2025/11/21 by Kostiantyn Drach, Drach, Kostiantyn, V. A. Kaloshin +1
Mathematics · Physics and Astronomy · #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2511.17452
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 ≥ 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.