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 Zd 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 Zd SFTs with (uniform) mixing properties. In this paper, we describe some progress on this question. We show that for alpha epsilon R-0(+) to be the topological entropy of a block gluing Z(2) SFT, it cannot be too poorly computable; in fact, it must be possible to compute approximations to a within arbitrary tolerance epsilon 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 Zd 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.