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

Transitive dendrite map with infinite decomposition ideal

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.
Citation: Vladimír Špitalský. Transitive dendrite map with infinite decomposition ideal. Discrete and 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, arXiv:1312.7426v1.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[12]

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

show all references

References:
[1]

G. Acosta, R. Hernández-Gutiérrez, I. Naghmouchi and P. Oprocha, Periodic points and transitivity on dendrites, preprint, arXiv:1312.7426v1.

[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.

[3]

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

[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.

[5]

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

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[12]

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

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