seminar: Natural boundary for the susceptibility function of generic piecewise expanding unimodal maps
speaker: David Sauzin (CNRS)abstract: [joint work with Viviane Baladi et Stefano Marmi]
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).
timetable:
Tue 5 Mar, 14:30 - 15:30, Sala Conferenze Centro De Giorgi
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