American Institute of Mathematical Sciences

Advanced
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 & 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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.   Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[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. Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[16]

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

[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.   Google Scholar

[18]

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

[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.  Google Scholar

[20]

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

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[26]

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

[27]

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

[28]

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

[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.  Google Scholar

[30]

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

[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. Google Scholar

[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.  Google Scholar

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 7.  Gehman dendrite
Figure Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

[1]

Daniel Gonçalves, Bruno Brogni Uggioni. Li-Yorke Chaos for ultragraph shift spaces. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2347-2365. doi: 10.3934/dcds.2020117
[2]

Jakub Šotola. Relationship between Li-Yorke chaos and positive topological sequence entropy in nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5119-5128. doi: 10.3934/dcds.2018225
[3]

Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $ \beta $-transformation. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267
[4]

Vladimír Špitalský. Transitive dendrite map with infinite decomposition ideal. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 771-792. doi: 10.3934/dcds.2015.35.771
[5]

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
[6]

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 &amp; Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
[7]

Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure &amp; Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393
[8]

Tiantian Ma, Zaihong Wang. Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1563-1581. doi: 10.3934/dcds.2013.33.1563
[9]

Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043
[10]

Teresa Faria, Eduardo Liz, José J. Oliveira, Sergei Trofimchuk. On a generalized Yorke condition for scalar delayed population models. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 481-500. doi: 10.3934/dcds.2005.12.481
[11]

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
[12]

Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085
[13]

Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure &amp; Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127
[14]

Chris Good, Robert Leek, Joel Mitchell. Equicontinuity, transitivity and sensitivity: The Auslander-Yorke dichotomy revisited. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2441-2474. doi: 10.3934/dcds.2020121
[15]

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
[16]

Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 137-160. doi: 10.3934/dcds.2005.12.137
[17]

Robert Roussarie. Putting a boundary to the space of Liénard equations. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 441-448. doi: 10.3934/dcds.2007.17.441
[18]

Wacław Marzantowicz, Piotr Maciej Przygodzki. Finding periodic points of a map by use of a k-adic expansion. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 495-514. doi: 10.3934/dcds.1999.5.495
[19]

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
[20]

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

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (140)
  • HTML views (56)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences