Ergodicity of non-uniformly expanding transitive group (or semigroup) actions (Q2814219)

Let \(\{f_1,f_2,\dots,f_k\}\), \(k\geq 1\), be a collection of \(C^1\) local diffeomorphisms of a compact manifold \(M\). The authors prove that if each diffeomorphism is conformal and the group (or semigroup) actions \(\mathcal F\) generated by \(\{f_1,f_2,\dots,f_k\}\) are non-uniformly expanding and transitive then \(\mathcal F\) is ergodic with respect to its quasi-invariant Lebesgue measure. The proof relies on the tools of hyperbolic preballs and hyperbolic times (see [\textit{J. F. Alves} and \textit{H. Vilarinho}, Ergodic Theory Dyn. Syst. 33, No. 3, 647--692 (2013; Zbl 1418.37052)]) and Lebesgue density points.NEWLINENEWLINEIt is worth noting that the notation in the paper is not always adequate and can confuse the reader. For example, \(\sigma\) is a shift and at the same time it is a real number \(\sigma\in(0,1)\), which is in the definition of hyperbolic preballs and hyperbolic times. Also \(k\) is a number of diffeomorphisms and at the same time it appears as an independent index, for example in Definition 3.1.