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Article Dans Une Revue Ergodic Theory and Dynamical Systems Année : 2014

Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps

Résumé

We consider the susceptibility function Psi(z) of a piecewise expanding unimodal interval map f with unique acim mu, a perturbation X, and an observable phi. Combining previous results (deduced from spectral properties of Ruelle transfer operators) with recent work of Breuer-Simon (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon), we show that density of the postcritical orbit (a generic condition) implies that Psi(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functions of Psi(z), associated to precritical orbits. If the perturbation X is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the nontangential limit of the Psi(z) as z tends to 1 exists and coincides with the derivative of the acim with respect to the map (linear response formula). Applying the Wiener-Wintner theorem, we study the singularity type of nontangential limits as z tends to e^{i\omega}. An additional LIL typicality assumption on the postcritical orbit gives stronger results.

Dates et versions

hal-00700880 , version 1 (24-05-2012)

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Viviane Baladi, Stefano Marmi, David Sauzin. Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps. Ergodic Theory and Dynamical Systems, 2014, 34 (Part3), pp.777-800. ⟨10.1017/etds.2012.161⟩. ⟨hal-00700880⟩
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