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

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.