and Aj11⊃Aj22⊃…⊃Ajnn. Let hjii be a homeomorphism corresponding to Ajii. There exists a positive real number ajnn such that if d<ajnn thenδjn+1n+1 for Ajn+1n+1⊂Ajnn is adjusted with the correspondence to the condition ξjn+1n+1<ajnn/8 . This will ensure that if uniform limit Rjnn∘hjnn→F exists, then(ajnn)n=1∞ is a Lebesgue sequence for {Rjnn∘hjnn}n=1∞ and (bn)n=1∞.Assume that the set T∈C is given a priori and F was obtained as its perturbation. Fix any i,n>0 and let γ>0 be such that ρ<γ. Then for any x∈D we haveT is fixed, we have ρ<1/n for each i=0,1,…,n, provided that γ was sufficiently small . Therefore, taking δjnn, ξjnn sufficiently small, we can require that if uniform limit Rjnn∘hjnn→F exists, theni=0,1,…,n. For each jn∈N we pick a real number cjnn>0 in such a way that cjnn<18ckn−1 where Akn−1⊃Ajnn and additionally, if d<cjnn for some x,y∈D, thenx^,y^∈lim← then there is M∈N0 such that d>γ for some γ>0. Take n>M such that 1/n<γ. Thend((Rjnn∘hjnn)n−M(xn+1),(Rjnn∘hjnn)n−M(yn+1))<1/4n then we have a contradiction with the choice of γ, therefore d>cjnn, meaning that {cjnn}n=1∞ is a measure sequence for F. We additionally require that for each n∈N0 we haveAssume that a map g∈A=∩n=1∞An. Then there are indices i=in such that g∈Ain=B where jn:=jinn. Consider the associated sequence of homeomorphisms hjnn:D→D. For any n<m we havehjnn form a Cauchy sequence in C. Thus there exists a map Fg obtained as the uniform limit of the maps hjnn. But then Fg|I=g∼ as g∼ is a uniform limit of maps f∼jn=hjnn|I. Furthermore hjnn|Cr=hjrr|Cr for all n⩾r and therefore Fg(Cr)⊂∪i=1∞Ci.Now we will turn our attention to the implications of the construction. Assume that the above inductive construction has been performed and fix any F:D×A→D×A byf,g∈A and ε>0 there is δ>0 such that if ρ<δ then there are n and jn such that 2−n+4<ε and additionally ρ<4δjnn<2−n+2 and ρ<4δjnn<2−n+2. Namely, for sufficiently small δ we have f,g∈B⊂Ajnn. This shows that F is continuous.Let us define a map F=(R(Fg(x)),g)=(g∼(x),g) for each x∈I. If we fix any set of positive Lebesgue measure U⊂D2∖D1 then R∘Fg(U)=R(U) and R(U) has positive one‐dimensional Lebesgue measure on I. But then by Fubini's theorem there is a set Sg⊂U such that λ(Sg)>0 and R(Sg) is contained in the set of generic points of g, in particular any point x∈Sg under iteration of Fgrecovers the Lebesgue measure onI, that is, the measure 1n∑i=0nδ(Fgi(x)) converges in weak* topology to the Lebesgue measure on I.Now we will deduce properties (c) and (d). By the definition it holds that Fi:=R∘Ffi and take any set U⊂intD of positive Lebesgue measure. We can write U=∪j=0∞Uj as a disjoint union of sets Uj such that j is the minimal index such that Fij(Uj)⊂D2. Note that for any j>0 we have Fij(Uj)=Rj(Uj)⊂D2∖I. In particular, if Y⊂Fij(Uj) is such that λ(Y)=0 then also λ(R−j(Y))=0, where the latter formula makes sense, because R−1 is well defined on D2∖I. But then Ffi(⋃j=0∞Rj(U∼j))⊂∪i=1∞Ci for some sets U∼j⊂Uj satisfying λ(Uj)=λ(U∼j) and thereforej. But then there is a set S∼i of full Lebesgue measure in D such that for each x∈S∼i there is N∈N0 such that FiN(x)∈S. This means that every point in S∼i is eventually transferred into a generic point of fi, which means that the orbit of x under Fi recovers the one‐dimensional Lebesgue measure on I. This shows that the Lebesgue measure on I is a physical measure for each F and it is unique physical measure for a dense set of functions g∈Cλ(I) (this dense set corresponds with the maps {fi}i=0∞ from the start of the construction). In fact it is unique each time when generic points of g contain the set S and may have (but not necessarily has) other physical measures in remaining cases.But now consider the special case of map D^:=lim←. Now we are going to define a map Θ:D^→D×A byDenote g∈∩n=1∞Ainn. We can write g as the second coordinate in lim←, since it is a constant sequence of gs; thus we can also write Θg:=Θ:D^g→D, where D^g:=lim←. Since we have satisfied its assumptions, by Theorem 1 from [Θ is well defined. Furthermore, by Theorem 2 from [Θ is a homeomorphism for each g∈A, because it is a composition of a homeomorphism with projection onto the first coordinate in the inverse limit defined by homeomorphisms lim←.where m 1 from  Θ is welm 2 from , it hold∈D^ thenNote that if ε>0 and n sufficiently large 1 denotes the natural projection to the first coordinate)g∈∩i=1∞Ajil and every x^ such that ∈D^. Thus as a consequence, for δ sufficiently small, all f,g∈∩i=1∞Ajil and d<δ we have1),Θ1)<3ε which proves that Θ is continuous.But by  we have+1so by  we obtaif we define a homeomorphism Φf:=Θf∘F^∘Θf−1:D→D. Abusing the notation, for the following inverse limit spaces we will identify f with the interval map f|I. Denote Λf:=Φf(I^f), where I^f:=lim← and note that by Corollary Λf is the pseudo‐arc for every f∈A. We can write Φ:=(Θ×id)∘F^∘(Θ×id)−1 and put Φf=Φ showing that the family Φf is varying continuously. This also shows that the family {Λf}f∈A varies continuously in Hausdorff metric.For each μ^f be an invariant measure induced on the inverse limit I^f using Lebesgue measure λ on I and define a push‐forward measure μf=(Θf)∗μ^f. Formally, the measure μ^f is defined on the space I^f⊂lim←, however we can also view it as a measure on the space D^. Let us show that measures μ^f vary continuously in the weak* topology in D^. By the definition μ^f(πn−1(B))=λ(B∩I) for any Borel set B⊂D and every n∈N0  such that πn(P^g)=P. Denote ν:=1|P|∑q∈Pδq and ν^g:=1|P^g|∑q^∈P^gδq^. Fix any ε>0 and let us assume that 2−n<ε/2. There is γ>0 such that if ρ<γ, x^,y^∈lim← and d(πn(x^),πn(y^))<2γ then d<ε. We may also assume that for any two consecutive points p,q∈P with respect to the ordering in I we have d<γ/2 and that γ>0 is sufficiently small so that if x^∈lim←, y^∈lim← satisfy πn(x^)=πn(y^), then d<ε.By Theorem ails see ). Take aρ<γ we haveq^∈P^ satisfies q^∈πn−1(B(πn(B^),2γ) then there is z^∈B^ such that d)<2γ and therefore d<ε. This givesD<ε  denotes the Prokhorov metric on M(I) defined in the end of Subsection q^∈P^g there is p^∈P^f such that πn(q^)=πn(p^) and therefore D<ε. This gives that D<3ε provided that ρ<γ. Indeed, the function f↦μ^f is continuous.Note that since α∈C, then by identifying Θ to the projection on the first coordinate α∘Θ∈C and we have already proven that for any fi→f from A,If f↦μf where each μf is the push‐forward measure on D defined by μf:=Θ∗μ^f. It is clear from the definition that the support of μf is Λf⊂D.□We therefore have thatProceedings of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not‐for‐profit Charity registered with the UK Charity Commission. 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