Entropy on regular trees

Abstract

We show that the limit in our definition of tree shift topological entropy is actually the infimum, as is the case for both the topological and measure-theoretic entropies in the classical situation when the time parameter is Z. As a consequence, tree shift entropy becomes somewhat easier to work with. For example, the statement that the topological entropy of a tree shift defined by a one-dimensional subshift dominates the topological entropy of the latter can now be extended from shifts of finite type to arbitrary subshifts. Adapting to trees the strip method already used to approximate the hard square constant on Z 2 , we show that the entropy of the hard square tree shift on the regular k-tree increases with k, in contrast to the case of Z k . We prove that the strip entropy approximations increase strictly to the entropy of the golden mean tree shift for k = 2, . . . , 8 and propose that this holds for all k ≥ 2. We study the dynamics of the map of the simplex that advances the vector of ratios of symbol counts as the width of the approximating strip is increased, providing a fairly complete description for the golden mean subshift on the k-tree for all k. This map provides an efficient numerical method for approximating the entropies of tree shifts defined by nearest neighbor restrictions. Finally, we show that counting configurations over certain other patterns besides the natural finite subtrees yields the same value of entropy for tree SFT's.

An analogue of topological sequence entropy for Markov hom tree-shifts

“…Note that the limit in (1.3) exists [24] and is actually the infimum of log |Bn(T A )| |∆n| [25]. For T being a golden-mean tree ( 2), the limit of (1.3) also exists, and more general results for the existence of the limit (1.3) can be found in [5].…”

“…[20,7]). Its topological behavior and entropy have been extensively studied during the last decade: see [2,3] for the topological behavior, [1] for the classification theory and [4,24,25] for entropy. It is worth noting that since amenability is no longer true for T , that is, |∆ n \∆ n−1 |/|∆ n | does not tend to 0 as n → ∞ (see [7]), where ∆ n denotes the set of all vertices of T whose distance from 0 is at most n, T A has interesting properties which are different from those of shift spaces defined on N or on amenable groups.…”

Characterization and Topological Behavior of Homomorphism Tree-Shifts

“…In other words, T X is the set consists of configurations whose projection on any infinite path is in X. Petersen and Salama [14,15] first proposed the class of treeshifts and demonstrated that the topological entropy of X is no larger than the topological entropy of T X . An immediate result is X being topologically mixing implies T X is of positive topological entropy.…”

Substreetutions and more on trees

“…Usually, entropy is given by the exponential growth rate of the number of configurations the system sees with respect to their length. For trees, it is naturally expected a super-exponential growth rate (see [16]). In [14], Sturmian trees are colored trees with an affine growth.…”