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February  2015, 35(2): 771-792. doi: 10.3934/dcds.2015.35.771

Transitive dendrite map with infinite decomposition ideal

Vladimír Špitalský 1,

1. 

Department of Mathematics, Faculty of Natural Sciences, Matej Bel University, Tajovského 40, 974 01 Banská Bystrica

Received  November 2013 Revised  April 2014 Published  September 2014

By a result of Blokh from 1984, every transitive map of a tree has the relative specification property, and so it has finite decomposition ideal, positive entropy and dense periodic points. In this paper we construct a transitive dendrite map with infinite decomposition ideal and a unique periodic point. Basically, the constructed map is (with respect to any non-atomic invariant measure) a measure-theoretic extension of the dyadic adding machine. Together with an example of Hoehn and Mouron from 2013, this shows that transitivity on dendrites is much more varied than that on trees.
Keywords: dense periodicity, infinite decomposition ideal, Transitive map, adding machine, dendrite..
Mathematics Subject Classification: Primary: 37B05, 37B20, 37B40; Secondary: 54H2.
Citation: 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
References:
[1]

G. Acosta, R. Hernández-Gutiérrez, I. Naghmouchi and P. Oprocha, Periodic points and transitivity on dendrites,, preprint, ().   Google Scholar

[2]

L. Alsedà, S. Kolyada, J. Llibre and L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc., 351 (1999), 1551-1573. doi: 10.1090/S0002-9947-99-02077-2.  Google Scholar

[3]

S. Baldwin, Entropy estimates for transitive maps on trees, Topology, 40 (2001), 551-569. doi: 10.1016/S0040-9383(99)00074-9.  Google Scholar

[4]

F. Balibrea and L'. Snoha, Topological entropy of Devaney chaotic maps, Topology Appl., 133 (2003), 225-239. doi: 10.1016/S0166-8641(03)00090-7.  Google Scholar

[5]

J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems, 17 (1997), 505-529. doi: 10.1017/S0143385797069885.  Google Scholar

[6]

A. M. Blokh, On transitive mappings of one-dimensional branched manifolds (Russian), Differential-difference equations and problems of mathematical physics (Russian), 3-9, 131, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1984.  Google Scholar

[7]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414; erratum: Trans. Amer. Math. Soc., 181 (1973), 509-510. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[8]

M. Dirbák, L'. Snoha and V. Špitalský, Minimality, transitivity, mixing and topological entropy on spaces with a free interval, Ergodic Theory Dynam. Systems, 33 (2013), 1786-1812. doi: 10.1017/S0143385712000442.  Google Scholar

[9]

L. Hoehn and C. Mouron, Hierarchies of chaotic maps on continua, Ergodic Theory Dynam. Systems, (2013), 1-17. doi: 10.1017/etds.2013.32.  Google Scholar

[10]

K. Kuratowski, Topology. Vol. II, Academic Press, New York-London, Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968.  Google Scholar

[11]

S. B. Nadler, Continuum Theory. An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992.  Google Scholar

[12]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

show all references

References:
[1]

G. Acosta, R. Hernández-Gutiérrez, I. Naghmouchi and P. Oprocha, Periodic points and transitivity on dendrites,, preprint, ().   Google Scholar

[2]

L. Alsedà, S. Kolyada, J. Llibre and L'. Snoha, Entropy and periodic points for transitive maps, Trans. Amer. Math. Soc., 351 (1999), 1551-1573. doi: 10.1090/S0002-9947-99-02077-2.  Google Scholar

[3]

S. Baldwin, Entropy estimates for transitive maps on trees, Topology, 40 (2001), 551-569. doi: 10.1016/S0040-9383(99)00074-9.  Google Scholar

[4]

F. Balibrea and L'. Snoha, Topological entropy of Devaney chaotic maps, Topology Appl., 133 (2003), 225-239. doi: 10.1016/S0166-8641(03)00090-7.  Google Scholar

[5]

J. Banks, Regular periodic decompositions for topologically transitive maps, Ergodic Theory Dynam. Systems, 17 (1997), 505-529. doi: 10.1017/S0143385797069885.  Google Scholar

[6]

A. M. Blokh, On transitive mappings of one-dimensional branched manifolds (Russian), Differential-difference equations and problems of mathematical physics (Russian), 3-9, 131, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1984.  Google Scholar

[7]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414; erratum: Trans. Amer. Math. Soc., 181 (1973), 509-510. doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[8]

M. Dirbák, L'. Snoha and V. Špitalský, Minimality, transitivity, mixing and topological entropy on spaces with a free interval, Ergodic Theory Dynam. Systems, 33 (2013), 1786-1812. doi: 10.1017/S0143385712000442.  Google Scholar

[9]

L. Hoehn and C. Mouron, Hierarchies of chaotic maps on continua, Ergodic Theory Dynam. Systems, (2013), 1-17. doi: 10.1017/etds.2013.32.  Google Scholar

[10]

K. Kuratowski, Topology. Vol. II, Academic Press, New York-London, Państwowe Wydawnictwo Naukowe Polish Scientific Publishers, Warsaw, 1968.  Google Scholar

[11]

S. B. Nadler, Continuum Theory. An Introduction, Monographs and Textbooks in Pure and Applied Mathematics, 158, Marcel Dekker, Inc., New York, 1992.  Google Scholar

[12]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

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