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

TL;DR

This paper investigates the boundary behavior of the susceptibility function in certain dynamical systems, showing it generally has a natural boundary on the unit circle under typical conditions, with implications for linear response theory.

Contribution

It establishes that the susceptibility function of generic piecewise expanding unimodal maps has a natural boundary on the unit circle, combining spectral theory and recent techniques to analyze its complex structure.

Findings

Susceptibility function has a strong natural boundary on the unit circle for generic maps.

Under horizontal perturbations, the limit of the susceptibility function at 1 exists and matches the linear response derivative.

Additional assumptions yield detailed insights into the singularity types of the susceptibility function.

Abstract

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.