Ergodic theorems for actions of certain maps

Specification and thermodynamical properties of semigroup actions

“…and it measures the maximal entropy rate in the semigroup. Finally we observe that it follows from [12] that the topological entropy of the semigroup action, according to Definition 3.2, in the case the generators are expanding is given by h top ((G, G 1 ), X) = log deg g 1 + deg g 2 + deg g 3 3 = log 10 3 > 0.…”

Convergence of Spherical Averages for Actions of Free Groups

“…We obtain that Tĝ−1 e 1 ψẽ 1 = Tĝ−1 e 1 gẽ 1 ϕ t(ẽ 1 ) = Tĝ−1 e 2 gẽ 2 ϕ t(ẽ 2 ) = Tĝ−1 e 2 ψẽ 2 , (8) and (19). Namely, (11) holds for a = 6, b = 2 and n 0 being the maximum of n(v) over all vertices of R.…”

“…For free groups we have cU P 2n−1 ϕ ≤ (P * ) n P n ϕ for any nonnegative ϕ. In the case of Fuchsian groups the inequality is more complicated and in particular containing an error term A n ϕ, see (11) in Section 6 below. The underlying geometric meaning of this inequality, Lemma 7.6, is that for a majority of thickened paths the following holds.…”

A note on non-commutative ergodic theorems for actions of hyperbolic groups

“…In order to work with the group actions we will consider a construction of the covering Markov group, which is originally due to Grigorchuk [18], J.-P. Thouvenot (oral communication), and Bufetov [5].…”

“…Among others we would like to mention the papers of Oceledetz [27], Tempelman [31], Bowen [3,4], Series [30], Arnold and Krylov [2], Ornstein [26], Guivarch [21], Grigorchuk [18,19], Nevo and Stein [24,25], Bufetov [5][6][7][8] (see also [9]). The non-commutative generalizations of the aforementioned results are due to the work of Grabarnik, Katz and Shwartz [15,16], as well as Anantharaman-Delaroche [1], see also [29].…”