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Roberto Castorrini

Publications and source records attributed to Roberto Castorrini.

10 recordsLinked to original sources
arXiv2026

Maharam-Pollicott-Ruelle resonances and self-similar translation flows on abelian covers

We study self-similar translation flows on $\mathbb Z^d$-covers of compact translation surfaces. Our main goal is to investigate their ergodic properties with respect to general Maharam measures. To this end, we develop a renormalization approach based on a family of twisted transfer operators associated with the renormalizing pseudo-Anosov map acting on anisotropic spaces of distributions. We describe the discrete spectrum of these operators in terms of the action of the pseudo-Anosov on suitable twisted cohomology groups. We further show that the resonant states of the dual operator corresponding to peripheral eigenvalues give rise to Maharam distributions which are invariant under the translation flow. Motivated by this correspondence, we refer to these eigenvalues as Maharam-Pollicott-Ruelle resonances. As applications, we derive asymptotic formulas for ergodic integrals of smooth observables at Maharam-generic points, prove a central limit theorem for the associated Frobenius cocycle, and compute the Hausdorff dimension of Maharam measures.

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arXiv2025

Four collapsing one-dimensional particles: a dynamical system approach of the spherical billiard reduction

We consider a system of four one-dimensional inelastic hard spheres evolving on the real line $\mathbb{R}$, and colliding according to a scattering law characterized by a fixed restitution coefficient $r$. We study the possible orders of collisions when the inelastic collapse occurs, relying on the so-called $\mathfrak{b}$-to-$\mathfrak{b}$ mapping, a two-dimensional dynamical system associated to the original particle system which encodes all the possible collision orders. We prove that the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping is a piecewise projective transformation, which allows one to perform efficient numerical simulations of its orbits. We recover previously known results concerning the one-dimensional four-particle inelastic hard sphere system and we support the conjectures stated in the literature concerning particular periodic orbits. We discover three new families of periodic orbits that coexist depending on the restitution coefficient, we prove rigorously that there exist stable periodic orbits for the $\mathfrak{b}$-to-$\mathfrak{b}$ mapping for restitution coefficients larger than the upper bounds previously known, and we prove the existence of quasi-periodic orbits for this mapping.

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arXiv2025

Limit theorems for toral partially hyperbolic endomorphisms

Under natural assumptions on the observable, we prove a Central Limit Theorem, a Berry-Esseen Theorem, and a quantitative Local Limit Theorem for a broad class of partially hyperbolic endomorphisms of the two-dimensional torus. Our results apply, but are not limited to, skew-products and their perturbations, and they remain valid even when the system admits multiple, though finitely many, absolutely continuous ergodic invariant measures.

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arXiv2024

Stability of Fixed Points for Nonlinear Selfconsistent Transfer Operators via Cone Contractions

In this paper we investigate the action of self-consistent transfer operators (STOs) on Birkhoff cones and give sufficient conditions for stability of their fixed points. Our approach relies on the order preservation properties of STOs that can be established via the study of their differential. We focus on the study of STOs arising from strongly coupled maps both deterministic and noisy. Our approach allows for explicit estimates that we use to give examples of STOs with multiple stable fixed points some of which are shown to be far from the asymptotic behaviour of the corresponding system of finite coupled maps and give information only on long transients for the finite dimensional system.

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arXiv2024

The differential of self-consistent transfer operators and the local convergence to equilibrium of mean field strongly coupled dynamical systems

We consider the differential of a self-consistent transfer operator at a fixed point of the operator itself and show that its spectral properties can be used to establish a kind of local exponential convergence to equilibrium: probability measures near the fixed point converge exponentially fast to the fixed point by the iteration of the transfer operator. This holds also in the strong coupling case. We also show that for mean field coupled systems satisfying uniformly a Lasota-Yorke inequality the differential does also. We present examples of application of the general results to self-consistent transfer operators based on deterministic expanding maps considered with different couplings, outside the weak coupling regime.

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arXiv2024

Horocycle flows on abelian covers of surfaces of negative curvature

We consider the unit speed parametrization of the horocycle flow on infinite Abelian covers of compact surfaces of negative curvature. We prove an asymptotic result for the ergodic integrals of sufficiently regular functions. In the case of constant curvature, where the unit speed and the uniformly contracting parametrizations of horocycles coincide, we recover a result by Ledrappier and Sarig. Our method, which does not use symbolic dynamics, is based on a general Fourier decomposition for Abelian covers and on the study of spectral theory of weighted (and twisted) transfer operators for the geodesic flow acting on appropriate anisotropic Banach spaces. Finally, as a byproduct result, we obtain a power deviation estimate for the horocycle ergodic averages on compact surfaces, without requiring any pinching condition as in previous results.

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arXiv2024

Dynamical Zeta functions for differentiable parabolic maps of the interval

This paper explores the domain of meromorphic extension for the dynamical zeta function associated to a class of one-dimensional differentiable parabolic maps featuring an indifferent fixed point. We establish the connection between this domain and the spectrum of the weighted transfer operators of the induced map. Furthermore, we discuss scenarios where meromorphic extensions occur beyond the confines of the natural disc of convergence of the dynamical zeta function.

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arXiv2023

Discontinuities cause essential spectrum on surfaces

Two-dimensional maps with discontinuities are considered. It is shown that, in the presence of discontinuities, the essential spectrum of the transfer operator is large whenever it acts on a Banach space with norm that is stronger than \(L^\infty\) or \(BV\). Three classes of examples are introduced and studied, both expanding and partially expanding. In two dimensions there is complication due to the geometry of the discontinuities, an issue not present in the one-dimensional case and which is explored in this work.

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arXiv2023

Thermodynamic formalism for piecewise expanding maps in finite dimension

We study weighted transfer operators associated to a piecewise expanding map on a compact manifold, and a piecewise Holder weight, acting on Sobolev spaces. We bound the essential spectral radius in terms of a topological pressure for a subadditive potential. Under a new small boundary pressure condition, we improve the estimate by establishing a variational principle for piecewise expanding maps and subadditive potentials. Version v2 is the electronic copy of the version to appear in DCDS

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