Length Spectrum Rigidity for piecewise analytic Bunimovich Billiards

TL;DR

This paper proves spectral rigidity theorems for piecewise analytic Bunimovich billiards, showing that their spectral data uniquely determine their shape, and computes dynamical invariants for smooth cases.

Contribution

It introduces new spectral rigidity results for Bunimovich billiards, extending understanding of how shape influences spectral and dynamical properties.

Findings

Spectral rigidity theorems for piecewise analytic Bunimovich stadia

Explicit Lyapunov exponents for smooth Bunimovich stadia

Calculation of Peierls' Barrier function for specific length spectra

Abstract

In the paper, we establish Squash Rigidity Theorem - the dynamical spectral rigidity for piecewise analytic Bunimovich squash-type stadia. We also establish Stadium Rigidity Theorem - the dynamical spectral rigidity for piecewise analytic Bunimovich stadia whose flat boundaries are a priori fixed. In addition, for smooth Bunimovich squash-type stadia we compute the Lyapunov exponents along the maximal period two orbit, as well as the value of the Peierls' Barrier function from the maximal marked length spectrum associated to the rotation number $\frac{2n}{4n+1}$.