An ergodic theorem for the action of a free semigroup (Q2783051)

Let \(\overline T:= {T_1,T_2,\dots,T_m}\) be a finite or countable collection of measurable maps from a measurable space \(X\) into itself, and let \(\overline p(x):= {p_1(x),p_2(x),\dots,p_m(x)}\), \(x\in X\), be a probability distribution, which is assumed to depend measurably on the point \(x\in X\). The triple \((X,\overline T, \overline p)\) defines the so-called iterated function system, which means that on each time step the probability to choose the map \(T_i\) is equal to \(p_i\). Usually in papers dedicated to this topic the probability distribution \(\overline p\) does not depend on \(x\), but the model with probabilities depending on the space position was discussed in the literature as well. On the other hand, the collection of maps \(\overline T\) generates on \(X\) the action of the semigroup \(G\) with \(m\) generators \({T_1,T_2,\dots,T_m}\). For a function \(f\colon X\to\mathbb R^1\) one can define the Markov operator corresponding to the iterated function system: \(Mf(x):= \sum_i p_i(x)f(T_ix)\). The main result of the paper is the following individual ergodic theorem. Let \(\mu\) be an invariant measure for the Markov operator \(M\); then for each \(1\leq q<\infty\) and any function \(f\in L^q(X,\mu)\) Cesàro means of the action of the operator \(M\) on the function \(f\) converge for \(\mu\)-a.a. \(x\in X\) to some \(M\)-invariant function \(\tilde f\in L^q(X,\mu)\), i.e. \(n^{-1}\sum_{k=0}^{n-1}M^kf \to \tilde f\) as \(n\to\infty\). Additionally, if the dynamical system \((G,X,\mu)\) is ergodic, then the function \(\tilde f\) is a constant almost everywhere with respect to the measure \(\mu\).NEWLINENEWLINEFor the entire collection see [Zbl 0981.00006].