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

作者：Baladi V^*; Marmi S; Sauzin D

来源：Ergodic Theory and Dynamical Systems, 2014, 34(03): 777-800.

DOI：10.1017/etds.2012.161

For a piecewise expanding unimodal interval map f with unique absolutely continuous invariant probability measure mu, a perturbation X, and an observable phi, the susceptibility function is Psi(phi)(z) = Sigma(infinity)(k=0) z(k) integral X(x)phi'(f(k))(x)(f(k))'(x)d mu. Combining previous results [V. Baladi, On the susceptibility function of piecewise expanding interval maps. Comm. Math. Phys. 275 (2007), 839-859; V. Baladi and D. Smania, Linear response for piecewise expanding unimodal maps. Nonlinearity 21 (2008), 677-711] (deduced from spectral properties of Ruelle transfer operators) with recent work of BreuerSimon [Natural boundaries and spectral theory. Adv. Math. 226 (2011), 4902-4920] (based on techniques from the spectral theory of Jacobi matrices and a classical paper of Agmon [Sur les s ries de Dirichlet. Ann. Sci. Ec. Norm. Super. (3) 66 (1949), 263-310]), we show that density of the postcritical orbit (a generic condition) implies that Psi(phi)(z) has a strong natural boundary on the unit circle. The Breuer-Simon method provides uncountably many candidates for the outer functions of Psi(phi)(z), associated with precritical orbits. If the perturbation X is horizontal, a generic condition (Birkhoff typicality of the postcritical orbit) implies that the non-tangential limit of Psi(phi)(z) as z -> 1 exists and coincides with the derivative of the absolutely continuous invariant probability measure with respect to the map ('linear response formula'). Applying the Wiener-Wintner theorem, we study the singularity type of non-tangential limits of Psi(phi)(z) as z -> e(i omega) for real omega. An additional 'law of the iterated logarithm' typicality assumption on the postcritical orbit gives stronger results.

• 出版日期2014-6