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New chaotic planar attractors from smooth zero entropy interval maps
 Jan P Boroński^{1, 2}Email author and
 Jiří Kupka^{1}
https://doi.org/10.1186/s1366201505659
© Boroński and Kupka 2015
 Received: 12 March 2015
 Accepted: 6 July 2015
 Published: 26 July 2015
Abstract
We show that for every positive integer k there exists an interval map \(f:I\to I\) such that (1) f is LiYorke chaotic, (2) the inverse limit space \(I_{f}=\lim_{\leftarrow}\{f,I\}\) does not contain an indecomposable subcontinuum, (3) f is \(C^{k}\)smooth, and (4) f is not \(C^{k+1}\)smooth. We also show that there exists a \(C^{\infty}\)smooth f that satisfies (1) and (2). This answers a recent question of Oprocha and the first author from (Proc. Am. Math. Soc. 143(8):36593670, 2015), where the result was proved for \(k=0\). Our study builds on the work of Misiurewicz and Smítal of a family of zero entropy weakly unimodal maps. With the help of a result of Bennett, as well as Blokh’s spectral decomposition theorem, we are also able to show that each \(I_{f}\) contains, for every integer i, a subcontinuum \(C_{i}\) with the following two properties: (i) \(C_{i}\) is \(2^{i}\)periodic under the shift homeomorphism, and (ii) \(C_{i}\) is a compactification of a topological ray. Finally, we prove that the chaotic attractors we construct are topologically distinct from the one presented by P Oprocha and the first author.
Keywords
 weakly unimodal map
 arclike
 LiYorke chaotic
 indecomposable continuum
 zero topological entropy
MSC
 54H20
 37B45
 37E05
1 Introduction
The celebrated CartwrightLittlewoodBell fixed point theorem [1, 2] asserts that any homeomorphism of \(\mathbb{R}^{2}\) must fix a point in an invariant plane nonseparating compact and connected set (continuum). The original work of Cartwright and Littlewood, motivated by problems in differential equations, in which they were led to consider invariant sets whose frontiers were indecomposable, brought more focus in the mathematical literature to the interplay of topology and dynamics in continua, of which the study of planar attractors generated by inverse limits of arcs became an important theme. The well known connection established by Barge and Martin in the 1980s states that chaotic (in the sense of positive entropy) interval maps generate planar dynamical systems with attractors that must contain an indecomposable subcontinuum (see e.g. [3, 4]). The connection is in fact a characterization for all piecewise monotone graph maps [5], but there had also been an aspect of it left over: must weak chaos (i.e. chaos in the sense of Li and Yorke) imply indecomposability in the inverse limit space? Recently, however, Oprocha and the first author showed in [6] that there exists a LiYorke chaotic interval map F such that the inverse limit space \(I_{F}=\lim_{\leftarrow}\{F,I\}\) does not contain an indecomposable subcontinuum. Since the map F constructed in [6] was not continuously differentiable, the following question was posed.
Problem 1
[6]
Is there a \(k>0\) and a LiYorke chaotic interval map φ such that φ is \(C^{k}\)smooth and the inverse limit space \(I_{\varphi}=\lim_{\leftarrow}\{\varphi,I\}\) does not contain an indecomposable subcontinuum? Must \(I_{\varphi}\) have a periodic structure similar to the continua described in [6]?
In the present paper, by employing an entirely different approach, we answer the question in the affirmative. First we prove the following result.
Theorem 1
 (1)
f is LiYorke chaotic,
 (2)
\(I_{f}=\lim_{\leftarrow}\{f,I\}\) does not contain an indecomposable subcontinuum,
 (3)
f is \(C^{k}\)smooth,
 (4)
f is not \(C^{k+1}\)smooth.
Theorem 2
 (1)
f is LiYorke chaotic,
 (2)
\(I_{f}=\lim_{\leftarrow}\{f,I\}\) does not contain an indecomposable subcontinuum.
It is noteworthy to mention that the LiYorke chaotic map F without an indecomposable subcontinuum in \(\lim_{\leftarrow}\{F,I\}\) constructed in [6] was not piecewise monotone, having countably many intervals of monotonicity. It was shown therein that, by an arbitrarily small perturbation, the map could be modified without increasing entropy to a map \(F_{\epsilon}\), as to generate various indecomposable subcontinua in \(\lim_{\leftarrow}\{F_{\epsilon},I\}\), including Knaster’s pseudoarc. An appropriate perturbation \(F_{S}\) could also result in the attractor \(\lim_{\leftarrow}\{F_{S},I\}\) being nonSuslinean. Our approach is different, as it is based on a careful selection of weakly unimodal maps. Although, in general, such maps may have positive entropy, and therefore may generate indecomposable subcontinua of their inverse limit spaces, there is a class of zero entropy and LiYorke chaotic weakly unimodal maps. The class was studied by Misiurewicz and Smítal [7] who, among other results, showed that the class contains a nonempty subclass of \(C^{\infty}\)smooth maps (see Section 2). In Section 3, we shall observe that the class also contains maps with other degrees of differentiability. In addition, since weakly unimodal maps are piecewise monotone, we shall be able to use the result of Barge and Diamond [5] that relates entropy of piecewise monotone graph maps to topological structure of the related inverse limit spaces (see next section for more details). In Section 4 we further study the topological structure of our attractors. We show that they have a similar periodic structure to the attractors in [6].
Theorem 3
 (i)
\(C_{i}\) is \(2^{i}\)periodic under the shift homeomorphism, and
 (ii)
\(C_{i}\) is a compactification of a topological ray.
2 Preliminaries

\(h(I_{f})=I_{f}\),

\(h_{I_{f}}=\sigma_{f}\) (up to conjugacy), and

there is an open set N containing \(I_{f}\) such that \(\bigcap_{n=1}^{\infty}h^{n}(N)=I_{f}\).
The following result of Barge and Diamond will be important to our results. Note that interval maps are special cases of graph maps.
Lemma 4
(Barge and Diamond [5])
Suppose \(f:G\to G\) is a piecewise monotone graph map. f has zero topological entropy if and only if \(\lim_{\leftarrow}\{f,G\}\) does not contain an indecomposable subcontinuum.
 (1)the setconsists of more than one point,$$J_{f}:= \bigl\{ x\in I \mid f(y) \leq f(x) \text{ for every }y\in I \bigr\} $$
 (2)
for each \(n\in\mathbb{N}\), f has a periodic point of period \(2^{n}\),
 (3)
f has no periodic points of other periods.
The family \(\mathcal{F}\) is nonempty and any map that satisfies the properties (2) and (3) is said to be of type \(2^{\infty}\). In [7] the following two results are proved (see also [14] for an alternative proof of the first result, and the comments on p.674 in [15] concerning [7]).
Lemma 5
[7]
Any map \(f\in\mathcal{F}\) has zero topological entropy and is chaotic in the sense of Li and Yorke.
Lemma 6
[7]
Let \(\mathcal{F}_{0}\subseteq\mathcal{F}\) be a family of \(C^{\infty}\) interval maps f satisfying \(f(0) = f(1) = 0\). Then \(\mathcal {F}_{0}\neq\emptyset\).
3 Main results
With the help of the following result we are able to ensure the existence of the desired LiYorke chaotic map in each differentiability class that we claim in our main theorems. Our proof of (iv) of the next result closely follows that of Theorem 3 in [7].
Lemma 7
 (i)
\(f(0)=f(1)=0\),
 (ii)
f is \(C^{k}\)smooth,
 (iii)
f is not \(C^{k+1}\)smooth.
 (iv)
there exists \(\bar{c}\in(0,1]\) such that \(\bar{c}\cdot f\in \mathcal{F}\).
Proof
Let us prove (iv). It is easy to see that, for any \(c\in(0,1]\), \(\varphi_{c} := c\cdot f\) is a weakly unimodal map and that \(J_{\varphi _{c}}\) is a nondegenerate interval (by the choice of \(e_{1}\) and \(e_{2}\)). It remains to show that \(\varphi_{\bar{c}}\) is of type \(2^{\infty}\) for some \(\bar{c}\).
It is well known that \(h(\varphi_{1} )>0\) and the smaller c is, the smaller \(h(\varphi_{c})\) we can get. For a sufficiently small c, periodic points of \(\varphi_{c}\) are just fixed points and, hence, \(h(\varphi_{c})=0\). Consequently, \(M := \{c\in(0,1] \mid h(\varphi _{c})>0\}\) is nonempty and \(\bar{c} = \inf M >0\).
It is well known (see e.g. [16]) that \(\varphi _{c}\in M\) implies that \(\varphi_{c}\in N\) where \(N:= \{f\in I\to I \mid f \text{ satisfies (2) of the definition of }\mathcal{F}\}\). By Proposition 2.1 in [17], the set N is closed, since its elements are \(C^{1}\). Therefore, \(\varphi_{\bar{c}} \in N\). We also claim that \(\varphi_{\bar{c}}\) satisfies (3) of the definition of \(\mathcal{F}\). Otherwise, \(\varphi_{\bar{c}}\) would have a periodic point of period other than \(2^{n}\), and hence (again [16]) \(h(\varphi_{\bar{c}})>0\). But this would contradict the choice of \(\bar{c}\), because the set M is open (see e.g. [18]). This completes the proof. □
Theorem 1
 (1)
f is LiYorke chaotic,
 (2)
\(I_{f}=\lim_{\leftarrow}\{f,I\}\) does not contain an indecomposable subcontinuum,
 (3)
f is \(C^{k}\)smooth,
 (4)
f is not \(C^{k+1}\)smooth.
Proof of Theorem 1
Let \(k\in\mathbb{N}\) be fixed and let f be a map guaranteed by Lemma 7. Since \(f\in\mathcal{F}\) we see that f is also LiYorke chaotic and has zero entropy. Since f is piecewise monotone and has zero entropy, by Lemma 4, we also see that \(\lim_{\leftarrow}\{f,I\}\) does not contain an indecomposable subcontinuum. □
Theorem 2
 (1)
f is LiYorke chaotic,
 (2)
\(I_{f}=\lim_{\leftarrow}\{f,I\}\) does not contain an indecomposable subcontinuum.
Proof of Theorem 2
By Lemmas 5 and 6 there is a \(C^{\infty}\)smooth, LiYorke chaotic map f with zero entropy. Again, by Lemma 4, the inverse limit space \(\lim_{\leftarrow}\{f,I\}\) does not contain an indecomposable subcontinuum. □
At the end of this section we note that all the attractors obtained by Theorem 1 and Theorem 2 are Suslinean by main Theorem 1 in [20]. As a consequence each of them can be embedded into the plane in such a way that all its points are accessible from the complement, by the results in [21].
4 More on the topological structure of the attractors
Although the attractors constructed in the present paper have a similar periodic structure to the two attractors constructed in [6] (see Theorem 12), they are topologically distinct from the latter. We shall show that the attractors described in the present paper in Theorem 1 and Theorem 2 are compactifications of a ray, whereas the attractors in [6] are not. As in [6], we shall use the following result of Bennett.
Theorem 8
(Bennett) (the proof can be found in [8])
 (1)
\(g([d,b])\subset[d,b]\),
 (2)
\(g_{[a,d]}\) is monotone, and
 (3)
there is \(n\in\mathbb{N}\) such that \(g^{n} ([a,d]) = [a,b]\).
Lemma 9
 (1)
\(f([d,b])\subset[d,b]\),
 (2)
\(f_{[a,d]}\) is monotone, and
 (3)
there is an \(n\in\mathbb{N}\) such that \(f^{n} ([a,d]) = [a,b]\).
Proof
Let \(f\in\mathcal{F}\) be fixed. By (1) of the definition of \(\mathcal {F}\), there exists a nondegenerate interval \([e_{1},e_{2}]\) such that \(f(x)=m=\max_{y\in I} \{f(y)\}\) for each \(x\in[e_{1},e_{2}]\).
Because \(f_{[0,e_{1}]}\) is nondecreasing, the choice of a implies that there exists a strictly decreasing sequence \(\{a_{i}\}_{i=1}^{\infty}\subseteq[a,e_{1}]\) converging to a such that \(a_{1}=e_{1}\) and \(a_{i} = f(a_{i+1})\) for \(i\in\mathbb{N}\).
By using the notation of the proof above, one can see that either \(a=0\) or \(a>0\) and we have \(f([0,a]) = [0,a]\). Thus, one can change a to be equal to 0 and because \(f^{1}((m,0])=\emptyset\) the following corollary of Theorem 8 can be obtained. So we strengthen Lemma 9.
Corollary 10
For every \(f\in\mathcal{F}\) with \(f(0)=0\), there is a topological ray L such that \(\overline{L}=I_{f}\).
Note that the proof of Lemma 9 holds the following result originally proved in [6], Remark 10.
Corollary 11
There is a LiYorke chaotic zero entropy map \(\varphi:I\to I\) such that \(I_{\varphi}\) contains the pseudoarc.
Proof
Fix an \(f\in\mathcal{F}\) and let a, d, and m be as in the proof of Lemma 9. Let \(f_{H}:[0,a]\to[0,a]\) be the Henderson map [22], rescaled to \([0,a]\). Now set \(\varphi(x)=f_{H}(x)\) for \(x\in[0,a]\) and \(\varphi(x)=f(x)\) for \(x\in[a,1]\). □
Lemma 12
Suppose \(f:I\to I\) is a map of type \(2^{\infty}\). Then the shift homeomorphism \(\sigma_{f}\) has a \(2^{i}\)periodic subcontinuum of \(I_{f}\), for every integer \(i>0\).
Proof
Lemma 13
Let \(f\in\mathcal{F}\). Then there exists a system \(\{J_{i}\}_{i\geq0}\) of generating intervals such that, for any \(i\geq0\), \(\lim_{\leftarrow}\{f^{2^{i}}_{J_{i}},J_{i}\}\) is a compactification of a topological ray.
Proof
We shall show that there exists a system \(\{J_{i}\}_{i\geq0}\) of generating intervals such that, for any \(i\geq0\), each \(f^{2^{i}}_{J_{i}}\) satisfies assumptions of Theorem 8. The existence of a system of generating intervals is well known. Within this proof we sketch a construction allowing us to find a particular one that will be useful to us. Let \(L_{0}:= [u_{0},v_{0}]\) denote the nondegenerate interval for which \(f([u_{0},v_{0}]) = m :=\max\{f(x) \mid x\in I\}\) and \(a_{0}\) denote the only fixed point in \([v_{0},1]\). Its existence follows from the ordering of periodic orbits of nonchaotic interval maps [16].
We claim that the system \(\{J_{i}\}_{i\in\mathbb{N}}\) is a system of generating intervals. By e.g. [9], p.26 and [7], p.3, \(m\in\bar{\omega}\) where \(\bar{\omega}\) is the unique infinite ωlimit set of f. It is well known (e.g. [9]) that \(\bar{\omega}\) is contained in the closure of periodic points, thus \(\lim_{i\to\infty} a_{i} = m\). Consequently (3) implies \(\bigcap_{i\in\mathbb{N}} J_{i}= \{m\}\).
As a consequence of the above we obtain the following result announced in the first section of our paper.
Theorem 3
 (i)
\(C_{i}\) is \(2^{i}\)periodic under the shift homeomorphism, and
 (ii)
\(C_{i}\) is a compactification of a topological ray.
Problem 2
Does Lemma 13 hold for any map from \(\mathcal{F}\)?
At the very end of this paper we would like to mention some facts showing that the inverse limit spaces constructed within this work are topologically distinct from those mentioned in [6].
 (1)
\(K_{1}\) is homeomorphic to \(K_{2}\),
 (2)
\(K_{1}\) is the union of a topological ray \(R_{1}\) and \(X^{1}_{0}\) that compactifies \(R_{1}\); i.e. \(\overline{R}_{1}\setminus R_{1}=X^{1}_{0}\),
 (3)
\(K_{2}\) is the union of a topological ray \(R_{2}\) and \(X^{1}_{1}\) that compactifies \(R_{2}\); i.e. \(\overline{R}_{2}\setminus R_{2}=X^{1}_{1}\), and
 (4)
\(K_{1}\cap K_{2}=R_{1}\cap R_{2}=\{\hat{p}\}\), where \(\hat{p}\) is the fixed point of \(\sigma_{f}\).
We also note that each our attractor has an absolute end point; i.e. a point at which it is locally connected and which does not separate it (see [24] for more on absolute end points). However, the attractor \(X_{f}\) from [6] does not have such a point.
The above remarks lead to the following questions.
Problem 3
Suppose g is a LiYorke chaotic weakly unimodal map of type \(2^{\infty}\). Is \(I_{g}\) homeomorphic to a ray limiting onto one of the attractors described in [6] or a subcontinuum of one of them?
Problem 4
Suppose f and g are two LiYorke chaotic weakly unimodal maps of type \(2^{\infty}\) that are in two different differentiability classes, as guaranteed by Theorem 1 and Theorem 2. Are \(I_{f}\) and \(I_{g}\) homeomorphic?
One should note that for an arbitrary nondegenerate metric continuum P, there is an uncountable family of topologically distinct metric compactifications of a topological ray L, such that \(\overline {L}\setminus L=P\) [25].
Declarations
Acknowledgements
The authors express many thanks to Henk Bruin and Sonja Štimac for the helpful feedback during the first author’s visit in Faculty of Mathematics at University of Vienna in May 2014. This author is grateful for the kind hospitality of the faculty during his stay in Vienna and acknowledges financial support of the DFG network grant Oe 538/31, OeAD and the Polish Ministry of Science and Higher Education. This work was supported by the European Regional Development Fund in the IT4Innovations Centre of Excellence project (CZ.1.05/1.1.00/02.0070). J Boroński also gratefully acknowledges the partial support from the MSK DT1 Support of Science and Research in the MoravianSilesian Region 2013 (RRC/05/2013).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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