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Singular cw-expansive flows
Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets
University of Carthage, Faculty of Sciences of Bizerte, Department of Mathematics, Jarzouna, 7021, Tunisia |
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.
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. |
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. |
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