Ergodic properties of Viana-like maps with singularities in the base dynamics (Q2846922)

The article under review is devoted to the analysis of statistical properties of two Viana-like maps. The first map \({F_1:S^1\times\mathbb{R}\to S^1\times\mathbb{R}}\) is defined as NEWLINE\[NEWLINE F_1(\theta, x)=(\beta\theta\;\mathrm{mod}\,1, Q_a(x)+\alpha\sin(2\pi\theta)), NEWLINE\]NEWLINE where \({Q_a(x)=a-x^2}\) is a Misiurewicz-Thurston quadratic map (i.e., \({a\in (0,2)}\) is chosen such that the critical point~\(Q_a\) is pre-periodic), \({\beta\geq\beta_a >2}\), and \({\alpha>0}\) is sufficiently small. The second map \({F_2: I\times\mathbb{R}\to I\times\mathbb{R}}\) is defined as NEWLINE\[NEWLINE F_2(\theta, x)=(Q^k_b(\theta), Q_a(x)+\alpha s(\theta)), NEWLINE\]NEWLINE where \({Q_b(\theta)=b-\theta^2}\), \(b\in(0,2]\), is another Misiurewicz-Thurston map, \(I\) is the interval \([Q_b^2(0), Q_b(0)]\), \({k\geq1}\) is an integer, \({s:I\to[-1,1]}\) is a coupling function, and \({\alpha>0}\) is also sufficiently small.NEWLINENEWLINEThe authors prove that the maps \(F_1\) and \(F_2\) admit unique absolutely continuous invariant probability measures \(\mu_1\) and \(\mu_2\) respectively for which stretched exponential decay of correlations and stretched exponential large deviations hold for some observables. Also the Central Limit Theorem, the Almost Sure Invariant Principle, the Local Limit Theorem and the Berry-Esseen Theorem hold for certain Hölder observables.