Non-uniformly expanding dynamics: Stability from a probabilistic viewpoint
Let \(f: M\to M\) be a smooth map defined on a finite-dimensional compact Riemannian manifold \(M\). On \(M\) fix a Riemannian volume \(m\) and call it Lebesgue measure. A map \(f\) is called a non-uniformly expanding map if there exists some constant \(c>0\) such that \[ \limsup_{n\to\infty}{1\over n}\sum_{j=0}^n\log Df(f^j(x))^{-1}\leq -c<0. \] The author considers SRB measures on \(M\) for a \(C^2\) map \(f\). Theorem 1.1. Let \(f\) be a \(C^2\) non-uniformly expanding local diffeomorphism. There is a finite number of SRB measures \(\mu_1,\ldots, \mu_p\) whose basins cover a full Lebesgue measure subset of \(M\). Moreover, every absolutely continuous \(f\)-invariant probability measure \(\mu\) may be written as a convex linear combination of SRB measures. The author also considers stability of the SRB measure. Let \(\mathcal U\) be an open set of a non-uniformly expanding \(C^2\) map in \(C^1\) topology. A map \(f_0\) is called statistically stable if every weak\(^*\) accumulation point of the SRB measures of \(f\in\mathcal U\) is a convex linear combination of the SRB measure of \(f_0\). Let \(f_0\) be a non-uniformly \(C^2\) local diffeomorphism, and there exists some \(C^1\) neighborhood of \(f_0\) where \(f\) satisfies non-uniformly expanding with the same \(c>0\). Fix \(0<c_0<c\), and define \(N_f(x)\) by \[ {1\over n}\sum_{j=0}^n\log Df(f^j(x))^{-1}< -c_0 \quad\text{for every }n\geq N_f(x). \] If for given \(\gamma>0\) there are a constant \(K\) and \(\delta>0\) such that for \(f\) which satisfies for \(f-f_0<\delta\) \[ \sum_{k\geq K}k m(\{N_f>k\})<\gamma, \] then \(m(\{N_f>k\}\) is said fast decay uniformly for \(f\) in a neighborhood of \(f_0\). Theorem 1.2. \(m(\{N_f>k\}\) has fast decay uniformly for \(f_0\), then \(f_0\) is statistically stable. The author also studies a random perturbation version of Theorem 1.2 (Theorem 1.3).
scientific article; zbMATH DE number 1882983