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June  2017, 37(6): 2957-2976. doi: 10.3934/dcds.2017127

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

Ghassen Askri

University of Carthage, Faculty of Sciences of Bizerte, Department of Mathematics, Jarzouna, 7021, Tunisia

Received  June 2015 Revised  January 2017 Published  February 2017

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.

Keywords: Dendrite, dendrite map, ω-limit set, periodic points, Li-Yorke pair, Li-Yorke chaos.
Mathematics Subject Classification: Primary: 37B45; Secondary: 37B99.
Citation: Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete and Continuous Dynamical Systems, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127
References:
[1]

G. Acosta, P. Eslami and L. Oversteegen, On open maps between dendrites, Houston. J. Math, 33 (2007), 753-770. 

[2]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.

[3]

L. Alseda, S. Kolyada, J. Libre and L. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc., 15 (1999), 221-237.  doi: 10.1090/S0002-9947-99-02077-2.

[4]

L. Alseda and X. Ye, No division and the set of periods for tree maps, Ergodic Theory Dynam. Systems, 15 (1995), 221-237.  doi: 10.1017/S0143385700008348.

[5]

D. Arévalo, W. J. Charatonik, P. P. Covarrubias and L. Simon, Dendrites with a closed set of endpoints, Top. App., 115 (2001), 1-17. 

[6]

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.

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L. S. Block and W. A. Coppel, Dynamics in One Dimension Lecture Notes in Math, 1513 Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0084762.

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A. Blokh, On transitive mappings of one-dimensional branched manifolds, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 131 (1984), 3-9. 

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A. Blokh, Dynamical systems on one-dimensional branched manifolds Ⅰ, (Russian), Teor. Funktsii Funktsional. Anal. i Prilozhen. , 46 (1986), 8-18; translation in J. Soviet Math. , 48 (1990), 500-508. doi: 10.1007/BF01095616.

[10]

A. Blokh, Dynamical systems on one-dimensional branched manifolds Ⅱ, (Russian)Teor. Funktsii Funktsional. Anal. i Prilozhen. , 47 (1987), 67-77; translation in J. Soviet Math. , 48 (1990), 668-674. doi: 10.1007/BF01094721.

[11]

A. Blokh, Dynamical systems on one-dimensional branched manifolds Ⅲ, (Russian) Teor. Funktsii Funktsional. Anal. i Prilozhen. , 48 (1987), 32-46; translation in J. Soviet Math. , 49 (1990), 875-883. doi: 10.1007/BF02205632.

[12]

A. Blokh, The connection between entropy and transitivity for one-dimensional mappings, (Russian) Uspekhi Mat. Nauk, 42 (1987), 209-210; translation in Russian Math. Surveys, 42 (1987), 165-166.

[13]

A. Blokh, Trees with snowflakes and zero entropy maps, Topology, 33 (1994), 379-396.  doi: 10.1016/0040-9383(94)90019-1.

[14]

A. Blokh, Pointwise-recurrent maps on uniquely arcwise connected locally arcwise connected spaces, Proc. Amer. Math. Soc., 143 (2015), 3985-4000.  doi: 10.1090/S0002-9939-2015-12589-0.

[15]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.

[16]

J. J. Charatonik, W. J. Charatonik and J. R. Prajs, Mapping hierarchy for dendrites, Dissertationes Math. (Rozprawy Mat.), 333 (1994), 52pp. 

[17]

W. J. Charatonik, E. P. Wright and S. S. Zafiridou, Dendrites with a countable set of endpoints and universality, Houston J. of Math, 39 (2013), 651-666. 

[18]

E. I. Dinaburg, The relation between topological entropy and metric entropy, Doklady Akademii Nauk SSSR, 190 (1970), 19-22. 

[19]

G. L. Forti, L. Paganoni and J. Smital, Strange triangular maps of the square, Bull. Austral. Math. Soc., 51 (1995), 395-415.  doi: 10.1017/S0004972700014222.

[20]

X. C. Fu and Z. M. Wang, The construction of chaotic subshifts, J. Nonlin. Dyn. Sci. Technol., 4 (1997), 127-132. 

[21]

J. L. G. Guirao and M. Lampart, Li and Yorke chaos with respect to the cardinality of the scrambled sets, Chaos Solitons Fractals, 24 (2005), 1203-1206.  doi: 10.1016/j.chaos.2004.09.103.

[22]

Z. Kocan, V. K. Kurkova and M. Malek, Entropy, horseshoes and homoclinic trajectories on trees, graphs and dendrites, Ergodic Theory Dynam. Systems, 31 (2011), p177.  doi: 10.1017/S0143385709001011.

[23]

M. Kuchta and J. Smital, Two-point scrambled set implies chaos, European Conference on Iteration Theory (Caldes de Malavella 1987), (1989), 427-430. 

[24]

K. Kuratowski, Topology. Vol. Ⅱ. , New edition, revised and augmented. Translated from the French by J. Jaworowski. Academic Press, Sceaux, 1992.

[25]

J. Mai and E. Shi, $\overline{R} = \overline{P}$ for maps of dendrites X with $Card E(X) < c$, Int. J. Bifurcation and Chaos, 19 (2009), 1391-1396.  doi: 10.1142/S021812740902372X.

[26]

M. Misiurewicz, Horseshoes for continuous mappings of an interval, Dynamical systems (Bressanone, 1978), 125-135, Liguori, Naples, 1980.

[27]

S. B. Nadler, Continuum Theory: An Introduction, Monogr. Textb. Pure Appl. Math., 158 (1992). 

[28]

J. Nikiel, A characterisation of dendroids with uncountably many endpoints in the classical sense, Houston J. Math., 9 (1983), 421-432. 

[29]

S. Ruette and L. Snoha, For graph maps, one scrambled pair implies Li-Yorke chaos, Proc. Amer. Math. Soc., 142 (2014), 2087-2100.  doi: 10.1090/S0002-9939-2014-11937-X.

[30]

A. N. Sharkovski, On cycles and the structure of continuous mappings, (Russian) Ukrain. Mat. Z., 17 (1965), 104-111. 

[31]

A. N. Sharkovski, The behavior of a map in a neighborhood of an attracting set, (Russian), Ukrain. Mat. Z. , 18 (1966), 60-83, English translation, Amer. Math. Soc. Translations, Series 2, 97 (1970), 227-258.

[32]

J. Smital, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282.  doi: 10.1090/S0002-9947-1986-0849479-9.

show all references

References:
[1]

G. Acosta, P. Eslami and L. Oversteegen, On open maps between dendrites, Houston. J. Math, 33 (2007), 753-770. 

[2]

R. L. Adler, A. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319.  doi: 10.1090/S0002-9947-1965-0175106-9.

[3]

L. Alseda, S. Kolyada, J. Libre and L. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc., 15 (1999), 221-237.  doi: 10.1090/S0002-9947-99-02077-2.

[4]

L. Alseda and X. Ye, No division and the set of periods for tree maps, Ergodic Theory Dynam. Systems, 15 (1995), 221-237.  doi: 10.1017/S0143385700008348.

[5]

D. Arévalo, W. J. Charatonik, P. P. Covarrubias and L. Simon, Dendrites with a closed set of endpoints, Top. App., 115 (2001), 1-17. 

[6]

F. Blanchard, E. Glasner, S. Kolyada and A. Maass, On Li-Yorke pairs, J. Reine Angew. Math., 547 (2002), 51-68.  doi: 10.1515/crll.2002.053.

[7]

L. S. Block and W. A. Coppel, Dynamics in One Dimension Lecture Notes in Math, 1513 Springer-Verlag, Berlin, 1992. doi: 10.1007/BFb0084762.

[8]

A. Blokh, On transitive mappings of one-dimensional branched manifolds, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 131 (1984), 3-9. 

[9]

A. Blokh, Dynamical systems on one-dimensional branched manifolds Ⅰ, (Russian), Teor. Funktsii Funktsional. Anal. i Prilozhen. , 46 (1986), 8-18; translation in J. Soviet Math. , 48 (1990), 500-508. doi: 10.1007/BF01095616.

[10]

A. Blokh, Dynamical systems on one-dimensional branched manifolds Ⅱ, (Russian)Teor. Funktsii Funktsional. Anal. i Prilozhen. , 47 (1987), 67-77; translation in J. Soviet Math. , 48 (1990), 668-674. doi: 10.1007/BF01094721.

[11]

A. Blokh, Dynamical systems on one-dimensional branched manifolds Ⅲ, (Russian) Teor. Funktsii Funktsional. Anal. i Prilozhen. , 48 (1987), 32-46; translation in J. Soviet Math. , 49 (1990), 875-883. doi: 10.1007/BF02205632.

[12]

A. Blokh, The connection between entropy and transitivity for one-dimensional mappings, (Russian) Uspekhi Mat. Nauk, 42 (1987), 209-210; translation in Russian Math. Surveys, 42 (1987), 165-166.

[13]

A. Blokh, Trees with snowflakes and zero entropy maps, Topology, 33 (1994), 379-396.  doi: 10.1016/0040-9383(94)90019-1.

[14]

A. Blokh, Pointwise-recurrent maps on uniquely arcwise connected locally arcwise connected spaces, Proc. Amer. Math. Soc., 143 (2015), 3985-4000.  doi: 10.1090/S0002-9939-2015-12589-0.

[15]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.

[16]

J. J. Charatonik, W. J. Charatonik and J. R. Prajs, Mapping hierarchy for dendrites, Dissertationes Math. (Rozprawy Mat.), 333 (1994), 52pp. 

[17]

W. J. Charatonik, E. P. Wright and S. S. Zafiridou, Dendrites with a countable set of endpoints and universality, Houston J. of Math, 39 (2013), 651-666. 

[18]

E. I. Dinaburg, The relation between topological entropy and metric entropy, Doklady Akademii Nauk SSSR, 190 (1970), 19-22. 

[19]

G. L. Forti, L. Paganoni and J. Smital, Strange triangular maps of the square, Bull. Austral. Math. Soc., 51 (1995), 395-415.  doi: 10.1017/S0004972700014222.

[20]

X. C. Fu and Z. M. Wang, The construction of chaotic subshifts, J. Nonlin. Dyn. Sci. Technol., 4 (1997), 127-132. 

[21]

J. L. G. Guirao and M. Lampart, Li and Yorke chaos with respect to the cardinality of the scrambled sets, Chaos Solitons Fractals, 24 (2005), 1203-1206.  doi: 10.1016/j.chaos.2004.09.103.

[22]

Z. Kocan, V. K. Kurkova and M. Malek, Entropy, horseshoes and homoclinic trajectories on trees, graphs and dendrites, Ergodic Theory Dynam. Systems, 31 (2011), p177.  doi: 10.1017/S0143385709001011.

[23]

M. Kuchta and J. Smital, Two-point scrambled set implies chaos, European Conference on Iteration Theory (Caldes de Malavella 1987), (1989), 427-430. 

[24]

K. Kuratowski, Topology. Vol. Ⅱ. , New edition, revised and augmented. Translated from the French by J. Jaworowski. Academic Press, Sceaux, 1992.

[25]

J. Mai and E. Shi, $\overline{R} = \overline{P}$ for maps of dendrites X with $Card E(X) < c$, Int. J. Bifurcation and Chaos, 19 (2009), 1391-1396.  doi: 10.1142/S021812740902372X.

[26]

M. Misiurewicz, Horseshoes for continuous mappings of an interval, Dynamical systems (Bressanone, 1978), 125-135, Liguori, Naples, 1980.

[27]

S. B. Nadler, Continuum Theory: An Introduction, Monogr. Textb. Pure Appl. Math., 158 (1992). 

[28]

J. Nikiel, A characterisation of dendroids with uncountably many endpoints in the classical sense, Houston J. Math., 9 (1983), 421-432. 

[29]

S. Ruette and L. Snoha, For graph maps, one scrambled pair implies Li-Yorke chaos, Proc. Amer. Math. Soc., 142 (2014), 2087-2100.  doi: 10.1090/S0002-9939-2014-11937-X.

[30]

A. N. Sharkovski, On cycles and the structure of continuous mappings, (Russian) Ukrain. Mat. Z., 17 (1965), 104-111. 

[31]

A. N. Sharkovski, The behavior of a map in a neighborhood of an attracting set, (Russian), Ukrain. Mat. Z. , 18 (1966), 60-83, English translation, Amer. Math. Soc. Translations, Series 2, 97 (1970), 227-258.

[32]

J. Smital, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc., 297 (1986), 269-282.  doi: 10.1090/S0002-9947-1986-0849479-9.

Figure 1.  $I, J$ form an arc horseshoe in the case $O_g (x) \nsubseteq C_{i_0}$
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Figure 2.  $I, J$ form an arc horseshoe in the case $O_g (x) \subseteq C_{i_0}$
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Figure 3.  $I, J$ is an arc horseshoe when $g^{k}(x_i) \in (c,g^{n}(x_i))$
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Figure 4.  $I, J$ form an arc horseshoe when $g^{p}(l_0 \cap (u,c)) \subset l_0$
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Figure 5.  Dendrite X with $E(X)$ closed and $E(X)^{\prime}$ is reduced to one point
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Figure 6.  Dendrite with a non-closed countable set of endpoints
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Figure 7.  Gehman dendrite
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Figure 8.  $l_1, l_2, l_3, l_4$ are cyclically permuted in the case $M\cap Fix(f) \neq \emptyset$.
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Figure 9.  $l_1, l_2, l_3, l_4$ are cyclically permuted in the case $M\cap Fix(f)=\emptyset$
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Figure 10.  Dendrite X with $E(X)$ closed and $E(X)^{\prime }$ infinite
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