Unmarked spectral rigidity of expanding circle maps

TL;DR

This paper proves a spectral rigidity result for smooth expanding circle maps, showing that under certain conditions, the length spectrum uniquely determines the map, using advanced mathematical tools and an iterative scheme.

Contribution

It establishes a length spectral rigidity theorem for expanding circle maps, a novel result linking spectrum and map uniqueness under specific conditions.

Findings

Spectral rigidity holds for certain expanding circle maps.

The proof employs Whitney extension, Livsic-type theorems, and an iterative scheme.

Small perturbations cannot preserve the length spectrum under the given assumptions.

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.