Interval maps generated by erasing substitutions
aa r X i v : . [ m a t h . D S ] M a r Interval maps generated by erasing block substitutions
A. Della Corte ∗ , M. Farotti † Abstract
We study discontinuous interval maps generated by the action of erasing block substi-tutions on the binary expansion. After establishing some general properties of thesemaps, we categorize erasing block substitutions in a hierarchy of classes displayingprogressively stronger erasing character. We investigate how this affects the dynamicsof the corresponding interval maps, showing that the richest dynamical behavior (De-vaney and Li-Yorke chaos, infinite topological entropy) is achieved at a precise step inthis hierarchy, which we name completely erasing substitutions.
KEYWORDS: Baire-1 maps; Erasing substitutions; Devaney chaos; Li-Yorke chaos; Topolog-ical entropy.MSC2020: 26A18, 37B40, 37B10, 37B20.
The study of dynamical systems has greatly benefited from the symbolic approach, andin particular from substitutive dynamics. Within this framework, several ideas havefound their neat, ideal formulation, generating techniques and results that proved fruit-ful, for instance, in ergodic theory, chaos theory, number theory and crystallography(a standard general reference on substitutive dynamical systems is [8]). While sub-stitutions and block substitutions are generally assumed to map symbols or words tononempty words, recently some attention has been devoted to the properties of erasing substitutions, which include the empty word in their range [6, 7, 18]. This interest isunderstandable, coming from a natural extension of the somewhat narrow original con-cept of substitution, and it will probably become wider, since a variety of real-worldprocesses that can be potentially formalized in a symbolic dynamical context easilyinclude erasing phenomena - think about DNA transcription and coding, informationtransmission with errors, model reduction for physical systems in which some states areintrinsically negligible. However, the generalization/adaptation of results establishedfor nonerasing substitutions to the erasing case is in general not trivial. ∗ Mathematics Division, School of Sciences and Technology, Università degli Studi di Camerino (Italy).Address: via Madonna delle Carceri 9A - Camerino (MC), Italy. Email: [email protected]: 0000-0002-1782-0270 (corresponding author). † Mathematics Division, School of Sciences and Technology, Università degli Studi di Camerino (Italy).Address: via Madonna delle Carceri 9A - Camerino (MC), Italy. Email: [email protected]. n this paper, we focus on erasing substitutions, and more precisely on the dynamicalproperties of maps defined by the action of erasing block substitutions on the binaryexpansion of reals in the unit interval (a model case of this type is studied in detailin [3]). The systematic use of base 2 was chosen for simplicity. Working in the realcontext involves some technicalities, mainly due to the ambiguity of the representationof dyadic rationals and to the identification of the finite words w m ( m ∈ N ) withthe infinite word w ∞ . However, once formalized appropriately, the maps generated inthis way look like rather natural hunting ground for interesting dynamical phenomena,since they lie at the boundary of the fast-expanding domain of topological dynamics.Indeed, they are typically Baire-1, not Darboux functions, and therefore they representparticularly simply-defined examples from classes of objects for which the study oftopological dynamical properties has begun in quite a recent past (for instance, [11]mostly focuses on Darboux, Baire-1 maps and [21, 19, 20] study topologically typicalBaire-1 maps).The paper is organized as follows: definitions and preliminary results are provided inSection 2, where, in particular, erasing substitutions are categorized in a hierarchyof classes displaying progressively stronger erasing character. In Section 3 we studysome general analytical and topological properties shared by the maps generated bythe members of all erasing classes if they verify some form of surjectivity (named here optimality condition ). In Section 4 we establish links between the erasing class of asubstitution and the dynamical behavior of the corresponding interval map (the mainresults are summarized in the table at the end of the Section). Finally, in Section 5some possible directions for further investigation are highlighted. We set I = [0 , and let Q = I ∩ (cid:8) n k : n, k ∈ N (cid:9) denote the dyadic rationals in I . Weset Q = Q \ { , } and Q = Q ∪ { } . We indicate by { , } ∗ the set of all finitebinary words and by { , } ∞ the set { , } ∗ ∪ { , } ω of all finite or infinite binarywords. We let ǫ denote the empty word and we set { , } + = { , } ∗ \ { ǫ } . For n ∈ N ,we set { , } ≤ n = [ N ∋ i ≤ n { , } i , { , } ≥ n = [ N ∋ i ≥ n { , } i ∪ { , } ω For n ∈ N and for u ∈ { , } ≥ n , we let u n denote the n -th digit of the binary word u and for u ∈ { , } n we indicate by | u | the length of the word u , i.e. the non-negativeinteger n , while we set | u | = ∞ if u ∈ { , } ω . For v ∈ { , } n and m ∈ N we let v m indicate the m -th power of the word v , that is the word u ∈ { , } mn such that u hn + r = v r for every positive integer r ≤ n and every non-negative integer h < m .We set v = ǫ . We will use natural numbers as subscripts (superscripts) in roundparentheses when they do not indicate digit number (word power).We will write concatenation of words using a multiplicative notation. Therefore, If u ∈ { , } n and v ∈ { , } ∞ , we write uv for the word u u . . . u n v v . . . . No-tice that, with concatenation as internal operation, the sets { , } ∗ and { , } + have espectively the structure of a free monoid and a free semigroup. Therefore, exploit-ing associativity, if (cid:0) u ( n ) (cid:1) n =1 ,...,N is a finite sequence of finite binary words, we let Q Ni =1 u ( i ) = u (1) u (2) . . . u ( N ) denote their concatenation. Moreover, if (cid:0) u ( n ) (cid:1) n ∈ N is aninfinite sequence of finite binary words, we let Q ∞ i =1 u ( i ) = u (1) u (2) . . . denote theirinfinite concatenation. We let u ∞ denote the infinite concatenation uuu . . . of u withitself.We say that v ∈ { , } ∞ is a prefix of a (possibly infinite) word w = a a . . . if either v = w ∈ { , } ω or w ∈ { , } ≥ k and there is a positive integer k such that v = a · · · a k .We say that v ∈ { , } ∗ is a suffix of a finite word w = a . . . a n if there is a positiveinteger k < n such that v = a n − k a n − k +1 . . . a n . Definition 2.1.
For k ∈ N , we say that w ∈ { , } ∞ has u ∈ { , } ∞ as a k -factor ifthere is a non-negative integer n such that w has a prefix of the form pu , with | p | = nk .Notice that | u | can also be ∞ .If w ∈ { , } ∞ , we let .w denote the real number P | w | i =1 w i i . We define binary expansion of x ∈ I any word u ∈ { , } ∞ such that x = 0 .u , so that Q is precisely the subsetof I whose points do not have a unique binary expansion. If x ∈ I \ { } , we let e x ∈ { , } ω denote the unique infinite binary expansion of x not ending with ∞ , sothat x = 0 . e x . Notice that e x is the unique binary expansion of x if and only if x / ∈ Q .For w ∈ { , } ∗ , we indicate by [ w ] the cylinder set generated by w , that is the set { x ∈ I : x = 0 .wv for some v ∈ { , } ∞ } . Notice that [ ǫ ] = [0 , . We indicate thestandard Lebesgue measure of a measurable set A ⊆ I by m ( A ) . Definition 2.2.
Herein by simple substitution we mean a map σ : { , } → { , } ∗ .For every integer k ≥ , by k -block substitution we mean a map σ : { , } k → { , } ∗ .We only consider the case in which there is exactly one block mapped to empty word: Definition 2.3.
For every integer k ≥ , by erasing k -block substitution we mean a k -block substitution σ such that there exists a unique w ∈ { , } k : σ ( w ) = ǫ .We will adopt the following conventions. The integer k > will always be the length ofthe blocks transformed by the erasing substitution σ . By w (1) , w (2) , . . . , w (2 k ) we willindicate the lexicographic enumeration of { , } k . For every i ∈ { , . . . , k } we set: σ (cid:0) w ( i ) (cid:1) = v ( i ) , v ( j ) = ǫ, v ( i ) ∈ { , } + for i = j (1)We will always indicate by w ǫ the word w j ∈ { , } k such that σ ( w j ) = ǫ . Definition 2.4.
We say that σ verifies the optimality condition (in short OC) if, forevery w ∈ { , } ω , there exists a sequence of integers ( h i ) i ∈ N ∈ { , , . . . , k } N suchthat w = ∞ Y i =1 v ( h i ) (2) Remark 1.
Notice that OC implies also that, for every finite binary word w , thereis a suitable concatenation of nonempty words v = v ( h ) v ( h ) . . . v ( h n ) such that w is aprefix of v . e will assume always σ ( ǫ ) = ǫ . The k -block substitution σ induces then naturally amap over W k = (cid:0) ∪ n ∈ N { , } nk (cid:1) ∪ { , } ω if we set, for every w ∈ W k , σ ( w ) = | w | k − Y i =0 σ ( w ki +1 w ki +2 . . . w k ( i +1) ) , where the concatenation index has to be intended to be up to ∞ if w ∈ { , } ω . Wecan exploit this to define a function f σ : I → I , which is determined by the symbolicaction of σ on the binary expansion of real numbers in the unit interval. Assuming w ǫ = 1 k , we set: f σ ( x ) = P | σ ( e x ) | h =1 ( σ ( e x )) h h = 0 .σ ( e x ) if x ∈ (0 , and e x = w ∞ ǫ if e x = w ∞ ǫ or x = 0 (3)In the following, we will have to pay attention to two technicalities: the ambiguity inthe binary representation of the elements of Q and the fact that the erasing characterof σ implies that the σ -image of some infinite words can be finite. Defining the set E σ = { x ∈ I ∩ Q : x = 0 .vw ∞ ǫ for some v ∈ { , } nk , n ∈ N } , it follows immediately that | σ ( e x ) | < ∞ = ⇒ x ∈ E σ , and the converse implication alsoholds unless w ǫ = 0 k . Remark 2. If w ǫ = 1 k , a complementary map f ′ σ can be introduced, replacing e x bythe unique infinite binary representation of x ∈ I not ending in ∞ . In this case, theproperty of f ′ σ are analogous to that of f σ in the particular case w ǫ = 0 k (with thedifference that the properties of left/right limits on Q for f σ correspond to propertiesof right/left limits for f ′ σ ). For this reason we will assume w ǫ = 1 k throughout. A particular subset of k -block substitutions are those which can be rewritten using k (generally distinct) simple substitutions, acting on the elements of a word u accordingto the congruence class modulo k of their indexes. More precisely, we introduce thefollowing Definition 2.5.
We say that the k -block substitution σ is alternating if there exist k simple substitutions σ , σ , . . . σ k such that the map σ e defined below is an extensionof σ on { , } ∞ : σ e ( u ) = n − Y j =0 k Y i =1 σ i ( u i + jk ) ! m Y i =1 σ i ( u i + nk ) (4)where | u | = nk + m ( n, m ∈ N , m < k ) and the first (last) product has to be taken asempty if n = 0 ( m = 0 ).Therefore, if σ is alternating, we can use Eq.(4) to extend the definition of σ to amorphism over all { , } ∞ setting σ ≡ σ e . We will exploit this fact in the followingso that, when dealing with an alternating substitution σ , we will write simply σ ( u ) (defined through the right hand side of Eq.(4)) for words of any length. Alternating ubstitution rules are quite well investigated and have also been generalized (see forinstance [4]). We will see that assuming σ alternating makes significantly less com-plicated to deduce combinatorially its dynamical properties. However, for the sake ofgenerality, we will not assume always this property. Remark 3.
When we do not assume σ alternating and v ∈ { , } nk + m ( < m < k ) ,we may write simply σ ( v ) for the word σ ( v . . . v nk ) , i.e. we implicitly drop the lastdigits of v so as to truncate it to a word in { , } nk . Of course this crude sort of“extension” of σ to { , } ∗ is not a morphic map, and this convention will be usedsimply to lighten the notation in cases in which we are not interested in what happensafter σ ( v . . . , v nk ) . The next step consists in defining some properties of σ , strengthening it simply beingerasing, which entail additional dynamical properties for the real map f σ associated toit. To do so we introduce some tools needed to settle words of any given length.Let w ∈ { , } + be such that nk < | w | < ( n + 1) k for some n ∈ N . A word u ∈ { , } ∗ such that u = wv and | u | = ( n + 1) k is said a k-rounding of w ; in this case v is saida k-extension of w . If | w | = nk ( n ∈ N ) we assume that ǫ and w are, respectively, theunique k -extension and the unique k -rounding of w . We set e ( w ) = { v : v is a k -extension of w } , r ( w ) = { v : v is a k -rounding of w } . For W ⊆ { , } ∗ , we set e ( W ) = [ w ∈ W e ( w ) , r ( W ) = [ w ∈ W r ( w ) . Let w ∈ { , } + . We set w [0] = { w } and, for any positive integer n : w [ n ] = { σ ( v ) : v ∈ r ( w [ n − ) } . Aimed at studying the dynamical properties of f σ , we want to classify block substi-tutions based on how “markedly erasing” they are. More precisely, let us give thefollowing Definition 2.6. A k -block substitution σ is said to be:1. Strongly erasing if for every w ∈ { , } ∗ there is n ∈ N such that: ǫ ∈ w [ n ] (5)So, when σ is strongly erasing, for every w ∈ { , } ∗ there exist a positive integer n and finite words r (0) , . . . , r ( n − , which we call erasing k -roundings of w , suchthat: i) r (0) is a k -rounding of w ; ii) for any integer j such that ≤ j ≤ n − , r ( j ) is a k -rounding of σ ( r ( j − ) ; iii) σ ( r ( n − ) = ǫ . The k -extensions corresponding tothe n k -roundings just defined, i.e. the n words { e ( i ) } i =0 ,...,n − , each belongingto { , } For every k -block substitution σ , we have:boundedly erasing = ⇒ completely erasing = ⇒ strongly erasing = ⇒ erasing Proof. The first and last implications are trivial, so we just have to prove the middleone. Assume then that σ is completely erasing. Since σ is alternating, we can rewriteit by means of the simple substitutions σ , . . . σ k as in Eq.(4). Applying σ to w ǫ wereadily deduce that σ i (( w ǫ ) i ) = ǫ for every i ∈ { , . . . , k } . (7)Take then any w ∈ { , } ∗ . Set w (0) = w and write | w | as n k + m , with n ∈ N and m ∈ { , . . . , k − } . Set w (1) = σ ( w (0) e (0) ) , where e (0) is the unique k -extensionof w (0) such that ( e (0) ) i = ( w ǫ ) m + i for every i ∈ { , . . . , k − m } . Because of Eq.(7),we have that σ ( w (0) e (0) ) = σ ( w (0) ) . We can proceed in this way for further ǫ ( w (0) ) − steps, setting, for every j ∈ { , . . . , ǫ ( w ) − } : | w ( j ) | = n j k + m j ( n j ∈ N , m j ∈ { , . . . , k − } ) , w ( j +1) = σ ( w ( j ) e ( j ) ) where | e ( j ) | = k − m j and (cid:0) e ( j ) (cid:1) i = ( w ǫ ) m j + i for every i ∈ { , . . . , k − m j } .Recalling Eq.(7), at each step we have σ ( w ( j ) e ( j ) ) = σ ( w ( j ) ) , so that σ ( w ( ǫ ( w ) − e ( ǫ ( w ) − ) = ǫ, which implies that ǫ ∈ w [ ǫ ( w )] .We now provide some examples of substitutions falling in each of the erasing classesintroduced in Definition 2.1. The substitution σ defined below is erasing, but notstrongly erasing. Indeed, 110 produces a cycle of order 2 of words having lengthexactly k (so that we do not have to choose any k -extension), that never reaches theempty word. Notice that σ verifies the optimality condition (2.4).The substitution σ defined below is strongly erasing, but not completely erasing.Indeed, σ is not alternating as, if so, odd-indexed 0s and even-indexed 1s should bemapped to ǫ , which would imply, since σ (00) = 0 , that even-indexed 0s are mappedto 0, but this is incompatible with σ (10) = 1 . Notice then that | σ ( w ) | ≤ ( | w | ) / for w having even length, while | σ ( w | ≤ ( | w | + 1) / for w having odd length, whichimplies that, -extending when needed with 1, we arrive in a finite number of iterates t length 1. At that point, -extending always with the digit 1 we arrive at ǫ in atmost 2 steps . Notice that σ verifies the optimality condition (2.4).The substitution σ defined below is completely erasing, but not boundedly erasing.Indeed σ is alternating, with all 0s going to ǫ , odd-indexed 1s going to and even-indexed 1s going to 1. Moreover | ( σ ) ( w ) | is strictly less than | w | for every finiteword w , because all odd-indexed elements of w go to ǫ in at most two iterations, whichimplies that σ is completely erasing. On the other hand, σ is not boundedly erasingas, taking for instance w = 1 m ( m ∈ N ), we have | ( σ ) ( w ) | = 1 m , so that ǫ ( w ) willdiverge with | w | . Notice that σ verifies the optimality condition (2.4).Finally, the substitution σ defined below is boundedly erasing. Indeed, it is alter-nating, as it can be rewritten as in Eq.(4) using the three simple substitutions σ i ( i = 1 , , ) defined by: σ , , (0) = ǫ , σ (1) = 010 , σ (1) = 001 and σ (1) = 000 .Observe that, for every word w of length 3, σ ( w ) is made of an integer number ofwords of length 3 which are all strictly smaller, in lexicographic order, than w . Sincethe smallest of these words, 000, is mapped to ǫ , this easily implies that ǫ ( w ) ≤ forevery w ∈ { , } ∗ . σ (000) = ǫ σ (00) = 0 σ (00) = ǫ σ (000) = ǫσ (001) = 00 σ (01) = ǫ σ (01) = 1 σ (001) = 000 σ (010) = 011 σ (10) = 1 σ (10) = 0 σ (010) = 001 σ (011) = 010 σ (11) = 0 σ (11) = 01 σ (011) = 001000 σ (100) = 10 σ (100) = 010 σ (101) = 110 σ (101) = 010000 σ (110) = 111 σ (110) = 010001 σ (111) = 110 σ (111) = 010001000 Erasing Strongly Completely Boundedlyerasing erasing erasing Remark 4. The substitution σ can be considered the simplest case of completelyerasing substitution. The related map f σ is thoroughly studied in [3]. We will see that, moving towards completely erasing substitutions, we get increasinglycomplex dynamics for the corresponding interval map, while boundedly erasing substi-tutions have to be considered an extreme case, as they are so efficient in chopping offdigits that the dynamical behavior becomes almost trivial.In general an erasing block substitution σ is a non-morphic map, in the following sense:if w ∈ { , } ω and u is a finite binary word, in general σ ( uw ) = σ ( u ) σ ( w ) . To addressthis point, let us introduce a countable family of maps induced by an erasing k -blocksubstitution σ and by the choice of a finite word u . With a slightly more complicated argument it can be shown that σ remains strongly erasing if we set σ (11) = 0 m for every positive integer m , meaning that this property does not imply that the substitutionis non-increasing with respect to word length. This is also shown by the example σ , which by Lemma 2.1is also strongly and completely erasing. ake u ∈ { , } ∗ , v ∈ { , } ∞ . Let us assume that σ u ( v ) = σ ( v ) = v . Let us define,then, the map σ u : { , } ∞ → { , } ∞ by means of the equality σ ( uv ) = σ ( uu (1) ) σ u ( v ) , (8)where u (1) is the unique prefix of the word v which is a k -extension of u . Set σ u ( w ) = σ u ( w ) . Notice that here, at the left hand side of Eq.(8), we apply the conventiondescribed in Remark 3, i.e. we “drop” the digits of v that exceed the longest of theprefixes of uv having length multimple of k .Iterating the procedure, for every positive integer n , we wet p (0) = u and define thewords p ( n ) and the maps σ nu : { , } ∞ → { , } ∞ inductively by means of the followingequalities: p (1) = σ (cid:0) uu (1) (cid:1) σ ( uv ) = σ (cid:0) p (1) u (2) (cid:1) σ u ( v ) . . .. . .p ( n − = σ (cid:0) p ( n − u ( n − (cid:1) σ n ( uv ) = σ (cid:0) p ( n − u ( n ) (cid:1) σ nu ( v ) (9)where, at each stage, u ( i ) is the unique prefix of σ i − u ( v ) which is a k -extension of p ( i − .Notice that σ ǫ coincides with σ . Remark 5. Also here we apply the convention described in Remark 3, i.e. we “drop”the digits of uv whenever this is necessary to get a word having length multiple of k ensuring that σ i ( uv ) makes sense for ≤ i ≤ n . This means that we may writesimply σ iu ( v ) for the word σ iu ( v ′ ) , where v ′ is the longest prefix of v for which thelast expression makes sense. As observed before, this convention will be used simplyto lighten the notation in cases in which we are not interested in what happens after σ iu ( v ′ ) . In case σ is alternating there is no need of a convention of the type described in Remarks3 and 5, because in that case, thanks to Eq.(4), σ ( w ) already makes sense for wordsof any length, and therefore a simpler definition can be given for σ u . Indeed, for everypositive integer n and every v ∈ { , } ∞ , we define σ nu ( v ) as the word verifying σ n ( uv ) = σ n ( u ) σ nu ( v ) (10)Notice finally that, both in Eqs.(9) and in Eq.(10), the map σ nu does not coincide with ( σ u ) n , that is the n -th iterate of the map σ u . Instead, { σ nu } n ∈ N is a countable set ofgenerally distinct maps indexed by the positive integers. f σ In this Section we want to address some analytical properties verified by f σ when σ isan erasing k -block substitution. Throughout the Section, σ is a k -block erasingsubstitution defined by Eq. (1) .Lemma 3.1. If σ verifies the optimality condition (2.4), then f σ : I → I is onto. roof. Take y ∈ I . By the optimality condition, there exists a sequence h = h , h , . . . ∈{ , , . . . , k } N such that u = Q ∞ i =1 v ( h i ) , y = 0 .u and v h i = ǫ for every i ∈ N .Suppose that w = Q ∞ i =1 w ( h i ) does not end in ∞ . Then we can take x ∈ I such that e x = ∞ Y i =1 w ( h i ) (11)so that σ ( e x ) = Q ∞ i =1 σ (cid:0) w ( h i ) (cid:1) = Q ∞ i =1 v ( h i ) = u . Since σ ( e x ) = u ∈ { , } ω , we have x / ∈ E σ . It follows that f σ ( x ) = 0 .σ ( e x ) = 0 .u = y. Suppose instead that w ends in ∞ and let a ∈ { , } ∗ be the shortest word suchthat w = a ∞ (notice that this implies that w ǫ = 0 k , as w ǫ does not appear in theright hand side of (11)). Then we cannot find any x ∈ I such that e x = w . However,we can insert infinitely many times w ǫ in w as a k -factor, obtaining a point whoseimage is y . More precisely, set: S y = { b ∈ { , } ω : b = a Q ∞ i =1 n i w m i ǫ } , where ( m i ) i ∈ N and ( n i ) i ∈ N are sequences of non-negative integers both not ultimately 0, and | a | + P ij =1 n j is a multiple of k for every i ∈ N . Recalling that σ ( w ǫ ) = ǫ , it followsthat ∅ 6 = { x ∈ I : e x ∈ S y } ⊆ f − σ ( y ) .In the following it will be useful to have a symbol for the points considered in thesecond part of the previous proof, so we set F σ = { y ∈ Q ∩ I : y = 0 .wv ∞ (0) , w = n Y j =1 v ( i j ) for some n ∈ N , and v (0) = σ (0 k ) } . Lemma 3.2. The map f σ is continuous on the set C σ = I \ ( Q ∪ E σ ) , and thereforeBaire-1. Moreover, f σ is left-continuous on Q \ { } . Proof. Take x ∈ C σ and fix n ∈ N . Then e x = Q ∞ h =1 w ( i h ) ( i h ∈ { , . . . k } ) and e x doesnot have w ∞ ǫ as a k -factor. Since | v ( j ) | ≥ if w ( j ) = w ǫ , there is a prefix p of e x such that | p | = mk ( m ∈ N ) and | σ ( p ) | ≥ n . Since x has a unique binary expansion, for every y ∈ I such that | x − y | < − mk , we have that e x, e y coincide at least up to the first mk digits. Then | f σ ( x ) − f σ ( y ) | ≤ − ( | σ ( p ) | ) ≤ − n . Since we assume w ǫ = 1 k , recallingthat σ acts on the 1-periodic expansion of dyadic rationals, the argument above showsthat f σ is left continuous on Q \ { } (notice that, assuming instead w ǫ = 0 k , we wouldget right continuity on Q \ { } ).Finally, interval maps with countably many discontinuities are pointwise limit of con-tinuous functions, that is Baire-1 (see for instance [22]).Notice that, in the particular case w ǫ = 0 k , the sets E σ , F σ , Q and I \ C σ coincide. Remark 6. For every pair of integers m, n > there are irrational points x suchthat e x has a prefix of type pw mǫ ( p ∈ { , } nk ) . Two points sharing such a prefix arearbitrarily close if m → ∞ , but their f σ -images are not necessarily arbitrarily close,because | σ ( pw mǫ ) | is independent of m . Therefore f σ | C σ is in general not uniformlycontinuous, so that it is impossible to extend it to a continuous function on I . e recall that a map g : A → B ( A, B ⊆ R ) is called Darboux if the image of everyinterval is an interval. From the point of view of combinatorial dynamics, Darbouxfunctions have some of the properties of continuous functions, in particular for whatconcerns the existence of periodic points (see e.g. [12], where Darboux plus Baire-1 isshown to imply the Sharkowsky ordering of periodic points).However, the general interval map f σ is not Darboux unless σ is constructed quite adhoc . Take indeed y ∈ Q such that e y = (cid:16)Q Nn =1 w ( i n ) (cid:17) ∞ for some N ∈ N and w i N ends with 0. Let p ∈ { , . . . , k } be such that w ( p ) ∈ { , } k is the unique word oflength k coinciding up to the digit k − with w ( i N ) , but ending with 1. Supposing that w ǫ = 0 k , the right and left limit of f σ at y are written as: lim x → y − f σ ( x ) = f σ ( y ) = 0 . N Y n =1 v ( i n ) ! ( σ (1 k )) ∞ (12) lim x → y + f σ ( x ) = 0 . N − Y i =1 v ( i n ) ! v ( p ) ( σ (0 k )) ∞ (13)Clearly f σ can be Darboux only if these two real numbers coincide for every y ∈ Q ,which will be not true in general. The map f σ is in general not Darboux also when w ǫ =0 k . The simplest example to see that is given by the substitution σ defined in Section2, for which, when y ∈ Q , the left limit lim x → y − f σ ( x ) = f σ ( y ) does not belongto [lim inf x → y + f σ ( x ) , lim sup x → y + f σ ( x )] (see [3] for further details). Therefore, forour general object, we cannot expect a priori any relation between periodic points ofdifferent order, so that we will have to provide an explicit proofs of basic combinatorialdynamical properties. To proceed further, let us introduce some technical tools. Definition 3.1. Let Ξ ⊂ N N be an uncountable set of sequences such that a, b ∈ Ξ , a = b = ⇒ ∀ M > , ∃ n ∈ N : n > M and a n = b n (14)Set Ω = Ξ ∩ { , } N and call Ξ + (Ω + ) the uncountable subset of Ξ ( Ω ) such that noone of its elements is ultimately 0. Definition 3.2. Take y ∈ I \ { } and write e y = Q ∞ i =1 w ( h i ) . Let ( h i j ) j ∈ N be thesubsequence of all h i such that w ( h ij ) = w ǫ . Then we define: y ◦ = ξ ∈ I such that e ξ = ∞ Y i =1 u ( j ) , where u ( j ) = w ( h ij ) (15)In other words, y ◦ is the point whose binary expansion (ending in ∞ , if there is achoice) coincides with that of y after having erased all the (irrelevant, from the pointof view of σ ) occurrences of w ǫ which are k -factors, so that e y ◦ nk +1 . . . e y ◦ ( n +1) k = w ǫ for every non-negative integer n . Definition 3.3. For u ∈ { , } ω such that u = Q ∞ i =1 u ( i ) (with u ( i ) ∈ { , } k ), and forevery a ∈ Ξ , we set: ξ u ( a ) = 0 . ∞ Y i =1 w a i ǫ u ( i ) (16) emma 3.3. For every x ∈ S σ = f σ ( I ) \ Q , the set f − σ ( x ) is uncountable. Proof. Suppose first that x ∈ S σ \ F σ . Then there is w ∈ { , } ω such that w doesnot end in ∞ and x = 0 .σ ( w ) , so that there exists y ∈ I such that w = e y , and thus y ∈ f − σ ( x ) . Moreover e y does not have w ∞ ǫ as a k -factor, because if so then | σ ( w ) | would be finite so that x ∈ Q against the hypothesis. Since every occurrence of w ǫ as a k -factor is mapped by σ to the empty word, we have f σ ( y ◦ ) = f σ ( y ) , so that y ◦ ∈ f − σ ( x ) .Let us write y ◦ as Q ∞ i =1 w ( h i ) ( w ( h i ) = w ǫ for every i ). By definition of the map ξ ,we have that, for every a ∈ Ξ , y ( a ) := ξ f y ◦ ( a ) = 0 . Q ∞ i =1 w a i ǫ w ( h i ) , which implies thatevery point y ( a ) ( a ∈ Ξ ) is in f − σ ( x ) . Recalling the definition of Ξ , it follows that the ξ f y ◦ -image of every sequence a ∈ Ξ is obtained by e y ◦ inserting, infinitely many times,finite and nonempty concatenations of w ǫ with itself, so that we are ensured that, for u = e y ◦ , the product in the right hand side of Eq.(16) will never end with ∞ (both if w ǫ = 0 k or not). Since in the binary expansion of y ◦ there is never w ǫ as a k -factor, itfollows that ξ f y ◦ is injective, which implies that f − σ ( x ) is uncountable.Suppose now that x ∈ S σ ∩ F σ . Since x ∈ F σ , there exists a word w ∈ { , } ω ending in ∞ such that .σ ( w ) = x . Notice that x ∈ S σ implies that | σ ( w ) | = ∞ , sothat we can exclude w ǫ = 0 k . Consider now the restriction ξ + w to Ξ + of the map ξ w defined in Definition 3.3. Since every a ∈ Ξ + is not ultimately 0, we have that w ( a ) := Q ∞ i =1 w a i ǫ w ( h i ) will never end in ∞ , so that w ( a ) = ^ ξ w ( a ) . Moreover, σ ( w ( a ) ) = σ ( w ) ,so that ξ w ( a ) belongs to f − σ ( x ) for every a ∈ Ξ + . The same argument as above showsthat ξ + w is injective, which concludes the proof. Corollary 1. If σ verifies the optimality condition 2.4, then for every x ∈ I the set f − σ ( x ) is uncountable. Proof. Under the assumption, for every x ∈ I there are an infinite binary word u and y ∈ I such that .σ ( e y ) = 0 .u = x . Then the thesis easily follows from the argumentused in the proof of the previous Lemma.A relatively simple characterization of the topological structure of the f σ -preimagesof singletons can be achieved strengthening the optimality condition. Even so, thereare a few technical issues, the main being that the points belonging to Q can be“reached” in general in infinitely many ways. Indeed, each of them has two infinitebinary expansions plus a countable family of finite ones (all ending in a finite stringof 0s except at most one) which, from a symbolic point of view, constitute distinctrepresentations. The different cases are covered in the following Lemma, which is themain technical tool needed to prove that, outside Q , the closures of the fibers arealways Cantor sets. Lemma 3.4. Suppose that σ verifies the optimality condition 2.4 and that, for every w ∈ { , } ω , the sequence h , h . . . verifying Eq.(2) is unique. If y ∈ I and x is a limitpoint of f − σ ( y ) , then one of the following cases occurs:1. x ∈ f − σ ( y ) x = 0 .vw ∞ ǫ , with | v | = nk for some non-negative integer n . . y ∈ F σ and x = 0 .a Q ∞ i =1 n i w m i ǫ , where a ∈ { , } ∗ , { m i } is a sequence of non-negative integers which is ultimately zero and n i are positive integers such that | a | + P ij =1 n j is a multiple of k for every i ∈ N . Proof. We divide the proof in four cases.1. y / ∈ Q ∪ F σ .Then y has a unique binary expansion u and, by hypothesis, there exists a uniquesequence h , h , . . . such that u = Q ∞ i =1 v ( h i ) . We can assume v ( h i ) = ǫ for every i ∈ N . Moreover, since y / ∈ F σ , w = Q ∞ i =1 w ( h i ) does not end in ∞ . Therefore,the point x = 0 .w belongs to f − σ ( y ) and it coincides with x ◦ . By uniqueness ofthe sequence h , h . . . , it follows that f − σ ( y ) can be written as [ a ∈ N N ξ f x ◦ ( a ) (17)where ξ f x ◦ : N N → I is the map defined in 3.3. If z is a limit point of f − σ ( y ) ,there exists a sequence { a ( n ) } n ≥ of elements of N N such that ξ f x ◦ ( a ( n ) ) −−−−→ n →∞ z .Notice that, since f x ◦ does not end with ∞ , neither does f ξ f x ◦ ( a ) for every a ∈ N N .We indicate by a ( n ) i the i -th element of the sequence a ( n ) . Let us first prove that,for every i ∈ N , there exists a i = lim n →∞ a ( n ) i ≤ ∞ . Indeed, suppose that m is the smallest positive integer such that a ( n ) m admits two distinct sublimits, i.e.such that there exist two subsequences n h and n j such that a ( n h ) m → a m and a ( n j ) m → a m . Set q = w a ǫ w ( h ) . . . w a m − ǫ w ( h m − ) w min { a i ,a i } ǫ and δ = | a i − a i | .Then the points { ξ f x ◦ ( a ( n h ) ) } h ∈ N , { ξ f x ◦ ( a ( n j ) ) } j ∈ N (18)ultimately belong respectively to the (closed) intervals [ qw δǫ w ( h m ) ] and [ qw ( h m ) ] ,which, recalling that w ǫ = w h m , have nonempty intersection only if either 1) δ = 0 or 2) δ ∈ N and w ( h m ) = p and w ǫ = p , or alternatively w ( h m ) = p and w ǫ = p for some p ∈ { , } k − . We can assume the former (in the other casethe reasoning is completely analogous). A point z in [ qw δǫ w ( h m ) ] ∩ [ qw ( h m ) ] mustthen admit the two binary forms z = 0 .qp ∞ and z = 0 .qp ∞ . The first onecan be only achieved if δ = 1 , w ( h m ) = 0 k and w ( h j ) = w ( h m ) for all j > m . Thisin turn implies that the second binary form, ending in ∞ , can be only obtainedif w ǫ = 1 k . But then w ( h m ) and w ǫ differ on every digit, which contradicts theexistence of the word p required before. It follows that it has to be δ = 0 , whichimplies a i = a i .Assume now that it is nonempty the set A of i ∈ N such that a i = ∞ , and set z = 0 .u . Then u = pw ∞ ǫ , (19)where p ∈ { , } ∗ is the word p = w a ǫ w ( h ) w a ǫ w ( h ) . . . w a ¯ i − ǫ w ( h ¯ i − ) with ¯ i =min A . Therefore, it falls under point 2. in the statement of the Lemma.Suppose instead that lim n →∞ a ( n ) i = ∞ for every integer positive i . Then u = Q ∞ i =1 w a i ǫ w ( h i ) , which implies that z ∈ f − σ ( y ) . Therefore, it falls under point 1. n the statement of the Lemma. Notice also that in this case, since u is obtainedfrom e x by inserting (infinitely many times) finite concatenations of w ǫ with itselfin a word not having w ǫ as a k -factor, u cannot end in ∞ either if w ǫ = 0 k or ifnot, so that u = e z .2. y ∈ Q \ F σ .By hypothesis y has two binary expansions, e y and u ∈ { , } ω such that u = v ∞ .Then: i) by assumption there exist two words w (1) , w (2) ∈ { , } ω not having w ǫ as a k -factor, such that σ ( w (1) ) = e y and σ ( w (2) ) = u ; ii) there is a countablefamily of finite integer sequences J = ∪ j ∈ N { j i } i =1 ...m j such that there are words w (3)( j ) = Q m j i =1 w ( j i ) ( m j ∈ N ) having the property that σ (cid:16) w (3)( j ) (cid:17) = v g j for suitablenon-negative integers g , g , · · · ∈ N . Notice that the family J is infinite, becausethe optimality condition ensures that there are arbitrarily large integers h suchthat we can write v h as a concatenation of the words v ( i ) = σ ( w ( i ) ) . Recallingthe definition of f σ (Eq. (3)), it follows that any point of type .w (3)( j ) belongs to f − σ ( y ) .Then the set f − σ ( y ) decomposes as S ∪ S ∪ S , where S = S a ∈ N N ξ w (1) ( a ) , S = S a ∈ N N ξ w (2) ( a ) S = S a ∈ N m ,j ∈ N { x ∈ I : x = 0 . (cid:0)Q m j i =1 w a i ǫ w ( j i ) (cid:1) w ∞ ǫ } , where ξ w ( i ) ( i = 1 , ) is the map defined in 3.3. Since a limit point of a finite unionof sets is limit point of (at least) one of the sets, we can repeat the argument usedin Case 1. separately for S and S . A similar argument shows that also the set S has always limit points of the form (19).3. y ∈ F σ \ Q .Then there is no x ∈ I such that f σ ( x ) = y and e x does not have w ǫ as a k -factor. Indeed, by definition of F σ , there is a ∈ { , } ∗ such that .σ ( a ∞ ) = y .Recalling that f σ is defined by the action of σ on the 1-periodic expansion ofdyadic rationals, it follows that x ∈ f − σ ( y ) = ⇒ x = 0 .a ∞ Y i =1 n i w m i ǫ (20)where { m i } is a sequence of non-negative integers which is not ultimately zeroand n i are positive integers such that | a | + P ij =1 n j is a multiple of k for every i ∈ N . Let us define the set B as: B = { b ∈ { , } ∗ : b = a N Y i =1 n i w m i ǫ for some N ∈ N . } (21)Recalling Case 1., it follows that the set B σ = { .bw ∞ ǫ , b ∈ B} (22) elongs to f − σ ( y ) , and its points falls under point 2. in the statement of theLemma. Moreover, it follows that the set B σ = { .b ∞ , b ∈ B} (23)belongs to f − σ ( y ) , and its points falls under point 3. in the statement of theLemma.4. y ∈ Q ∩ F σ .Suppose first that w ǫ = 0 k . By assumption y has two infinite binary expansions, e y and u ∈ { , } ω such that u = v ∞ for some finite word v . Since y ∈ F σ and isa dyadic rational, we have that σ (0 k ) = 0 N or σ (0 k ) = 1 N for some N , N ∈ N .In the following we consider the first case, a similar argument applies in the otherone switching the roles of e y and u .Then: i) there exists a word w (1) ∈ { , } ω not having w ǫ as a k -factor, such that σ ( w (1) ) = e y ; ii) moreover, recalling that f σ is defined by the action of σ on the 1-periodic expansion of dyadic rationals and that σ (0 k ) is a string of 0s, there is no x ∈ I such that σ ( e x ) = u and e x does not have w ǫ as a k -factor, while, by definitionof F σ , there is c ∈ { , } ∗ such that σ ( c ∞ ) = u = v ∞ ; iii) finally, there is acountable family of finite integer sequences J = ∪ j ∈ N { j i } i =1 ...m j such that thereare words w (2)( j ) = Q m j i =1 w ( j i ) ( m j ∈ N ) having the property that σ (cid:16) w (2)( j ) (cid:17) = v g j for suitable non-negative integers g j ∈ N . It follows that f − σ ( y ) decomposes as S ∪ S ∪ S , where S = S a ∈ N N ξ w (1) ( a ) , S = { x ∈ I : x = 0 .c Q ∞ i =1 n i w m i ǫ } S = S a ∈ N m ,j ∈ N { x ∈ I : x = 0 . (cid:0)Q m j i =1 w a i ǫ w ( j i ) (cid:1) w ∞ ǫ } , where { m i } and { n i } have the same properties of Case 3. and ξ w (1) is the mapdefined in 3.3. Then we can repeat the argument of Case 2. for S and S andthe argument of Case 3. for S .Finally, notice that, if w ǫ = 0 k , the sets S and S do not belong to f − σ ( y ) , since σ acts on the 1-periodic expansion of dyadic rationals. Theorem 1. In the hypotheses of Lemma 3.4, the closure of f − σ ( y ) is a Cantor space,as a topological subspace of I , for every y ∈ I \ Q .Proof. Suppose y / ∈ Q ∪ F σ and take the point x ◦ ∈ f − σ ( y ) . By the previous Lemma, f − σ ( y ) = S a ∈ N N ξ f x ◦ ( a ) , where ξ f x ◦ : N N → I is the map defined in 3.3, and Q = f − σ ( y ) \ f − σ ( y ) is a countable set. If a, b ∈ N N agree up to the n − th element, then | ξ f x ◦ ( a ) − ξ f x ◦ ( b ) | ≤ − nk − P ni =1 a i k , which implies that f − σ ( y ) is dense in itself, and sincethe points in Q are limit points of f − σ ( y ) by construction, so is f − σ ( y ) . Moreover if,in the binary expansion w a ǫ w ( h ) w a ǫ w ( h ) . . . of ξ f x ◦ ( a ) , we replace w ( h m ) by a word v ∈ { , } k which is distinct from both w ǫ and w ( h m ) (recall that we assume k ≥ ), we et a point which, by Lemma 3.4, does not belong to f − σ ( y ) . Since m can be arbitrarilylarge, it follows that f − σ ( y ) cannot contain any interval, so that it is nowhere denseand we can conclude.Suppose now that y ∈ F σ \ Q . Then in the last member of (20) we can choose asequence { h i } coinciding with { m i } up to an arbitrarily large index ¯ i to define a newpoint in f − σ ( y ) which will be arbitrarily close to x . This shows that f − σ ( y ) is dense initself, and since the points in B σ ∪ B σ are limit points of f − σ ( y ) by construction, so is f − σ ( y ) . Moreover, it can be seen with the same argument applied above that f − σ ( y ) does not contain any interval. Remark 7. Notice that we excluded dyadic rationals in the hypotheses of Theorem 1because the points belonging to the set S described in points 2. and 4. of Lemma 3.4are in general isolated points of the fiber. Theorem 2. There exists a measurable subset Γ ⊂ I such that f σ (Γ) is not measurable.Proof. Set M = { x ∈ I : e x does not have w ǫ as a k -factor } . Let V ⊂ I be a Vitali set.The optimality condition 2.4 implies that f σ ( M ) = I . Therefore, there is Γ ⊂ M : f σ (Γ) = V . Observe then that the points in M are not normal real numbers, in thesense (equivalent to the usual definition, see for instance [15]) that, partitioning theirbinary expansion in words of length k , the asymptotic frequency of the block w ǫ isevidently 0. Therefore we have m ( M ) = 0 (see for instance [9]). By completeness ofthe Lebesgue measure, we have m (Γ) = 0 .It is not trivial to say for which (if any) k -block erasing substitutions σ the set f − σ ( V ) can be Lebesgue-measurable and, if so, whether its measure can be strictly positive.Notice that Theorem 2 can also be obtained as an easy consequence of a classical resultby Purves ([17]), stating that, if f is a bi-measurable map from a standard Borel space X to a Polish space Y , then the cardinality of points having an uncountable f -preimageis at most countable. We preferred to provide a (simple) direct proof mainly to makeclearer the meaning of the questions above. f σ based on the erasing class In this section we will study the connection between the erasing class of σ and thedynamical properties of f σ . Throughout the Section, σ is a k -block erasingsubstitution defined by Eq. (1) . We start “getting rid” of the extreme case of boundedly erasing substitutions in thenext Lemma. Lemma 4.1. If σ is boundedly erasing, then σ does not verify the optimality condition(2.4). Moreover, there exists n ∈ N such that, for every x ∈ I such that its f σ -orbithas empty intersection with Q , we have f nσ ( x ) = 0 . roof. Since σ is boundedly erasing, there exists n = max w ∈{ , } ∗ ǫ ( w ) (as ǫ ( · ) ≥ isinteger and bounded by assumption). Then, for every u ∈ { , } ω and for every m > , σ n ( u . . . u m ) = ǫ , whence, by induction on m , we have ∀ u ∈ { , } ω , σ n ( u ) = ǫ. (24)Take w (0) ∈ { , } ω . Suppose by absurde that σ verifies the optimality condition.Then there exists w (1) ∈ { , } ω such that σ ( w (1) ) = w (0) . Iterating the argument, itfollows that we can construct a sequence of nonempty infinite words { w ( m ) } m ∈ N suchthat, for every m ∈ N , σ ( w ( m ) ) = w ( m − . But then the word w ( N ) contradicts (24)for N large enough. Take now x ∈ I such that its f σ -orbit is does not contain dyadicrationals other than 0 and 1. Suppose that f jσ ( x ) = 0 for every j ∈ { , . . . , n − } .Then σ n − ( e x ) = w ∞ ǫ which implies f nσ ( x ) = 0 . Recalling that 0 is a fixed point of f σ by definition, we have our thesis.In the following Lemma we prove a technical result which will be widely used in thefollowing. Lemma 4.2. If σ is strongly erasing and verifies the optimality condition 2.4, then,for every w ∈ { , } ∗ , u ∈ { , } ∞ , there is a positive integer h depending only on w and a word v ∈ { , } ∞ , such that:a) σ h ( wv ) has u as a prefix (in particular it coincides with u if | u | = ∞ );b) for every finite integer sequence { n , . . . , n h − } ∈ ( N ) h , the word σ iw ( v ) has e ( i ) w n i ǫ as a prefix for i ∈ { , . . . , h − } , where { e ( i ) } i =0 ,...,h − is a set of erasing k -extensions of w .Moreover:c) if σ is completely erasing, we have the same results and we can choose h = ǫ ( w ) . Proof. We start fixing w ∈ { , } ∗ and u ∈ { , } ∞ .1. | u | < ∞ .It follows from the optimality condition 2.4 that, for every finite sequence ofnon-negative integers { n , . . . , n h − } ∈ ( N ) h , we can find h finite binary words p (0) , p (1) , . . . , p ( h − such that, for suitable q ( i ) ∈ { , } ∗ ( i = 0 , . . . , h − ), wehave σ ( p (0) ) = uq (0) , σ ( p ( i ) ) = e ( h − i ) w n h − i ǫ p ( i − q ( i ) ∀ i = 1 , . . . , h − . It follows that the integer h and the word v = e (0) w n ǫ p ( h − verify the claim,because σ h ( wv ) has uq (0) as a prefix and, for every i ∈ { , . . . , h − } , σ iw ( v ) has e ( i ) w n i ǫ as a prefix. Notice that this occurrence of w n i ǫ will be a k -factor in σ ( i ) ( wv ) by construction.2. | u | = ∞ .It follows from the optimality condition 2.4, and in particular from Remark 1,that, for every finite sequence of non-negative integers { n , . . . , n h − } ∈ ( N ) h ,we can find h finite binary words p (0) , p (1) , . . . , p ( h − such that σ ( p (0) ) = u and σ ( p ( i ) ) = e ( h − i ) w n h − i ǫ p ( i − for i = 1 , . . . , h − . It follows that the integer h andthe word v = e (0) p ( h − verify the claim. inally, recalling that completely erasing substitutions are always strongly erasing, a)and b) are obviously true also for them, and since σ in this case is alternating, we cantake as erasing k -extension e ( i ) the word ( w ǫ ) k −| e ( i ) | +1 . . . ( w ǫ ) k , which readily impliesthat h can be taken equal to ǫ ( w ) , which proves point c). Lemma 4.3. If σ is strongly erasing and verifies the optimality condition 2.4, thenthere is an uncountable set D ⊂ I such that the f σ -orbit of every x ∈ D is dense in I .Moreover, D is dense in I . Proof. Let us indicate by { w ( n ) } n ∈ N an enumeration of all finite binary words. We willconstruct inductively a point x = 0 . e x with the property that there exists a sequence ofintegers { h n } n ∈ N such that σ h n ( e x ) has w ( n ) as a prefix for every n .By Lemma 4.2, there exists u (1) ∈ { , } + and h ∈ N such that σ h ( w (0) u (1) ) has w (1) as a prefix. Let { e (0)( j ) } j ∈{ ,...,h − } be the erasing k -extensions of w (0) . By Lemma4.2, for { n , . . . , n h − } ∈ ( N ) h , we can select u (1) so that the word σ jw (0) (cid:0) u (1) (cid:1) has e (0)( j ) w n j ǫ as a prefix for j ∈ { , . . . , h − } .Set p (1) = w (0) u (1) . Suppose that we arrived at the step i − and we have de-fined p ( i − = w (0) u (1) . . . u ( i − . At the i -th step we have that, again by Lemma 4.2,there exists u ( i ) ∈ { , } + and h i ∈ N such that σ h i ( p ( i − u ( i ) ) has w ( i ) as a prefix.Let { e ( i − j ) } i ∈{ ,...,h − } be the erasing k -extensions of w ( i − . By Lemma 4.2, for { n i , . . . , n ih i − } ∈ ( N ) h , we can select u ( i ) so that the word σ jp ( i ) (cid:0) u ( i ) (cid:1) has e ( i − j ) w n ij ǫ as a prefix for j ∈ { , . . . , h i − } . By the construction made in Lemma 4.2, we have σ h i ( p ( i ) ) = ǫ , and since p ( i − is always a prefix of p ( i ) , it follows that h i ≥ h i − forevery i . We can then define the word v as v = w (0) ∞ Y j =1 u ( j ) (25)so that by construction the σ -orbit of v has w ( n ) as a prefix for every n ∈ N . Noticefurther that different choices for n i ( i ∈ N ) produce different words in Eq.(25), whichproves that we have uncountably many choices for v . If, for every j ∈ N , we takethe sequence ( n ij ) i ∈ N in the set Ξ + , we are ensured that σ j ( v ) will not end in ∞ forevery j (either in case w ǫ = 0 k or not). Therefore, the point x = 0 .v = 0 . e x has a dense f σ -orbit and the set D of points with a dense f σ -orbit is uncountable. Finally, by thearbitrariness of the prefix w (0) , the set D is dense in I .Since the sequence of words (cid:0) w ( n ) (cid:1) n ∈ N can be arbitrary, in the previous proof we alsoproved the following Corollary 2. Suppose that σ is strongly erasing and verifies the optimality condition2.4. Then, if ( w ( n ) ) n ∈ N is any sequence of finite binary words, there is x ∈ [ w (0) ] and positive integers h , h . . . (with h i depending only on w ( i ) for every positiveinteger i ) such that f h n σ ( x ) ∈ [ w ( n ) ] for every n ∈ N . Moreover, exploiting point c)of Lemma 4.2, we can write e x as: e x = Q ∞ i =1 v ( i ) where, for every positive integer n , σ ǫ ( v (1) ...v ( n − ) v (1) ...v ( n − (cid:0) v ( n ) (cid:1) has w ( n ) as a prefix. emma 4.4. If σ is strongly erasing and verifies the optimality condition 2.4, then f σ has / -sensitive dependence on initial conditions. Proof. Take x ∈ I . If y ∈ I is such that e x and e y have in common a prefix p longenough, then | x − y | < δ for arbitrarily small δ > . By Lemma 4.2, there exists apositive integer n (depending only on p ) such that, for every u ∈ { , } ∞ , there is v ∈ { , } ∞ such that σ n ( pv ) has u as a prefix. Set u = 0 ∞ if .σ n ( e x ) > / and u = 1 ∞ if .σ n ( e x ) ≤ / . Then, if we take z ∈ I such that e z has pv as a prefix, wehave | f nσ ( x ) − f nσ ( z ) | ≥ / . The thesis follows from the arbitrariness of δ and of thepoint x ∈ I .A particular case of Corollary 2 is that it is possible to obtain a point x whose orbitvisits infinitely many times any cylinder [ w ] by simply imposing that { w ( n ) } in theproof of Lemma 4.3 is the constant sequence: w ( n ) ≡ w ∀ n ∈ N . Notice however that,assuming σ strongly erasing, this does not provide in general a periodic point, but onlya (uniformly) recurrent point. Indeed, for every prefix w ∈ { , } ∗ , the fact that σ isstrongly erasing implies that we can find a positive integer n and a word u ∈ { , } ∗ such that σ n ( wu ) = wp for some word p ∈ { , } Let be u ∈ { , } ω and w ∈ { , } ∗ such that σ h ( w ) = p ∈ { , } ∗ . Set ǫ i = ( w ∞ ǫ ) i for i ∈ N and w ( j ) = σ j ( w ) , n j k + m j = | w ( j ) | for ≤ j ≤ h , n j ∈ N , ≤ m j < k (notice that w (0) = w ). By the optimality condition, there is q (1) ∈ { , } ω such that σ (cid:0) q (1) (cid:1) = u . Therefore, σ ( aq (1) ) = σ ( a ) u for every a ∈ { , } nk , n ∈ N . Set v (1) = ǫ m h − +1 . . . ǫ k q (1) , so that, recalling Eq.(7), σ ( w ( h − v (1) ) = w ( h ) u = pu .Iterating the argument, we get the existence of words v ( j ) ( ≤ j ≤ h ) such that σ ( w ( h − j ) v ( j ) ) = w ( h − j +1) v ( j − , so that σ h ( wv ( h ) ) = pu , and therefore v = v ( h ) verifiesthe claim. Lemma 4.6. If σ is completely erasing and verifies the optimality condition 2.4, thenthe set of f σ -periodic points of period p , indicated by P p , is uncountable for every p ∈ N . Moreover, P = ∪ p ∈ N P p is dense in I . Proof. Take u (0) ∈ { , } ≥ k . Since σ is alternating, we can apply it to finite wordsof any length (without dropping any digit for words of length not multiple of k ). Set ǫ i = ( w ∞ ǫ ) i for i ∈ N . We will show an iterative construction leading to a periodicpoint belonging to [ u (0) ] . et | u (0) | = n k + m with n ∈ N and ≤ m < k . By Lemma 4.2, there existsa binary word u ∈ { , } ω such that σ ǫ ( u (0) ) (cid:0) u (0) u (cid:1) = u (0) Q ∞ j = m +1 ǫ j , so that theremust be p , s ∈ N such that σ ǫ ( u (0) ) (cid:16) u (0) u . . . u p (cid:17) = u (0) ǫ m +1 ǫ m +2 . . . ǫ m + s . Set then s = r k + q for r ∈ N , ≤ q < k and define w (0) = u (0) ǫ m +1 ǫ m +2 . . . ǫ m + s , u (1) = k − q Y j =1 ǫ m + s + j u . . . u p , where the product can be replaced by the empty word if q = 0 . Recalling Eq.(7),we have σ ( u (0) ) = σ ( w (0) ) , so that ǫ ( u (0) ) = ǫ ( w (0) ) . Since by construction s + k − q is a multiple of k , we have thus σ (cid:0) w (0) u (1) (cid:1) = σ (cid:0) u (0) u . . . u p (cid:1) and therefore σ ǫ ( w (0) ) ( w (0) u (1) ) = w (0) .Suppose now that there are i + 2 finite binary words w (0) , . . . , w ( i ) , u ( i +1) such that σ ǫ ( w (0) ) ( w (0) . . . w ( i ) u ( i +1) ) = w (0) . . . w ( i ) . Set | w (0) . . . w ( i ) u ( i +1) | = n i +1 k + m i +1 with n i +1 ∈ N and ≤ m i +1 < k . By Lemma 4.5, there exists a binary word u ∈ { , } ω such that σ ǫ ( w (0) ) (cid:0) w (0) . . . w ( i ) u ( i +1) u (cid:1) = w (0) . . . w ( i ) u ( i +1) Q ∞ j = m i +1 +1 ǫ j , so that theremust be p i +1 , s i +1 ∈ N such that σ ǫ ( w (0) ) (cid:16) w (0) . . . w ( i ) u ( i +1) u . . . u p i +1 (cid:17) = w (0) . . . w ( i ) u ( i +1) ǫ m i +1 +1 ǫ m i +1 +2 . . . ǫ m i +1 + s i +1 . Set then s i +1 = r i +1 k + q i +1 for r i +1 ∈ N , ≤ q i +1 < k and define w ( i +1) = u ( i +1) ǫ m i +1 +1 ǫ m i +1 +2 . . . ǫ m i +1 + s i +1 u ( i +2) = k − q i +1 Y j =1 ǫ m i +1 + s i +1 + j u . . . u p i +1 where the product can be replaced by the empty word if q i +1 = 0 . Since by construction s i +1 + k − q i +1 is a multiple of k , we have thus σ (cid:16) w (0) . . . w ( i ) w ( i +1) u ( i +2) (cid:17) = σ (cid:16) w (0) . . . w ( i ) u ( i +1) u . . . u p i +1 (cid:17) , and therefore σ ǫ ( w (0) ) ( w (0) . . . w ( i +1) u ( i +2) ) = w (0) . . . w ( i +1) .By induction on i we get the existence of a countable family of words verifying, forevery i ∈ N , σ ǫ ( w (0) ) (cid:16) w (0) . . . w ( i +1) (cid:17) = w (0) . . . w ( i ) . It follows that the word w = Q ∞ i =0 w ( i ) is σ -periodic with period ǫ ( w (0) ) . As observedmultiple times, in the applications of Lemma 4.2 (as well as of Lemma 4.5) we canensure that, for every h ∈ N , σ h ( w ) does not end in ∞ , so it follows that the point x = 0 . Q ∞ i =0 w ( i ) is an f σ -periodic point of period ǫ ( w (0) ) .The density of the set P of f σ -periodic points follows from the arbitrariness of the word u (0) . The uncountability of P p ( p ∈ N ) follow from the same argument used in Lemma4.3. emark 8. Notice that, in the previous proof, taking u (0) = w mǫ ( m ∈ N ) we obtainfixed points for f σ , and that for every m there are uncountably many of them. In the proof of Lemma 4.5 we actually also proved the following Corollary 3. If σ is completely erasing and verifies the optimality condition 2.4, thenfor every w, u ∈ { , } ∗ there is s ∈ N and v ∈ { , } ∗ such that, setting | w | = nk + m ( n ∈ N , ≤ m < k ), σ ǫ ( w ) ( wv ) = u Q si = m +1 ( w ∞ ǫ ) i . Notice that by construction σ w ( Q si = m +1 ( w ∞ ǫ ) i ) = ǫ .This in turn implies the following Lemma 4.7. If σ is completely erasing and verifies the optimality condition 2.4, then,for every n ∈ N , there is a word w ∈ { , } ∗ such that ǫ ( w ) = n . Proof. By Corollary 3, there is w (1) such that σ ( w (1) ) = w ǫ q (1) , with σ w ǫ ( q (1) ) = σ ( q (1) ) = ǫ . Applying further n − times the same argument one gets the existence of w (2) , . . . , w ( n ) such that σ ( w ( i +1) ) = w ( i ) q ( i ) with σ w ( i ) ( q ( i ) ) = ǫ for ≤ i ≤ n − , sothat σ n ( w ( n − ) = ǫ and therefore ǫ ( w ( n − ) = n .Lemmas 4.3, 4.4 and 4.6 imply the following Theorem 3. If σ is completely erasing and verifies the optimality condition 2.4, then f σ is Devaney chaotic. Remark 9. It is well-known that, in the definition of Devaney chaos, assuming sen-sitive dependence on initial conditions is redundant for continuous maps defined on(infinite) metric spaces ([1]), and on the interval the existence of one dense orbit im-plies both sensitive dependence and the existence of a dense set of periodic points ([23]).Because our general object is just Baire-1, we proved explicitly the three properties. Theorem 4. If σ is strongly erasing and verifies the optimality condition 2.4, then forevery w ∈ { , } ∗ , there is h ∈ N such that f hσ ([ w ]) = I , and therefore f σ is topologicallymixing.Proof. By Lemma 4.2, for every w ∈ { , } ∗ there is h ∈ N such that, for every y ∈ I there is v ∈ { , } ω such that y = 0 .σ h ( wv ) , where we can take σ iw ( v ) not ultimatelyconstantly 0 for every i ∈ { , . . . , h − } by exploiting point b) of Lemma 4.2 to insert,if necessary, the words w ǫ as k -factors. Therefore, f hσ ( x ) = y if wv = e x , which impliesthat f hσ ([ w ]) = I . Take now A, B open subsets of I . Since open sets in the interval arecountable union of (disjoint) open intervals, there is w ∈ { , } + such that [ w ] ⊂ A . If h ∈ N is that of the previous argument, by Lemma 3.1, f nσ ( A ) ∩ B = ∅ for every n ≥ h ,which means that f σ is topologically mixing.For continuous interval maps, Devaney chaos implies Li-Yorke chaos ([10]), and theexistence of a Li-Yorke pair implies the existence of an uncountable scrambled set([13]). In our case, since the map f σ is discontinuous on a countable subset of I , Li-Yorke chaos is a priori not given, and moreover the existence of just one Li-Yorke pairdoes not guarantee Li-Yorke chaos. However, assuming that σ is completely erasing,we can in fact prove that f σ exhibits Li-Yorke chaos. heorem 5. If σ is completely erasing and verifies the optimality condition 2.4, then f σ is Li-Yorke chaotic, meaning that there exists an uncountable scrambled subset ofits domain, that is a set S ⊂ I such that, for every x, y ∈ S : lim inf n →∞ | f nσ ( x ) − f nσ ( y ) | = 0 , lim sup n →∞ | f nσ ( x ) − f nσ ( y ) | > (26) Proof. Consider a sequence W = { w ( n ) } n ∈ N of finite binary words such that {| w ( n ) |} n ∈ N is an unbounded sequence of positive integers. By Corollary 2, we can find x ∈ I suchthat its f σ -orbit visits [ w ( n ) ] for every n ∈ N , and since σ is completely erasing, wecan write e x as: e x = Q ∞ i =1 v ( i ) where, for every positive integer n there is q ( n ) ∈ { , } ∗ such that σ ǫ ( v (1) ...v ( n − ) v (1) ...v ( n − (cid:16) v ( n ) (cid:17) = w ( n ) q ( n ) . (27)Moreover, by Corollary 3, we can choose q ( n ) such that ǫ ( w ( n ) q ( n ) ) = ǫ ( w ( n ) ) .Notice that (27) is the only property used to define v ( n ) , which means that in fact wecan define a family of points X ⊂ I as follows: X = { x = 0 . ∞ Y n =1 v ( n ) where v ( n ) is such that (27) holds for every n ∈ N } (28)In particular, by Lemma 4.5, we are free to choose v ( n ) so as to have that σ ǫ ( v (1) ...v ( n − ) − v (1) ...v ( n − (cid:16) v ( n ) (cid:17) has u ( n ) as a prefix , (29)where σ ( u ( n ) ) = w ( n ) q ( n ) and u ( n ) does not have w ǫ as a k -factor (notice that we donot have to insert a suffix after q ( n ) because we know that Eq.(27) holds).On the other hand, we can also make the choice σ ǫ ( v (1) ...v ( n − ) − v (1) ...v ( n − (cid:16) v ( n ) (cid:17) has w ǫ u ( n ) as a prefix . (30)We underline that, for every n ≥ , we can make choice (29) or choice (30) while stillretaining the property (27). We want to use this to construct an uncountable scrambledsubset of I . For α = α (1) α (2) · · · ∈ Ω (see Definition 3.1), let us define the set X α ⊂ X of points in I such that, for every x ∈ X α , e x = ∞ Y n =1 v ( n ) where, for every n : ( (29) holds if α ( n ) = 1 (30) holds if α ( n ) = 0 . (31)Pick now exactly one point x α from every set X α and set X Ω = S α ∈ Ω { x α } . We willshow now that X Ω is an uncountable scrambled set. Indeed, let α = β be two elementsof Ω . Since both f x α and f x β verify (27) for every n , we have that f nσ ( x α ) , f nσ ( x β ) ∈ [ w ( n ) ] . Since | w ( n ) | is unbounded, it follows that lim inf n →∞ | f nσ ( x α ) − f nσ ( x β ) | = 0 . oreover, only for finitely many integers n the word u ( n ) is shorter than 2 k , so that w ǫ is different from both u ( n )1 . . . u ( n ) k and u ( n ) k +1 . . . u ( n )2 k for large enough n . It follows fromEqs. (30) and (31) that, if n is such that α ( n ) = β ( n ) , then q = ǫ ( v (1) . . . v ( n − ) − implies | f qσ ( x α ) − f qσ ( x β ) | > − k . Since α and β are not ultimately coinciding, itfollows that lim sup n →∞ | f nσ ( x α ) − f nσ ( x β ) | > . Finally, X Ω is uncountable as α = β implies that x α = x β because by construction they have distinct f σ -orbits.Let us now devote our attention to the topological entropy of the maps f σ . The in-vestigation of topological entropy for discontinuous maps has become more intense inlast years, and in particular maps with dense discontinuities, as for instance Darboux,Baire-1 maps, have been studied also from this point of view (see [12]). In the fol-lowing result we will see that the topological entropy of f σ depends on how ǫ ( w ) goesasymptotically with | w | . The proof technique, which recalls the classical constructionof horseshoes developed originally for continuous interval maps ([14]), is based on theidea of “spreading” a set of points belonging to each interval from a suitable partitionof I over all the other intervals of the partition. Theorem 6. Suppose that σ is a completely erasing substitution verifying the optimal-ity condition 2.4. Then f σ has infinite topological entropy if, for every w ∈ { , } ∗ , lim | w |→∞ | w | ǫ ( w ) = ∞ . (32) Proof. For every w ∈ { , } + , set F ( | w | ) = max u ∈{ , } | w | ε ( u ) . For x , x ∈ I let usintroduce the metric: d N ( x , x ) := max { (cid:12)(cid:12) f iσ ( x ) − f iσ ( x ) (cid:12)(cid:12) : 0 ≤ i ≤ N } (33)We say that a subset S ⊂ I is ( n, ε ) -separated in the metric d N if for all x , x ∈ S , x = x we have that d N ( x , x ) ≥ ε and we indicate by | ( n, ǫ ) | the maximum cardinality ofan ( n, ε ) -separeted set. We recall that the topological entropy h of the map f σ can bewritten as ([5, 2]): h ( f σ ) = lim ε → (cid:18) lim n →∞ sup 1 n log | ( n, ε ) | (cid:19) Fix now ε = 2 − ( k +1) . Step 1. If f tσ ( x ) ∈ [ w ( i ) w ǫ ] and f tσ ( x ) ∈ [ w ( r ) w ǫ ] , for some ≤ t ≤ N ( i = r ), then d N ( x , x ) > ǫ . Indeed, we can observe that we will have the smallest possibledifference between f tσ ( x ) and f tσ ( x ) when w ( i ) and w ( r ) differ at the k -th digitand f tσ ( x ) = 0 .w ( i ) w ǫ ∞ , f tσ ( x ) = 0 .w ( r ) w ǫ ∞ (or vice-versa), so that we have | f tσ ( x ) − f tσ ( x ) | ≥ (cid:12)(cid:12) k − k (cid:12)(cid:12) ≥ · k = ǫ .Clearly for any finite word w we have ǫ ( w ) = ǫ ( ww ǫ ) . Set now t = F ( k ) . Step 2. For any i, r ∈ { , . . . , k } , there exists x ∈ [ w ( i ) w ǫ ] such that f tσ ( x ) ∈ [ w ( r ) w ǫ q ( r ) ] with σ w ( r ) w ǫ q ( r ) = ǫ .Indeed, let us fix i, r ∈ { , . . . , k } . Since by assumption ǫ ( w ) ≤ F ( | w | ) for every w ∈ { , } ∗ , we have ǫ ( w ( i ) ) = ǫ ( w ( i ) w ǫ ) ≤ t . By Lemma 4.7, there exists a word ∈ { , } ∗ such that ǫ ( p ) = t − ǫ ( w ( i ) w ǫ ) , and by Corollary 3, there is q ∈ { , } ∗ such that σ ǫ ( w ( i ) w ǫ ) ( w ( i ) w ǫ q ) = pp ′ , with p ′ ∈ { , } ∗ such that ǫ ( pp ′ ) = ǫ ( p ) , andtherefore ǫ ( w ( i ) w ǫ q ) = t . Then the thesis follows by applying Lemma 4.2, where w = w ( i ) w ǫ q , u = w ( r ) w ǫ and h = t . Step 3. We will define now a family of sets { S i ⊂ [ w ( i ) w ǫ ] } i =1 ,..., k , each containing nk ( n ∈ N ) distinct points of I , such that ∪ ≤ i ≤ k S i will be an ( n, ε ) -separated set.We will indicate by x il ( l ∈ { , . . . , nk } ) the points of S i , where the subscript l identifies their standard order as real numbers, and we will construct them iter-atively, requesting properties which will be ensured by increasingly long prefixesof their binary expansion. By Step 2., for i ∈ { , . . . , k } the points x i l can betaken such that: f tσ ( x i l ) ∈ S i for l ∈ { ( i − ( n − k + 1 , . . . , i ( n − k } (34)where i ∈ { , . . . , k } . Hence we can define the set S i i = { x : x ∈ S i , f tσ ( x ) ∈ S i } and we have (cid:12)(cid:12) S i i (cid:12)(cid:12) = 2 ( n − k . Moreover, notice that after further t iteratesof f σ , the prefix w i w ǫ of every point f tσ ( x i l ) is mapped again in the emptyword. Therefore for every i , i ∈ { , . . . , k } and for l ∈ { ( i − ( n − k +1 , . . . , i ( n − k } , Step 2. allows us to have: f tσ ( x i i l ) ∈ S i for l ∈ { ( i − ( n − k + 1 , . . . , i ( n − k } (35)where { x i i l } l =1 ,..., ( n − k ∈ S i i , the subscript l indicates their standard orderas reals and i ∈ { , . . . , k } . Hence we can define the set S i i i = { x : x ∈ S i , f tσ ( x ) ∈ S i , f tσ ( x ) ∈ S i } and we have (cid:12)(cid:12) S i i i (cid:12)(cid:12) = 2 ( n − k . We can proceed in this way for further n − steps in order to obtain that f jtσ ( x i ...i j l ) ∈ S i j +1 for l ∈ { ( i j +1 − ( n − j ) k + 1 , . . . , i j +1 ( n − j ) k } (36)for all j ∈ { , . . . , n − } , where { x i ...i j l } l =1 ,..., k ∈ S i ...i j , the subscript l indicatestheir standard order as reals and i j +1 ∈ { . . . , k } . We finally arrive at the set S i ...i j +1 = ( { x : x ∈ S i , f tσ ( x ) ∈ S i , . . . , f jtσ ( x ) ∈ S i j +1 } ) j ∈{ ,...,n − } such that (cid:12)(cid:12) S i ...i j +1 (cid:12)(cid:12) = 2 k . Step 4. Recalling Step 1., the last argument implies that S i ∈{ ,..., k } S i is an ( nt, ǫ ) -separated set, so that | ( nt, ε ) | ≥ ( n +1) k . From this it follows that h ( f σ ) ≥ lim k → + ∞ lim sup n →∞ log 2 ( n +1) k ( n + 1) t = lim k → + ∞ k log 2 F ( k ) (37)which immediately implies the thesis. heorem 6 can be applied to the model case map f σ , where σ is defined in Section2. Indeed, since, as observed before, all odd-indexed elements of w go to ǫ in at mosttwo iterations, the asymptotic behavior of ǫ ( w ) is logarithmic (for the optimal boundsee [3]).Attention has been devoted to the points around which the entropy concentrates, i.e. entropy points in the sense of [24]. We recall that an entropy point is a point x suchthat the topological entropy restricted to any of its closed neighborhoods K is positive,in symbols h ( f σ , K ) > . In case it always coincides with the entropy of the map onthe whole space, the point is called a full entropy point. It is known that, in caseof continuous maps on a compact metric space, every point is a full entropy point ifthe system is minimal and has positive topological entropy. When σ is completelyerasing, the system ( I , f σ ) is not minimal (there are periodic points). Still, we have thefollowing Theorem 7. Suppose that σ is completely erasing and verifies the optimality condition2.4. If Eq. (6) holds, then every point x ∈ I is a full entropy point for f σ .Proof. Take any x ∈ I . For every closed neighborhood K of x , there is w ∈ { , } ∗ such that e x = wv for some v ∈ { , } ω , and that [ w ] ⊂ K . If σ is completely erasingand verifies the optimality condition, by Theorem 4 we have f ǫ ( w ) σ ([ w ]) = I . Therefore,there exists S ⊂ [ w ] such that f ǫ ( w ) σ ( S ) = ∪ i S i (where S i are the same as in theprevious proof). Then the same argument as above implies the following bounds: h ( f σ , K ) ≥ h ( f σ , [ w ]) ≥ lim k → + ∞ k log 2 F ( k ) + ǫ ( w ) = ∞ (38)In [16], the author introduces a powerful concept which is useful when dealing withthe dynamical properties of Darboux, Baire-1 interval maps, i.e. that of almost fixedpoints . A point x ∈ I is an almost fixed point for a map f : I → I if it belongs to thetopological interior (in the space I with the natural topology) of at least one of the twosets: R − ( f, x ) = { y : f − ( y ) ∩ ( x − ε, x ) = ∅ ∀ ε > } ,R + ( f, x ) = { y : f − ( y ) ∩ ( x, x + ε ) = ∅ ∀ ε > } Among the results of [16] there is that, for Darboux, Baire-1 interval maps, the existenceof at least one almost fixed point has strong dynamical consequences, as it impliesinfinite topological entropy as well as periodic points of every period; moreover, in thesame assumptions, a fixed points exists in any open neighborhood of an almost fixedpoint. The scope of the concept is perhaps even wider, and it has possibly a dynamicalsignificance also for a subset of Baire-1, non-Darboux interval maps, including thosestudied herein. Here we limit ourselves to establish two very simple results linking theconcept of almost fixed point to the framework of erasing substitutions. Lemma 4.8. If σ is a k − block substitution, then the map f σ admits an almost fixedpoint only if σ is erasing. roof. If σ is not erasing, then the map f σ admits right and left limits, say x r and x l , at every x ∈ I (and they coincide outside Q ). In this case, R − ( f σ , x ) = { x l } and R + ( f σ , x ) = { x r } , so that their topological interior is empty. Lemma 4.9. If σ is an erasing k -block substitution verifying the optimality condition2.4, then the map f σ admits at least one almost fixed point, namely x = 0 .w ∞ ǫ . Proof. For every ε > , there is a positive integer m so large that every point in [ w mǫ ] is less than ε apart from x . By the optimality condition, there is v ∈ { , } ω suchthat σ ( v ) = σ ( w mǫ v ) = w ∞ ǫ . We can suppose that v . . . v k is strictly larger than w ǫ in lexicographic order (the argument is analogous if it is strictly smaller), so that x ∈ R + ( f σ , x ) . Since the optimality condition also ensures that [ σ ( v . . . v k )] belongsto R + ( f σ , x ) , the point x belongs to its topological interior. Notice that this alsoapplies if w ǫ = 0 k or k , as the interior is meant with respect to the natural topologyon I .Notice that, by Remark 8, when σ is completely erasing, in any open neighborhood of .w ∞ ǫ there is a fixed point for f σ .It seems no coincidence that (some of) the properties guaranteed by the existence ofan almost fixed point in the Darboux case are verified also for the maps f σ when σ iscompletely erasing. On the other hand, the fact that a further assumption is neededon ǫ ( · ) to get infinite entropy most probably means that there is no straightforwardstrengthening of the results of [16], but rather some suitable weakening of the Darbouxproperty has to be invoked/introduced. rasing + OC = ⇒ • f σ is continuous (non uniformly) on a set C σ such that | I \ C σ | = ℵ • f − σ ( x ) is uncountable for every x ∈ I \ Q ,• f σ is not bi-measurable• f σ is Baire-1 and (generally) not Darboux Strongly erasing + OC = ⇒ • The set of points with a dense f σ -orbit isuncountable and dense in I .• f σ has sensitive dependence on initialconditions• f σ is topologically mixing Completely erasing + OC = ⇒ • f σ exhibits Devaney chaos• f σ exhibits Li-Yorke chaos• f σ has infinite topological entropy as soon as ǫ ( · ) is asymptotically sublinear• f σ has almost fixed points Boundedly erasing = ⇒ • f σ has trivial dynamics: there is n such that 0attracts in n iterates every x ∈ I whose orbitdoes not intersect Q Summary of the relations between erasing class of σ and properties of f σ . A series of possible generalizations of the questions addressed in this paper appearnatural from either an analytical or a dynamical point of view. First of all, if wedrop the assumption of uniqueness of the sequence ( h i ) made in Lemma 3.4, then thetopological structure of the fibers becomes more intricate, because in general a fibercan be an uncountable union of sets each of which has a Cantor closure. It is notclear how this can affect the dynamics of the map f σ and more generally one can askwhich are the minimal assumptions on an erasing substitution σ to obtain the samedynamical properties proved here in the completely erasing case.Some natural questions also arise from the following simple argument: by construction, f σ (0 .w ) = f σ (0 .w ǫ w ) , so that the functional relation f σ ( x ) = f σ (cid:0) x − k + 0 .w ǫ (cid:1) alwaysholds. This induces a fractal structure in the graph of f σ , because its restriction to w ǫ ] is a horizontal compression plus a translation (the latter unless w ǫ = 0 k ) of thewhole graph. Since a fractal structure arises, it seems natural to ask what is the linkbetween the substitution σ and the Hausdorff dimension of the graph of f σ . We pointout that the estimate of the Hausdorff dimension of the graph is not trivial even in themodel case represented by the substitution σ defined in Section 2 (see [3]). References [1] John Banks, Jeffrey Brooks, Grant Cairns, Gary Davis, and Peter Stacey. OnDevaney’s definition of chaos. The American mathematical monthly , 99(4):332–334, 1992.[2] Rufus Bowen. Entropy for group endomorphisms and homogeneous spaces. Trans-actions of the American Mathematical Society , 153:401–414, 1971.[3] Alessandro Della Corte, Stefano Isola, and Riccardo Piergallini. 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