Recurrence to Vanishing Sets in Matrix-Driven Toral Dynamics

Abstract

 We studied recurrence phenomena on the -dimensional torus generated by linear and nonlinear dynamical mechanisms, with emphasis on sets defined through infinitely many close returns. First, we analyzed fractal subsets of the torus arising from coordinate-wise expanding maps associated with real parameters greater than one, where recurrence is governed by a general decay function. Using geometric and measure-theoretic techniques, we determined the Hausdorff dimension of these sets and identified how it depends on the expansion rates and the rate of decay of the target sizes.

We then introduced a unified framework that connects shrinking target problems with quantitative recurrence for toral transformations induced by real, non-singular matrices. Within this setting, we considered sequences of anisotropic target neighborhoods whose sizes vary independently along each coordinate direction. The targets were allowed to move dynamically according to uniformly Lipschitz vector-valued functions, leading to non-stationary and non-symmetric recurrence conditions. For this general class of moving hyperrectangular targets, we established precise dimension formulae for the associated limsup sets. These results reveal a robust relationship between matrix-driven toral dynamics, geometric shrinking mechanisms, and the fractal structure of recurrent orbits.