Piecewise \(C^2\) perturbation of Lasota-Yorke maps and their ergodic properties
A map \(T:[0,1]\to[0,1]=:X\) is said to be piecewise \(C^2\) (notation P\(C^2\)) if there exists a finite partition of \(X\) into intervals \(X_i\) such that \(T|_{X_i}\) is a \(C^2\) map and any iteration \(T^n\) of the map \(T\) is also a P\(C^2\)-map. A P\(C^2\)-map \(T\) is called piecewise expanding or PE-map (the author call it a Lasota-Yorke map) if for some positive integer \(n\) the expanding constant of the \(n\)-th iteration of the map \(T\) -- the number \(\lambda(T^n) :=\text{ess}\inf |DT^n|>1\), where \(DT\) stands for the derivative of the map \(T\). The author assumes also that any PE-map under consideration has a unique absolutely continuous invariant measure \(\mu\) and that the dynamical system \((T,X,\mu)\) is Bernoulli. Introducing a metric \(d\) in the space of P\(C^2\)-maps, describing the `distance' between the graphs of the maps, the author studies stability of ergodic properties of PE-maps under d-small perturbations. Namely, it is shown that any P\(C^2\)-map close enough to a PE-map \(T\) with the expanding constant \(\lambda(T)>2\) is also a PE-map. The stability of the leading eigenvalue of the corresponding Perron-Frobenius operator, considered in the space of functions of bounded variation, is studied as well. The autor also gives 2 examples showing that the assumption \(\lambda(T)>2\) is essential for the stability results mentioned above. Reviewer's remark: Note that very similar results are known in the case of random perturbations of PE-maps and in the case of coupled map lattices [see, e.g., \textit{M. Blank}, Discreteness and continuity in problems of chaotic dynamics, Am. Math. Soc. Monographs (1997; Zbl 0881.58047)]. In both cases it is shown that the assumption about the expanding constant is important but can be replaced by a much weaker assumption about the absence of periodic turning points of the PE-maps.
scientific article; zbMATH DE number 1894634