Entropies realizable by block gluing ℤ^d shifts of finite type

Abstract

In [9], Hochman and Meyerovitch gave a complete characterization of the set of topological entropies of ℤ^d shifts of finite type (SFTs) via a recursion-theoretic criterion. However, the ℤ^d SFTs they constructed in the proof are relatively degenerate and, in particular, lack any form of topological mixing, leaving open the question of which entropies can be realized within ℤ^d SFTs with (uniform) mixing properties. In this paper, we describe some progress on this question. We show that for α ∈ ℝ ⁺₀ to be the topological entropy of a block gluing ℤ² SFT, it cannot be too poorly computable; in fact, it must be possible to compute approximations to α within arbitrary tolerance ∈ in time ${2^{O(1/{\epsilon ^2})}}$. On the constructive side, we present a new technique for realizing a large class of computable real numbers as entropies of block gluing ℤ^d SFTs for any d > 2. As a corollary of our methods, we construct, for any N > 1, a block gluing ℤ^d SFT (d > 2) with entropy logN but without a full N-shift factor, strengthening previous work [6] by Boyle and the second author.