Entropy formula for random ℤ^{𝕜}-actions

Abstract

In this paper, entropies, including measure-theoretic entropy and topological entropy, are considered for random Z k -actions which are generated by random compositions of the generators of Z k -actions. Applying Pesin's theory for commutative diffeomorphisms we obtain a measure-theoretic entropy formula of C 2 random Z k -actions via the Lyapunov spectra of the generators. Some formulas and bounds of topological entropy for certain random Z k (or Z k + )-actions generated by more general maps, such as Lipschitz maps, continuous maps on finite graphs and C 1 expanding maps, are also obtained. Moreover, as an application, we give a formula of Friedland's entropy for certain C 2 Z k -actions.

A note on two types of Lyapunov exponents and entropies for Zk-actions

“…(3) (Dimension formula) If T is C 2 and there is no additional assumption of absolute continuity on µ, then ) and Mãné ([18]) to the random case. The proof in [28] is in a sense self-contained and elaborated. But now after getting the formula (2.4), we can directly obtain (2.15) and (2.16) by substitution.…”

“…But now after getting the formula (2.4), we can directly obtain (2.15) and (2.16) by substitution. Furthermore, we can see an application of the "random" entropy formula (2.16) to the "deterministic" Friedland's entropy of T in [28].…”

“…In recent years, in order to investigate the diverse and intricate dynamics of T , various of entropytype invariants to measure the complexity of T in different levels and from different points of views are introduced and investigated, such as directional entropy ( [19], [20] and [1]), and [28]), Fried average entropy ( [3] and [9]), slow entropy ( [10] and [9]) and Friedland entropy ( [4], [5], [2]), etc. Another type of entropy to describe the chaotic behavior of T is its random entropy (or say, the entropy of the random Z k -actions induced by T ).…”

Entropies of commuting transformations on Hilbert spaces

“…entropy formula and SRB measures for random transformations generated by finitely commutative transformations in infinite dimensional Hilbert spaces via its generators, which can be viewed as a generalization of the work in [7,8,24,25] to the infinite dimension spaces. However, the techniques and strategies are completely different due to the feature of infinite dimensional smooth dynamics.…”