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Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets

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  • Let X be a dendrite with set of endpoints $E(X)$ closed and let $f:~X \to X$ be a continuous map with zero topological entropy. Let $P(f)$ be the set of periodic points of f and let L be an ω-limit set of f. We prove that if L is infinite then $L\cap P(f)\subset E(X)^{\prime}$, where $E(X)^{\prime}$ is the set of all accumulations points of $E(X)$. Furthermore, if $E(X)$ is countable and L is uncountable then $L\cap P(f)=\emptyset$. We also show that if $E(X)^{\prime}$ is finite and L is uncountable then there is a sequence of subdendrites $(D_k)_{k ≥ 1}$ of X and a sequence of integers $n_k ≥ 2$ satisfying the following properties. For all $k≥1$,
      1. $f^{α_k}(D_k)=D_k$ where $α_k=n_1 n_2 \dots n_k$,
      2. $\cup_{k=0}^{n_j -1}f^{k α_{j-1}}(D_{j}) \subset D_{j-1}$ for all $j≥q 2$,
      3. $L \subset \cup_{i=0}^{α_k -1}f^{i}(D_k)$,
      4. $f(L \cap f^{i}(D_k))=L\cap f^{i+1}(D_k)$ for any $ 0≤q i ≤q α_{k}-1$. In particular, $L \cap f^{i}(D_k) ≠ \emptyset$,
      5. $f^{i}(D_k)\cap f^{j}(D_k)$ has empty interior for any $ 0≤q i≠ j<α_k $.
      As a consequence, if f has a Li-Yorke pair $(x,y)$ with $ω_f(x)$ or $ω_f(y)$ uncountable then f is Li-Yorke chaotic.

    Mathematics Subject Classification: Primary: 37B45; Secondary: 37B99.

    Citation:

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  • Figure 1.  $I, J$ form an arc horseshoe in the case $O_g (x) \nsubseteq C_{i_0}$

    Figure 2.  $I, J$ form an arc horseshoe in the case $O_g (x) \subseteq C_{i_0}$

    Figure 3.  $I, J$ is an arc horseshoe when $g^{k}(x_i) \in (c,g^{n}(x_i))$

    Figure 4.  $I, J$ form an arc horseshoe when $g^{p}(l_0 \cap (u,c)) \subset l_0$

    Figure 5.  Dendrite X with $E(X)$ closed and $E(X)^{\prime}$ is reduced to one point

    Figure 6.  Dendrite with a non-closed countable set of endpoints

    Figure 7.  Gehman dendrite

    Figure 8.  $l_1, l_2, l_3, l_4$ are cyclically permuted in the case $M\cap Fix(f) \neq \emptyset$.

    Figure 9.  $l_1, l_2, l_3, l_4$ are cyclically permuted in the case $M\cap Fix(f)=\emptyset$

    Figure 10.  Dendrite X with $E(X)$ closed and $E(X)^{\prime }$ infinite

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