Bowen’s Entropy-conjugacy Conjecture Is True up to Finite Index

For a topological dynamical system (X, f), consisting of a continuous map f : X → X, and a (not necessarily compact) set Z ⊂ X, Bowen [2] defined a dimension-like version of entropy, hX(f, Z). In the same work, he introduced a notion of entropyconjugacy for pairs of invertible compact systems: the systems (X, f) and (Y, g) are entropy-conjugate if there exist invariant Borel sets X ′ ⊂ X and Y ′ ⊂ Y such that hX(f,X \ X ′) < hX(f,X), hY (g, Y \Y ′) < hY (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.