# American Institue of Mathematical Sciences

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

## Li-Yorke chaos for dendrite maps with zero topological entropy and $\omega$-limit sets

 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.

Citation: Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127
##### References:
 [1] G. Acosta, P. Eslami, L. Oversteegen, On open maps between dendrites, Houston. J. Math, 33 (2007), 753-770. [2] R. L. Adler, A. G. Konheim, 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, 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, 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, L. Simon, Dendrites with a closed set of endpoints, Top. App., 115 (2001), 1-17. [6] F. Blanchard, E. Glasner, S. Kolyada, 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. [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. [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. [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. [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, J. R. Prajs, Mapping hierarchy for dendrites, Dissertationes Math. (Rozprawy Mat.), 333 (1994), 52pp. [17] W. J. Charatonik, E. P. Wright, 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, J. Smital, Strange triangular maps of the square, Bull. Austral. Math. Soc., 51 (1995), 395-415. doi: 10.1017/S0004972700014222. [20] X. C. Fu, Z. M. Wang, The construction of chaotic subshifts, J. Nonlin. Dyn. Sci. Technol., 4 (1997), 127-132. [21] J. L. G. Guirao, 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, 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, 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, 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, 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, L. Oversteegen, On open maps between dendrites, Houston. J. Math, 33 (2007), 753-770. [2] R. L. Adler, A. G. Konheim, 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, 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, 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, L. Simon, Dendrites with a closed set of endpoints, Top. App., 115 (2001), 1-17. [6] F. Blanchard, E. Glasner, S. Kolyada, 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. [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. [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. [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. [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, J. R. Prajs, Mapping hierarchy for dendrites, Dissertationes Math. (Rozprawy Mat.), 333 (1994), 52pp. [17] W. J. Charatonik, E. P. Wright, 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, J. Smital, Strange triangular maps of the square, Bull. Austral. Math. Soc., 51 (1995), 395-415. doi: 10.1017/S0004972700014222. [20] X. C. Fu, Z. M. Wang, The construction of chaotic subshifts, J. Nonlin. Dyn. Sci. Technol., 4 (1997), 127-132. [21] J. L. G. Guirao, 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, 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, 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, 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, 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.
$I, J$ form an arc horseshoe in the case $O_g (x) \nsubseteq C_{i_0}$
$I, J$ form an arc horseshoe in the case $O_g (x) \subseteq C_{i_0}$
$I, J$ is an arc horseshoe when $g^{k}(x_i) \in (c,g^{n}(x_i))$
$I, J$ form an arc horseshoe when $g^{p}(l_0 \cap (u,c)) \subset l_0$
Dendrite X with $E(X)$ closed and $E(X)^{\prime}$ is reduced to one point
Dendrite with a non-closed countable set of endpoints
Gehman dendrite
$l_1, l_2, l_3, l_4$ are cyclically permuted in the case $M\cap Fix(f) \neq \emptyset$.
$l_1, l_2, l_3, l_4$ are cyclically permuted in the case $M\cap Fix(f)=\emptyset$
Dendrite X with $E(X)$ closed and $E(X)^{\prime }$ infinite
 [1] Vladimír Špitalský. Transitive dendrite map with infinite decomposition ideal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 771-792. doi: 10.3934/dcds.2015.35.771 [2] Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047 [3] Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703 [4] Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393 [5] Tiantian Ma, Zaihong Wang. Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1563-1581. doi: 10.3934/dcds.2013.33.1563 [6] Teresa Faria, Eduardo Liz, José J. Oliveira, Sergei Trofimchuk. On a generalized Yorke condition for scalar delayed population models. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 481-500. doi: 10.3934/dcds.2005.12.481 [7] Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043 [8] Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085 [9] Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127 [10] Amer Rasheed, Aziz Belmiloudi, Fabrice Mahé. Dynamics of dendrite growth in a binary alloy with magnetic field effect. Conference Publications, 2011, 2011 (Special) : 1224-1233. doi: 10.3934/proc.2011.2011.1224 [11] Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557 [12] Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 137-160. doi: 10.3934/dcds.2005.12.137 [13] Robert Roussarie. Putting a boundary to the space of Liénard equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 441-448. doi: 10.3934/dcds.2007.17.441 [14] Wacław Marzantowicz, Piotr Maciej Przygodzki. Finding periodic points of a map by use of a k-adic expansion . Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 495-514. doi: 10.3934/dcds.1999.5.495 [15] Eduardo Liz, Victor Tkachenko, Sergei Trofimchuk. Yorke and Wright 3/2-stability theorems from a unified point of view. Conference Publications, 2003, 2003 (Special) : 580-589. doi: 10.3934/proc.2003.2003.580 [16] Isaac A. García, Jaume Giné, Jaume Llibre. Liénard and Riccati differential equations related via Lie Algebras. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2/3, September) : 485-494. doi: 10.3934/dcdsb.2008.10.485 [17] Jin Ma, Shige Peng, Jiongmin Yong, Xunyu Zhou. A biographical note and tribute to xunjing li on his 80th birthday. Mathematical Control & Related Fields, 2015, 5 (3) : i-iii. doi: 10.3934/mcrf.2015.5.3i [18] Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237 [19] Alexander Blokh, Michał Misiurewicz. Dense set of negative Schwarzian maps whose critical points have minimal limit sets. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 141-158. doi: 10.3934/dcds.1998.4.141 [20] James W. Cannon, Mark H. Meilstrup, Andreas Zastrow. The period set of a map from the Cantor set to itself. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2667-2679. doi: 10.3934/dcds.2013.33.2667

2016 Impact Factor: 1.099