Searcharxiv⌕ Search

arXiv subjects

Yair Hartman

Publications and source records attributed to Yair Hartman.

20 records · Page 2Linked to original sources
arXiv2012

Large Semigroups of Cellular Automata

In this article we consider semigroups of transformations of cellular automata which act on a fixed shift space. In particular, we are interested in two properties of these semigroups which relate to "largeness". The first property is ID and the other property is maximal commutativity (MC). A semigroup has the ID property if the only infinite invariant closed set (with respect to the semigroup action) is the entire space. We shall consider two examples of semigroups: one is spanned by cellular automata transformations that represent multiplications by integers on the one-dimensional torus and the other one consists of all the cellular automata transformations which are linear (when the symbols set is the ring of integers mod n). It will be shown that the two properties of these semigroups depend on the number of symbols s. The multiplication semigroup is ID and MC if and only if s is not a power of prime. The linear semigroup over the mentioned ring is always MC but is ID if and only if s is prime. When the symbol set is endowed with a finite field structure (when possible) the linear semigroup is both ID and MC. In addition, we associate with each semigroup which acts on a one sided shift space a semigroup acting on a two sided shift space, and vice versa, in such a way that preserves the ID and the MC properties.

math.DS↗
arXiv2012

An Abramov formula for stationary spaces of discrete groups

Let (G,mu) be a discrete group equipped with a generating probability measure, and let Gamma be a finite index subgroup of G. A mu-random walk on G, starting from the identity, returns to Gamma with probability one. Let theta be the hitting measure, or the distribution of the position in which the random walk first hits Gamma. We prove that the Furstenberg entropy of a (G,mu)-stationary space, with respect to the induced action of (Gamma,theta), is equal to the Furstenberg entropy with respect to the action of (G,mu), times the index of Gamma in G. The index is shown to be equal to the expected return time to Gamma. As a corollary, when applied to the Furstenberg-Poisson boundary of (G,mu), we prove that the random walk entropy of (Gamma,theta) is equal to the random walk entropy of (G,mu), times the index of Gamma in G.

math.DS↗