Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2509.06915

Isoperimetric-type inequalities for Mather's $β$-function of convex billiards

Stefano Baranzini·Misha Bialy·Alfonso Sorrentino
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this article we discuss pointwise spectral rigidity results for several billiard systems (e.g., Birkhoff billiards, symplectic billiards and $4$-th billiards), showing that a single value of Mather's $β$-function can determine whether a strongly convex smooth planar domain is a disk (or an ellipse, in the affine-invariant case of symplectic billiards). Evoking the famous question "Can you hear the shape of a billiard?", one could say that circular billiards can be heard by a single whisper! More specifically, we prove isoperimetric-type inequalities comparing the $β$-function associated to the billiard map of domain to that of a disk with the same perimeter or area, and investigate what are the consequences of having an equality. Surprisingly, this rigidity fails for outer billiards, where explicit counterexamples are constructed for rotation numbers $1/3$ and $1/4$. The results are framed within Aubry-Mather theory and provide a modern dynamical reinterpretation and extension of classical geometric inequalities for extremal polygons.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Stefano Baranzini, Misha Bialy, Alfonso Sorrentino. 2025-12-22. Isoperimetric-type inequalities for Mather's $β$-function of convex billiards. https://arxiv.org/abs/2509.06915

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Pivoting technique for the circle homeomorphism group

We adapt Gou{ë}zel's pivoting technique to the circle homeomorphism group. As an application, we give different proofs of Gilabert Vio's probabilistic Tits alternative and Malicet's exponential synchronization.

math.DS↗
arXiv2026

Asymmetry of a class of Mellin transforms

We introduce the quantity $μ_η$, defined for every complex $s$ in the critical strip, as a transformation of the Mellin transform associated to the functions $η$. We establish a sufficient condition on $η$ under which $μ_η(s)$ and $μ_η(1-s)$ cannot both vanish outside the critical line. An application is given to the case in which $η$ is the fractional part function, and the zeros of $μ_η$ coincide with the zeros of the Riemann zeta function.

math.DS↗
arXiv2026

Infinite Set of Resonances in the Linear Damped Oscillator Subject to Harmonic Forcing with Non-standard Frequency Modulation

It is shown that harmonic signals incorporating a type of weak non-standard frequency modulation (wNSFM) have interesting spectral properties, namely, time-dependent bandwidths that become increasingly broader with increasing time. As such, they represent a class of signals with frequency-time coupling in their spectra. Specifically, the weakly damped oscillator exhibits always two transient resonance captures involving two distinct harmonics possessing relatively high amplitudes over finite time intervals, while the overall response decays as $~t^{-1/2}$ as $t\rightarrow\infty$. Considering the undamped oscillator, it possesses two types of resonances, referred to as simple and non-simple resonances. Simple resonances correspond to finite-amplitude steady-state responses caused by two sustained resonance captures, in the form of two distinct modulated quasi-periodic responses, which, however are "activated" at different time instances. The necessary and sufficient conditions for non-simple resonances are given in the form of a theorem which predicts the existence of resonant harmonics and specifies the special phase conditions that the resonant harmonics must satisfy for constructive interference; the resulting undamped non-simple resonance grows unboundedly as $~t^{-1/2}$ as $t\rightarrow\infty$, in contrast to the classical resonance growth of the linear resonator with unmodulated harmonic excitation whose response grows as $~t$ as $t\rightarrow\infty$. These resonant responses are persistent to changes in the parameters of the wNSFM. Our results reveal interesting infinite sets of resonances in linear SDOF resonators under frequency-modulated excitations.

math.DS↗