Treffer: 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 -c0\). Fix \(00\) there are a constant \(K\) and \(\delta>0\) such that for \(f\) which satisfies for \(f-f_0k\})k\}\) 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).