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

Abstract:

For a topological dynamical system $(X,f)$, consisting of a continuous map $f : X \to X$, and a (not necessarily compact) set $Z \subset X$, 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 sets $X’ \subset X$ and $Y’ \subset Y$ 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.