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Central limit theorem for the modulus of continuity of averages of observables on transversal families of piecewise expanding unimodal maps. (English) Zbl 1387.37025

Summary: Consider a \(C^2\) family of mixing \(C^4\) piecewise expanding unimodal maps \(t \in [a,b] \mapsto f_t\), with a critical point \(c\), that is transversal to the topological classes of such maps. Given a Lipchitz observable \(\phi\) consider the function \[ \mathcal{R}_{\phi} (t)=\int \phi d \mu_t, \] where \(\mu_t\) is the unique absolutely continuous invariant probability of \(f_t\). Suppose that \(\sigma_t > 0\) for every \(t\in [a,b]\), where \[ \sigma_t^2 = \sigma_t^2(\phi) = \lim _{n \rightarrow \infty }\int \left(\frac{\sum_{j=0}^{n-1}\left(\phi \circ f_t^j - \int \phi d \mu_t\right)}{\sqrt{n}}\right)^2 d \mu_t. \] We show that \[ m\left\{t \in [a,b]:t+h\in [a,b]\text{and} \frac{1}{\psi (t)\sqrt{-\log |h|}}\left(\frac{{\mathcal{R}}_\phi (t+h)- \mathcal{R}_\phi(t)}{h}\right)\leqslant y \right\} \] converges to \[ \frac{1}{\sqrt{2\pi}} \int _{-\infty }^y e^{-\frac{s^2}{2}} ds, \] where \(\psi (t)\) is a dynamically defined function and \(m\) is the Lebesgue measure on \([a,b]\), normalized in such way that \(m([a,b])=1\). As a consequence, we show that \(\mathcal{R}_\phi\) is not a Lipchitz function on any subset of \([a,b]\) with positive Lebesgue measure.

MSC:

37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
37D50 Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37E05 Dynamical systems involving maps of the interval
37A05 Dynamical aspects of measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
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