Kostiantyn Drach (Universitat de Barcelona)

Title: Lyapunov spectral rigidity of expanding circle maps

Abstract: For an expanding circle map of degree at least 2, its Lyapunov spectrum is defined as the set of all Lyapunov exponents (multipliers) at periodic orbits. This set is analogous to the unmarked length spectrum of negatively curved metrics on surfaces of genus at least 2. Motivated by the unmarked length spectral rigidity conjecture for metrics, in the talk we will discuss whether the Lyapunov spectrum defines a smooth conjugacy class of the corresponding expanding circle map. In general this is false (we will present a counterexample). However, the following local rigidity holds: every C^r smooth expanding circle map f has a neighborhood (in C^r topology) such that any perturbation of f within this neighborhood that keeps the Lyapunov spectrum must be smoothly conjugate to f (subject to some sparsity assumption on the spectrum on f). The proof uses a novel iterative scheme which we will outline in the talk. This is joint work in progress with Vadim Kaloshin.