Bowen's entropy-conjugacy conjecture is true up to finite index

Mike Boyle (1) , Jerome Buzzi (2) , Kevin Mcgoff (3)

1 Department of Mathematics
2 LM-Orsay - Laboratoire de Mathématiques d'Orsay
3 Department of Mathematics

For a topological dynamical system consisting of a continuous map f, and a (not necessarily compact) subset Z of X, Bowen (1973) defined a dimension-like version of entropy, h_X(f,Z). In the same work, he introduced a notion of entropy-conjugacy for pairs of invertible compact systems: the systems (X,f) and (Y,g) are entropy-conjugate if there exist invariant Borel subsets X' of X and Y' of Y such that h_X(f,X\setminus X') < h_X(f,X), h_Y(g,Y \setminus Y') < h_Y(g,Y), and (X',f|_{X'}) is topologically conjugate to (Y',g|_{Y'}). Bowen conjectured that two mixing shifts of finite type are entropy-conjugate if they have the same entropy. We prove that two mixing shifts of finite type with equal entropy and left ideal class are entropy-conjugate. Consequently, in every entropy class Bowen's conjecture is true up to finite index.

Mots clés: symbolic dynamics; subshifts of finite type; topological entropy; dimensional entropy; entropy-conjugacy; topological conjugacy; left ideal class; resolving maps

Domaines: Systèmes dynamiques [math.DS]

Mike Boyle, Jerome Buzzi, Kevin Mcgoff. Bowen's entropy-conjugacy conjecture is true up to finite index. 2013. ⟨hal-00871620⟩