# Limit Sets, Attractors and Chaos

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## Abstract

The aim of this paper is a study on relations between \(\omega \)-chaos and the structure of \(\omega \)-limit sets. We propose a definition of \(\hat{\omega }\)-chaos which requires stronger relations between limit sets of points from tuples. We present example with relatively simple dynamics (almost equicontinuous system) which is \(\omega \)-chaotic and propose further restrictions on the conditions in the definition.

## Keywords

\(\omega \)-Chaos Almost equicontinuous Entropy Interval Scrambled set## Mathematics Subject Classification

37B05 37B10 37B20## 1 Introduction

The qualitative theory of dynamical systems focuses on the understanding how the trajectories of all points from the state space behave in long time. There appear situations when the motion is periodic or almost periodic however also more complicated, irregular patterns can appear during evolution of a point. Roughly speaking, the last case can usually be identified with some kind of chaos, and was intensively studied by many authors from different points of view, introducing several notions of chaos (e.g. see [1]).

The notion of omega chaos was introduced by Li [15] in 1993 for interval maps and was later generalized for continuous maps on compact metric spaces. An uncountable subset of the state space is called \(\omega \)-scrambled if for any nondiagonal pair in the set: (i) difference of omega limit sets is uncountable, (ii) intersection of omega limit sets is nonempty and (iii) each omega limit set is not contained in the set of periodic points. It provides yet another characterization of the situation when interval map has positive entropy or other chaotic phenomena, equivalent in this one-dimensional context.

Omega chaos is still not well understood and many open questions remains unsolved. The main difficulty in studies on this notion is that it more relies on geometric structure of attractors in the space, rather than evolution of single points. Simply, points with completely different, unrelated trajectories can lead at the end to the same omega limit set (e.g. two points with dense trajectory). In [16] authors discussed omega chaos for the circle maps and [11, 12] studied cardinality of omega scrambled sets and relations to the chaos in the sense of Li and Yorke. Later in [4, 23] authors discussed Lebesgue measure of omega scrambled sets for continuous maps on the interval. Recently in [14] the specification property was used to study omega chaos and [19] relates limit shadowing with \(\omega \)-chaos. In [8] the notion of \(\omega \)-chaos was transferred from individual dynamics to collective one. For further reading on the topic see e.g. [9] and [22]. An important notion tightly bonded with \(\omega \)-chaos is minimality. It is worthy to note at this point, that by the definition, \(\omega \)-chaos is not present in minimal dynamical systems. So intuitively it seems that \(\omega \)-chaos may not appear if the structure of minimal systems is not rich. This is somehow misleading.

In [6] authors construct space which is \(\omega \)-chaotic but the only minimal sets are periodic orbits. Furthermore, \(\omega \)-chaos can appear in dynamical system with one minimal subset, even in the case when the minimal system is rigid and dynamics is almost equicontinuous (see Sect. 5). To avoid the above situation and induce a more complicated structure of limit sets we will modify the definition of \(\omega \)-chaos. Let us first recall mathematically strict definition of \(\omega \)-chaos introduced first by Li in [15].

### **Definition 1.1**

*-scrambled set*for dynamical system (

*X*,

*f*) if, for any two \(x\ne y\) in \(\Omega \),

- (1)
\(\omega _{f}(x){\setminus }\omega _{f}(y)\) is uncountable,

- (2)
\(\omega _{f}(x) \cap \omega _{f}(y) \ne \emptyset \) and

- (3)
\(\omega _{f}(x) {\setminus } Per(f) \ne \emptyset \).

*f*is \(\omega \)-

*chaotic*if there is an uncountable \(\omega \)-scrambled set. By an \(\omega \)-

*chaotic*pair we mean any \(\omega \)-scrambled set of cardinality 2.

It was pointed out in [15] that the third condition is superfluous when \(X=I=[0,1]\). In [13] it was shown, however, that even in dimension one there are systems where the third condition is essential. Recent progress in theory of dynamical systems revealed big advantages of studies of dynamics of tuples, especially when studying topological entropy (e.g. see [20] for a survey on recent results). Following this approach, we will require that tuples in scrambled set satisfy a stronger property, which is a natural generalization of chaotic pairs (in particular, 2-tuple is chaotic pair in the standard sense).

### **Definition 1.2**

*scrambled*

*n*-

*tuple*if the following three conditions are satisfied:

- (1)
\(\left( \omega _{f}(x_{\pi (1)})\cap \ldots \cap \omega _{f}(x_{\pi (n-1)})\right) {\setminus } \omega _{f}(x_{\pi (n)})\) is uncountable for any permutation \(\pi \) of the set \(\left\{ 1,\ldots ,n\right\} \),

- (2)
\(\bigcap _{i=1}^n \omega _{f}(x_i) \ne \emptyset \) and

- (3)
\(\omega _{f}(x_i) {\setminus } Per(f) \ne \emptyset \) for \(i=1,\ldots ,n\).

*-scrambled set*if for any \(n\ge 2\), any choice of distinct points \(x_1,\ldots ,x_n\in Q\) forms and \(\omega \)-scrambled

*n*-tuple \((x_1,\ldots ,x_n).\) If an uncountable \(\hat{\omega }\)-scrambled set exists, the map

*f*is called \(\hat{\omega }\)

*-chaotic*.

### *Remark 1.3*

- (i)
\(\left( \omega _{f}(x_{\pi (1)})\cap \ldots \cap \omega _{f}(x_{\pi (n-1)})\right) {\setminus } \omega _{f}(x_{\pi (n)})\) contains an uncountable minimal set for any permutation \(\pi \) of the set \(\left\{ 1,\ldots ,n\right\} \),

- (ii)
\(\bigcap _{i=1}^n \omega _{f}(x_i)\) contains an uncountable minimal set.

Observe that for \(n=2\) Definition 1.2 immediately gives Definition 1.1. On the other hand it is possible to construct an example of \(\omega \)-chaotic map with \(\omega \)-chaotic pairs without \(\omega \)-chaotic triples; the construction is just simplification of the proof of Proposition 1 from [23] while picking the initial set *A* as two point set of sequences containing infinitely many zeros and ones.

In the article (in Sect. 3) we will show that every interval map with positive topological entropy is \(\hat{\omega }\)-chaotic, obtaining yet another condition equivalent to positive topological entropy. We strongly believe that this definition (after update proposed in Remark 1.3) should lead to dynamics with complicated structure of limit sets (both theoretically and geometrically, which should be visible in computer simulations).

## 2 Definitions and Notations

*dynamical system*we mean a pair (

*X*,

*f*), where

*X*is a compact metric space with a metric

*d*, and

*f*is a continuous map from

*X*to itself. The

*orbit*of \(x \in X\) is the set \(\text{ O }^+(x):=\{f^{k}(x): k\ge 0\}\), where as usual \(f^k\) stands for the

*k*fold composition of

*f*. For \(x \in X\) the \(\omega \)-

*limit set*is the set

*invariant*under

*f*if \(f(A) \subset A\) and

*minimal*if it is nonempty, closed and invariant under

*f*and it does not contain any proper subset which satisfy these three conditions. When

*X*is a minimal set, then we say that the dynamical system (

*X*,

*f*) is

*minimal*. It is known that (

*X*,

*f*) is minimal iff each \(x\in X\) has dense orbit or equivalently \(\omega _f (x) =X\) for any \(x \in X\). If \(x\in \omega _f(x)\) then

*x*is

*recurrent*.

We say that *f* is *(topologically) transitive* if for any two nonempty open sets \(U,V \subset X\) there exist \(n>0\) such that \(f^{n}(U)\cap V \ne \emptyset \) and is *(topologically) weakly mixing* if for any three nonempty open sets \(U,V,W \subset X\) there exist \(n>0\) such that \(f^{n}(W)\cap U \ne \emptyset \) and \(f^{n}(W)\cap V \ne \emptyset \) (see [2] for a list of equivalent conditions). If for every two nonempty sets \(U,V \subset X\) there exist \(N>0\) such that \(f^{n}(U)\cap V \ne \emptyset \) for all \(n>N\) then we say that *f* is *(topologically) mixing*.

Now let us present some standard notation related to symbolic dynamics. Let \(\mathcal {A}\) be any finite set (an alphabet) and let \(\mathcal {A}^*\) denote the set of all finite words over \(\mathcal {A}\). In what follows, for simplicity of notation, we assume that symbol \(0\in \mathcal {A}\). For any word \(w \in \mathcal {A}^*\) we denote by |*w*| the length of *w*, that is the number of letters which form this word. If *w* is empty word then we put \(|w|=0\). An *infinite word* is a mapping \(w: \mathbb {N}\rightarrow \mathcal {A}\), hence we may view it as an infinite sequence \(w_1 w_2, \ldots \) where \(w_i \in \mathcal {A}\) for any \(i \in \mathbb {N}\). The set of all infinite words over alphabet \(\mathcal {A}\) is denoted by \({\mathcal {A}}^{\mathbb {N}}\). By \(0^{\infty }\) we will denote the infinite word \(0^{\infty }=000 \ldots .\) If \(x \in \mathcal {A}^\mathbb {N}\) and \(i \le j\) are integers then we denote \(x_{[i,j)}=x_{i}x_{i+1} \ldots x_{j-1}\) and by \(\mathcal {L}(X)\) we denote *the language* of subshift *X*, that is, the set \(\mathcal {L}(X):=\{x_{[0,k)}: x \in X, k\ge 0 \}\). If a word \(u\in \mathcal {A}^*\) appears in \(z\in {\mathcal {A}}^{\mathbb {N}}\) (the same for \(z\in \mathcal {A}^*\)), then we denote it by \(u \sqsubset z\) and say that *u* is a *subword* of *z*. If \(u_k\) is a sequence of words such that \(|u_k|\longrightarrow \infty \) then we write \(z=\lim _{k\rightarrow \infty } u_k\) if the limit \(z=\lim _{k\rightarrow \infty } u_k 0^\infty \) exists in \(\mathcal {A}^\mathbb {N}\).

*k*is the length of maximal common prefix of

*x*and

*y*, that is \(k=\max \{ i: x_{[0,i)}=y_{[0,i)}\}.\) By \(\Sigma ^{+}_{n}\) we denote the dynamical system \( (\{0, \ldots , n-1\}^{\mathbb {N}} , \sigma )\), where \(\sigma \) is a shift map defined by

*cylinder set*) and by \(C_{A}[w]=C[w]\cap A\) we denote

*trace of cylinder set*

*C*[

*w*] in a set \(A\subset \Sigma _n^+\). The collection of all cylinder sets form a basis of the topology of \(\Sigma ^{+}_{n}\).

Observe that \(x \in \Sigma _n^+\) is a minimal point for \(\sigma \) if for any open and nonempty set *U* there exist a positive number *m* such that for any \(i>0\) there exist \(j \in [i, i+m]\) such that \(\sigma ^{j}(x) \in U\). By *N*(*U*, *V*) we denote set of all positive numbers *i* such that \( \sigma ^{i}(U) \cap V \ne \emptyset .\)

A Hausdorff space *X* is *perfect* if it has no isolated points, and a *Cantor space* if it is non-empty, compact, totally disconnected, perfect metrizable space. We say that a subset in a Hausdroff space is a *Cantor set* if it is a Cantor space with respect to relative topology, and a *Mycielski set* if it can be presented as a countable union of Cantor sets.

We present here a simplified version of Mycielski’s theorem ([17], Theorem 1) that will play a crucial role in the proof of Theorems 4.1 and 4.3.

### **Theorem 2.1**

(Mycielski) Let *X* be perfect complete metric space and let \(R_n \subset X^n\) be residual subsets in \(X^n\) for each \(n\in \mathbb {N}\). Then there is a dense Mycielski set \(S \subset X\) with the property that \((x_1,x_2, \ldots , x_n) \in R_n\) for any \(n \in \mathbb {N}\) and any pairwise distinct points \(x_1,x_2, \ldots ,x_n \in S\).

### *Remark 2.2*

Clearly, we may apply Mycieski theorem when \(R_n\) are defined only for \(n\in F\) where \(F\subset \mathbb {N}\). Simply, for \(n\not \in F\) we can put \(R_n=X^n\).

## 3 Construction of \(\hat{\omega }\)-Chaotic Set in Full Shift

In what follows, we will need the following simple fact (e.g. see [18, Lemma 3.8]):

### **Lemma 3.1**

*X*,

*f*) the following conditions are equivalent:

- (1)
(

*X*,*f*) is topologically weakly mixing, - (2)
there is a point \(x\in X\) with dense orbit such that for any non-empty and open neighbourhood

*U*of*x*there is an integer \(n>0\) such that \(n,n+1\in N(U, U)\).

### **Lemma 3.2**

- (i)
\(X_k\) is minimal and infinite (thus uncountable),

- (ii)
\(X_k\) is weakly mixing,

- (iii)
\(X_m\cap X_n =\emptyset \) provided that \(m\ne n\).

### *Proof*

*j*, both \(a^{(k)}_{r+j}\) and \(b^{(k)}_{r+j}\) can be presented as concatenations of \(a^{(k)}_r\) and \(b^{(k)}_r\). We prove this fact by induction on

*j*. Indeed, for \(j=0\) the claim follows by the definition. Now assume, that the claim holds for some \(j \ge 0\). By the definition

To prove (i) that \(X_k\) is minimal we show that any point *x* from \(X_k\) is a minimal point. To do so, fix any \(z^{(k)}\), its open neighbourhood *U* and let \(p>0\) be an integer such that \(C[u]\subset U\), where \(u=z_{[0,p)}^{(k)}\). There exist \(l>0\) such that \(|a^{(k)}_l| \ge |u|\). Observe that from the above claim we get that, \(z^{(k)}=u_1u_2u_3 \ldots \), where \(u_i \in \{a^{(k)}_{l-1}, b^{(k)}_{l-1} \}\) for every \(i=1,2, \ldots \). Denote \(N=6|b^{(k)}_{l}| \ge 2|b^{(k)}_{l+1}|\). Observe that for any *i*, word \(z^{(k)}_{[i,i+N)}\) contains \(b^{(k)}_{l+1}\) or \(a^{(k)}_{l+1}\) as a subword and both of them contain \(a^{(k)}_{l}\) as a subword. This shows that \(z_{[i, i+N)}^{(k)}\) has *u* as a subword, and hence there is \(j \in [i,i+N)\) such that \(\sigma ^{j}(z^{(k)})\in C[u] \subset U\).

We will prove (ii) by reducing to Lemma 3.1. Let us fix any non-empty open neighbourhood \(z^{(k)}\in U \subset X_k\). There exist \(r>0\) such that \(C[a^{(k)}_r] \subset U\).

*r*we have \(|b_{r}^{(k)}|=|a_{r}^{(k)}|+1\). It is obviously true for \(r=0\). Now assume, that the following equalities \(|b_{r}^{(k)}|=|a_{r}^{(k)}|+1\) is true for some \(r>0\). By the definition

*U*was arbitrarily chosen, \(X_k\) is weakly mixing by Lemma 3.1.

Now we continue with the construction of an \(\hat{\omega }\)-chaotic map. Let \(x\in \Sigma _2^+\) and let \(P=\left\{ p_i\right\} _{i=1}^\infty =\left\{ 2,3,5,\ldots \right\} \) be an increasing sequence consisting of all prime numbers.

### **Lemma 3.3**

- (i)
\(z=0^\infty \),

- (ii)
there is

*j*such that \(x_j=1\) and \(z=0^m y\) for some \(m\ge 0\) and \(y\in X_j\), - (iii)
there is

*j*such that \(x_j=1\) and \(z=0^s w 0^\infty \) for some \(w\in \mathcal {L}(X_j)\) and some \(s \ge 0\),

### *Proof*

- (a)
First case is when \(u_k=0^k\), for every

*k*. Then we immediately get \(z=\lim _{k\rightarrow \infty } u_k=0^{\infty }\) which is condition (i). - (b)Another possibility is when \(u_k=0^{m}\phi _x (n)_{[0,p)}\) for some \(m\ge 0\), \(n>0\) and \(p\le |\phi _x (n)|\). Thenwhich proves the setting of (ii), that is \(z=0^{m}y\) for some \(y \in X_j\).$$\begin{aligned} \lim _{n \rightarrow \infty } \phi _x(n)_{[0,p)} = \lim _{n \rightarrow \infty } {a^{(j)}_{n}}_{[0,p)}={z^{(j)}}_{[0,p)} \in \mathcal {L}(X_j) \end{aligned}$$
- (c)
Similarly we show that it may happen that \(z=w0^{\infty }\) for some \(w \in \mathcal {L}(X_j)\) which appears when \(u_k={\phi _x (n)}_{[|\phi _x (n)|-p, |\phi _x (n)|)}0^{k-p}\) for every

*k*, where*n*depends on*k*and*p*is fixed and has the same value for all*k*. Note that \({\phi _x (n)}_{[|\phi _x (n)|-p, |\phi _x (n)|)}\) is finite hence is the same for infinitely many*n*(and as a result infinitely many*k*). In other words, since \(z=\lim _{k\rightarrow \infty } u_k\) exists, we obtain that \(z={\phi _x (n)}_{[|\phi _x (n)|-p, |\phi _x (n)|)}0^\infty \) for some*n*. This is the case (iii) with \(s=0\), provided that \(x_j=1\), \({\phi _x (n)}=a_{\psi _2(n)}^{(j)}\) and its suffix is*w*. - (d)The remaining situation that we have in (iii) is \(z=0^s w0^{\infty }\) for some \(s>0\). Note that if \(x_j\) is 1 and
*n*has decomposition into primes \(n=p_j^r....\) where, \(p_j\) is smallest prime in decomposition, we may obtain \(u_k\) of the formwhere$$\begin{aligned} u_k=0^{t}\phi _x(n) 0^{n-1} w_k=0^{t}a^{(j)}_r 0^{n-1} w_k \end{aligned}$$*t*is the same for all*k*and words \(w_k\) are possibly changing with*k*. But*n*tends to infinity, so without loss of generality we may assume that \(w_k\) is an empty word. If*r*increases with*n*then \(|a^{(j)}_r| \rightarrow \infty \), which reduces to the case (ii). The other situation is that*r*stabilizes, but then \(z=0^t a^{(j)}_r 0^\infty \) which is the case (iii).

### **Lemma 3.4**

### *Proof*

*j*there is \(\varepsilon >0\) such that if \(d_n((x_1,\ldots ,x_n), (y_1,\ldots , y_n))<\varepsilon \) then \(x_k(j)=y_k(j)\) for \(k=1,2,\ldots , n\). Note that \(R_n\) is also dense, because if we fix any nonempty words \(w_1,\ldots , w_n\) and put \(v_i=w_i w_{i+1}\ldots w_n w_1\ldots w_{i-1}\), then \(|v_1|=|v_2|=\cdots =|v_n|\) and clearly

*Q*is a Cantor set, hence there is also \(x_{n+1}\in Q\) such that \(x_1,\ldots , x_{n+1}\) are pairwise distinct. Then there is

*i*such that \(1=x_1(i)=x_2(i)=\cdots = x_{n}(i) \ne x_{n+1}(i)\). The proof is completed. \(\square \)

### **Lemma 3.5**

*Q*is provided by Lemma 3.4). If \(x_1,\ldots ,x_n\in Q\) then there are \(k,s\ge 1\) such that

### *Proof*

*k*, since then \(d(z_{x_1},z_{x_2})<\varepsilon \). Note that

Let *Q* be the Cantor set provided by Lemma 3.4. Since \(\zeta \) is continuous, \(\zeta (Q)\) is compact. But \(\zeta \) is also injective, hence *Q* does not have isolated points. This shows that \(\zeta (Q)\) is a perfect subset in a totally disconnected space, hence is a Cantor set.

Now, fix any \(x_1,\ldots ,x_n\in Q\). There is *k* such that \(x_1(k)=x_2(k)=\cdots = x_n(k)=1\). Then by the definition of \(z_{x_i}\) we obtain that \(X_k \subset \omega _{\sigma }(z_{x_i})\) for \(i=1,\ldots ,n\). Similarly, there is *s* such that \(x_1(s)=x_2(s)=\cdots = x_{n-1}(s)=1\) and \(x_n(s)=0\). Then \(X_s \subset \omega _{\sigma }(z_{x_i})\) for \(i=1,\ldots ,n-1\) and by Lemma 3.3 we obtain that \(X_s \cap \omega _{\sigma }(z_{x_n})=\emptyset \). Indeed, conditions (3.3) and (3.4) are satisfied and the proof is finished. \(\square \)

By Lemma 3.5 (see also Lemma 3.2) we directly obtain the following. Simply, it is enough to put \(D=\zeta (Q)\).

### **Theorem 3.6**

- (1)
\(\left( \omega _{\sigma }(z_1)\cap \ldots \cap \omega _{\sigma }(z_{n-1})\right) {\setminus } \omega _{\sigma }(z_n)\) contains an uncountable weakly mixing minimal set,

- (2)
\(\bigcap _{i=1}^n \omega _{\sigma }(z_i)\) contains an uncountable weakly mixing minimal set (hence is nonempty).

*D*is an \(\hat{\omega }\)-scrambled set for \(\sigma \).

## 4 Dimension One and Beyond

### **Theorem 4.1**

Let \(f:[0,1]\rightarrow [0,1]\) be transitive. Then there exists a dense Mycielski \(\hat{\omega }\)-scrambled set \(M\subset [0,1]\).

### *Proof*

Let \(\{V_j: j \in \mathbb {N}\}\) be countable base for the topology of [0, 1] consisting of nonempty open sets of [0, 1], and for each \(j \in \mathbb {N}\), let \(U_j \subset (0,1)\) be a nonempty open interval with \(\overline{U_j} \subset V_j\). Let *D* be a Cantor set provided by Theorem 3.6 and let \(C_1, C_2, \ldots \subset D\) be pairwise disjoint Cantor sets.

Since *f* is a transitive interval map, there exists \(p\in (0,1)\) such that for every \(\varepsilon >0\) and every *j* there is *n* such that \((\varepsilon ,p-\varepsilon )\subset f^n(U_j)\) and \((p+\varepsilon ,1-\varepsilon )\subset f^{n+1}(U_j)\). In particular, there is *n* and a Cantor set \(C_j'\) such that \(f^n(U_j)\cap C_j=C_j'\). By [24, Remark4.3.6], there exists a Cantor set \(D_j\subset U_j\) such that \(f^n|_{D_j}\) is one-to-one and \(f^n(D_j)\subset C_j'\subset C_j\).

Putting \(M=\bigcup _{j=1}^{\infty } D_j\) we obtain a dense Mycielski set. But for every *n* we have \(\omega _f(x)\,{=}\,\omega _f(f^n(x))\), and sets \(C_j\) were pairwise disjoint, hence *M* is \(\hat{\omega }\)-scrambled. \(\quad \square \)

Before proceeding, we will need the following result by Oxtoby and Ulam (see [21, Thm. 9]).

### **Theorem 4.2**

(Oxtoby and Ulam) Let \(B\subset [0,1]^k\) and suppose that there exists a sequence \(\left\{ S_n\right\} _{n=1}^\infty \) of perfect sets \(S_n\subset B\) such that \(\bigcup _{n\in \mathbb {N}} S_n\) is dense in \(I^k\). Then there exists a homeomorphism \(\eta :[0,1]^k \rightarrow [0,1]^k\) such that \(\eta |_{\partial [0,1]^k}=id\) and \(\mathscr {L}(\eta (B))=1\), where \(\mathscr {L}\) denotes the Lebesgue measure on \([0,1]^k\).

### **Theorem 4.3**

For every \(n\ge 2\) there exists a continuous \(\hat{\omega }\)-chaotic map \(F:[0,1]^n\rightarrow [0,1]^n\) and a dense Mycielski \(\hat{\omega }\)-scrambled set \(M\subset [0,1]^n\) for *F* such that its Lebesgue measure \(\mathscr {L}(M)=1\).

### *Proof*

Take a mixing map \(f:[0,1]\rightarrow [0,1]\) (e.g. the standard tent map) and denote \(G=f\times \cdots \times f:[0,1]^n\rightarrow [0,1]^n\). Since there is a fixed point \(p\in (0,1)\) for *f*, the set \(J=[0,1]\times \left\{ p\right\} ^{n-1}\) is invariant under *G* and we may identify \(G|_J\) with a map acting on [0, 1]. Let \(D\subset J\) be a Cantor \(\hat{\omega }\)-scrambled set for \(G|_J\) (hence also for *G*) provided by Theorem 3.6. Taking a Cantor subset of *D* if necessary, we may assume that \(D\subset (0,1)^n\). Observe that since *f* is mixing, for every open set \(U\subset [0,1]^n\) and every \(\varepsilon >0\) there is \(k>0\) such that \((\varepsilon ,1-\varepsilon )^n\subset G^k(U)\), so there is \(k>0\) such that \(D\subset G^k(U)\). Repeating all the arguments from the proof of Theorem 4.1 we obtain a dense Mycielski \(\hat{\omega }\)-scrambled set *B* for *G*. By Oxtoby-Ulam theorem we obtain a homeomorphism \(\eta :[0,1]^k \rightarrow [0,1]^k\) such that \(\eta |_{\partial [0,1]^k}=id\) and \(\mathscr {L}(\eta (B))=1\). Since \(\hat{\omega }\)-scrambled sets are preserved by topological conjugacy, the proof is completed by putting \(F=\eta \circ G \circ \eta ^{-1}\) \(\square \)

Clearly \(T=f\times f\) is a special type of triangular map, hence we have also the following.

### **Corollary 4.4**

There exists a transitive triangular map \(T:[0,1]^2\rightarrow [0,1]^2\) with a dense Mycielski \(\hat{\omega }\)-scrambled set \(M\subset [0,1]^2\).

A computer simulation, showing how \(\omega \)-limits sets of a triple in a map \(T=f\times f\) when *f* is the standard tent map can look like is presented on Fig. 1. Points \(x_1,x_2,x_3\in [0,1]^2\) were chosen in such a way that \((0,0)\in \bigcap _{i=1}^3\omega _T(x_i)\).

## 5 Almost Equicontinuous Example

The aim of this section is to construct a compact set *X* and an \(\hat{\omega }\)-chaotic homeomorphism *F* on it which is almost equicontinuous (however is not transitive). We will construct the space \(\mathcal {X}\) (and the map *F*) in a few steps. Let us first recall the definition of equicontinuity and almost equicontinuity (see [10])

### **Definition 5.1**

*X*,

*f*) be a dynamical system.

- (1)The set \(\mathcal {E} \subseteq X\) of
*equicontinuous points*of (*X*,*f*) is defined by$$\begin{aligned} x \in \mathcal {E} \Leftrightarrow \forall _{\varepsilon >0} \ \exists _{\delta >0} \ \forall _{ y \in B_{\delta }(x)} \ \forall _{ n \ge 0} \ d\left( f^n(x), f^n(y)\right) < \varepsilon . \end{aligned}$$ - (2)
We say that (

*X*,*f*) is*equicontinious*if \(\mathcal {E}=X.\) - (3)
We say that (

*X*,*f*) is*almost equicontinious*if \(\mathcal {E}\) is a residual set.

*g*(Denjoy map) defined on the extended circle:

- (1)
each interval \(I^n_j\) is mapped homeomorphicaly onto \(I_{j+1}^n\),

- (2)
if all intervals \(I^n_j\) are collapsed back into single points \(x^{(n)}_j\) then

*g*reverts back to the map*R*.

*C*is an open set).

*j*and for every

*m*there is \(N_m\) such that

*z*in \(\mathbb {S}^1\) which does not belong to the orbit of any \(x_0^{(n)}\). Since orbit of

*z*under

*R*is dense, we can find sequence of integers \(a^{(n,m)}\) increasing with respect to

*m*such that:

- (i)
\(\left[ a^{(n,m)},a^{(n,m)}+3m\right] \cap \left[ a^{(i,j)},a^{(i,j)}+3j\right] =\emptyset \) provided that \((n,m)\ne (i,j)\)

- (ii)
\(\left| R^{a^{(n,m)}}(z)-x^{(n)}_{-m}\right| <4^{-m}\).

- (iii)
\(\left| R^{a^{(n,m)}}(z)-x^{(n)}_{-m}\right| < \left| R^{a^{(n,m)}}(z)-x^{(i)}_{j}\right| \) for \(i\le N_m\) ,\(|j|\le N_m+2m\), \((i,j)\ne (n,-m)\).

*i*,

*j*.

Denote by \(\hat{z}\in C\) the unique point such that \(\pi (\hat{z})=z\) and for any *n* denote by \(b^n_m\) one of the endpoints of \(I^n_m\) in such a way that \(g(b^n_m)=b^n_{m+1}\) for every \(m\in \mathbb {Z}\). Simply it is enough to fix \(b^n_0\) as one of the endpoints of \(I^n_0\) and then put \(b_m^n=g^m(b_0^n)\) for every \(m\in \mathbb {Z}\). Since we have freedom when choosing \(a^{(n,m)}\), in particular we can choose “the side” of \(x_m^n\) approached by orbit of *z*, we may assume that \(b_m^n\) is closer to \(\hat{z}\) than the other endpoint of \(I^n_m\).

Define a sequence \(\xi _j\) by replacing segments of the orbit of *z* for iterations \(j=a^{(n,m)},\ldots ,a^{(n,m)}+2m\), putting \(\xi _j=g^{j-a^{(n,m)}}(b^n_m)\) for all these *j* and keeping \(\xi _j=g^j(z)\) otherwise. Furthermore, we consider only pairs (*n*, *m*) with \(m\ge n\). For \(j=a^{(n,m)},\ldots ,a^{(n,m)}+2m\) the distance between \(\xi _j\) and \(g^j(z)\) is bounded from the above by \(\left| R^{a^{(n,m)}}(z)-x^{(n)}_{-m}\right| \) increased by the diameters of intervals inserted into points between \(R^{j}(z)\) and \(R^{j-a^{(n,m)}}(x^{(n)}_{-m})\). But by condition (iii) and (5.1) these diameters are bounded from the above, hence \(|\xi _j-g^j(z)|<4^{-m+1}\) for \(j=a^{(n,m)},\ldots ,a^{(n,m)}+2m\). This shows that the constructed sequence is asymptotic to the orbit of *z*, that is \(\lim _{j\rightarrow \infty } |\xi _j-g^j(z)|=0\).

*n*such that

*Y*is compact (and as a result \(\mathcal {X}\) is compact too, since \(\sim \) is a closed equivalence relation).

*G*is a continuous map.

### **Lemma 5.2**

Dynamical system \((\mathcal {X},G)\) is almost equicontinuous.

### *Proof*

*n*be such that \(4^{-n}<\varepsilon \). Let \(\gamma >0\) be such that \((t_i-\gamma ,t_i+\gamma )\cap (t_{i-1},t_{i+1})=\left\{ t_i\right\} \) and \(\gamma <2^{-n} t_i\), \(\gamma <\varepsilon \). Then \(d(\alpha ,\beta )<\gamma \) implies \(\alpha _{[0,n]}=\beta _{[0,n]}\). Let

*G*is equicontinuous at each point \([\alpha ,x,t]_\sim \) with \(|t|<1\). In particular the set of equicontinuity points is an open dense set, completing the proof. \(\square \)

### **Lemma 5.3**

Dynamical system \((\mathcal {X},G)\) is \(\hat{\omega }\)-chaotic.

### *Proof*

*S*satisfies conditions (2) and (3) in Definition 1.2. On the other hand, by the definition of

*Q*, for any choice of \(\alpha ^1,\ldots ,\alpha ^n\in Q\) we can always find

*i*such that \(\alpha ^1_i=\alpha ^2_i=\cdots =\alpha ^{n-1}_i=1\) and \(\alpha ^{n}_i=0\). But then the set of accumulation points of the sequence \(\left\{ p^{\alpha ^j}_i\right\} _{i=1}^\infty \) for \(j<n\) contains the set \(I^n_0\) while none of the points from interior of \(I^n_0\) can be obtained as a limit of a subsequence of \(\left\{ p^{\alpha ^n}_i\right\} _{i=1}^\infty \). This shows that

It is enough to combine Lemmas 5.2 and 5.3 to obtain the following.

### **Theorem 5.4**

There exists an almost equicontinuous \(\hat{\omega }\)-chaotic dynamical system.

### *Remark 5.5*

Note that the set *S* in the proof of Lemma 5.3 is open (because \(t=0\) is isolated in the sequence \(\left\{ t_n\right\} _{n\in \mathbb {Z}}\)), but \(G^n(S)\cap S=\emptyset \) for every \(n>0\). In particular \((\mathcal {X},G)\) is not transitive.

### *Remark 5.6*

We can identify antipodal points in \(\mathcal {X}\) by putting relation \([(\alpha ,x,t)]_\sim \approx [(\beta ,x,s)]_\sim \) when \(|s|=|t|=1\), obtaining this way a system with exactly one minimal subset.

- (ii’)
\(\bigcap _{i=1}^n \omega _{f}(x_i)\) contains an uncountable family of pairwise disjoint uncountable minimal set.

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