Nicola Bertozzi | Searcharxiv

arXiv2026

Exponential Mixing for Hyperbolic Flows on Non-Compact Spaces

We introduce a family of hyperbolic flows on non-compact phase spaces that includes the geodesic flow on the modular surface. For these systems we prove exponential decay of correlations for sufficiently regular observables with respect to its SRB measure. Our approach follows the dynamical method of Dolgopyat and subsequent developments for suspension flows with uniformly hyperbolic Poincaré maps satisfying a uniform non-integrability condition. To fit this framework, we construct a suspension model via a triple inducing scheme that yields a uniformly hyperbolic Poincaré map with a countable Markov partition. We show that the resulting roof function is cohomologous to one that is constant along stable leaves and satisfies the required non-integrability and tail conditions. As an application, we recover a dynamical proof on Ratner's exponential mixing for the geodesic flow on the modular surface.

math.DS↗

arXiv2024

Asymptotic behaviour of the confidence region in orbit determination for hyperbolic maps with a parameter

When dealing with an orbit determination problem, uncertainties naturally arise from intrinsic errors related to observation devices and approximation models. Following the least squares method and applying approximation schemes such as the differential correction, uncertainties can be geometrically summarized in confidence regions and estimated by confidence ellipsoids. We investigate the asymptotic behaviour of the confidence ellipsoids while the number of observations and the time span over which they are performed simultaneously increase. Numerical evidences suggest that, in the chaotic scenario, the uncertainties decay at different rates whether the orbit determination is set up to recover the initial conditions alone or along with a dynamical or kinematical parameter, while in the regular case there is no distinction. We show how to improve some of the results in \cite{maro.bonanno}, providing conditions that imply a non-faster-than-polynomial rate of decay in the chaotic case with the parameter, in accordance with the numerical experiments. We also apply these findings to well known examples of chaotic maps, such as piecewise expanding maps of the unit interval or affine hyperbolic toral transformations. We also discuss the applicability to intermittent maps.

nlin.CD↗