Skip to Main content Skip to Navigation

Invariant measures in symbolic dynamics : a topological, combinatorial and geometrical approach

Abstract : In this work we study some dynamical properties of symbolic dynamical systems, with particular emphasis on the role played by the invariant probability measures of such systems. We approach the study of the set of invariant measures from a topological, combinatorial and geometrical point of view. From a topological point of view, we focus on the problem of orbit equivalence and strong orbit equivalence between dynamical systems given by minimal actions of Z, through the study of an algebraic invariant, namely the dynamical dimension group. Our work presents a description of the dynamical dimension group for two particular classes of subshifts: S-adic subshifts and dendric subshifts. From a combinatorial point of view, we are interested in the problem of balance in minimal uniquely ergodic systems given by actions of Z. We investigate the behavior regarding balance for substitutive, S-adic and dendric subshifts. We give necessary conditions for a minimal substitutive system with rational frequencies to be balanced on its factors, obtaining as a corollary the unbalance in the factors of length at least 2 in the subshift generated by the Thue-Morse sequence. Finally, from the geometrical point of view, we investigate the problem of realization of Choquet simplices as sets of invariant probability measures associated to systems given by minimal actions of amenable groups on the Cantor set. We introduce the notion of congruent monotileable amenable group, we prove that every virtually nilpotent amenable group is congruent monotileable, and we show that for a discrete infinite group G with this property, every Choquet simplex can be obtained as the set of invariant measures of a minimal G-subshift.
Complete list of metadatas

Cited literature [137 references]  Display  Hide  Download
Contributor : Abes Star :  Contact
Submitted on : Friday, June 26, 2020 - 11:52:09 AM
Last modification on : Friday, August 21, 2020 - 5:58:31 AM


Version validated by the jury (STAR)


  • HAL Id : tel-02882021, version 1



Paulina Alejandra Cecchi Bernales. Invariant measures in symbolic dynamics : a topological, combinatorial and geometrical approach. Combinatorics [math.CO]. Université Sorbonne Paris Cité; Universidad de Chile, 2019. English. ⟨NNT : 2019USPCC039⟩. ⟨tel-02882021⟩



Record views


Files downloads