# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 2015, 35 (2) : 771-792. doi: 10.3934/dcds.2015.35.771
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