Unmarked spectral rigidity of expanding circle maps

Abstract

For a smooth expanding map $f$ of the circle, its (unmarked) length spectrum is defined as the set of logarithms of multipliers of periodic orbits of $f$. This spectrum is analogous to the set of lengths of all closed geodesics on negatively curved surfaces -- the classical length spectrum. In the paper, we prove a length spectral rigidity result for expanding circle maps. Namely, we show that a smooth expanding circle map $f$ of degree $d \ge 2$, under certain assumptions on the sparsity of its length spectrum, cannot be perturbed with an arbitrarily small perturbation (depending on $f$) so that its length spectrum stays the same. The proof uses the Whitney extension theorem, a quantitative Livsic-type theorem, and a novel iterative scheme.