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

For a piecewise expanding unimodal interval map f with unique absolutely continuous invariant probability measure μ, a perturbation X, and an observable φ, the susceptibility function is Ψφ(z)=∑∞k=0zk∫X(x)φ′(fk)(x)(fk)′(x)dμ. 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 Breuer–Simon [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 séries de Dirichlet. Ann. Sci. Éc. Norm. Supér. (3) 66 (1949), 263–310]), we show that density of the postcritical orbit (a generic condition) implies that Ψφ(z) has a strong natural boundary on the unit circle. The Breuer–Simon method provides uncountably many candidates for the outer functions of Ψφ(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 Ψφ(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 Ψφ(z) as z→eiω for real ω. An additional ‘law of the iterated logarithm’ typicality assumption on the postcritical orbit gives stronger results.