arXiv subjects

Gustavo Pessil

Publications and source records attributed to Gustavo Pessil.

3 recordsLinked to original sources

arXiv2026

On quantization and the classical variational principle for the metric mean dimension

This paper investigates the relationship between quantization of measures and metric mean dimension of topological dynamical systems. We introduce the concept of mean quantization dimension for invariant probability measures and establish a classical variational principle: the metric mean dimension of a dynamical system is equal to the maximum mean quantization dimension among all invariant measures. This approach effectively characterizes the complexity of systems with infinite entropy by identifying a measure that captures information across all scales; and yields a fundamental property that allows for the exchange of limits and suprema in the Lindenstrauss-Tsukamoto variational principles, a feat that most known entropy-like maps fail to achieve due to convexity. Nevertheless, we show that the Katok and Shapira entropies do satisfy this property and, therefore, a classical variational principle for the metric mean dimension, for which maximizing measures always exist.

math.DS↗

arXiv2024

Metric mean dimension, H\"older regularity and Assouad spectrum

Metric mean dimension is a geometric invariant of dynamical systems with infinite topological entropy. We relate this concept with the fractal structure of the phase space and the H\"older regularity of the map. Afterwards we improve our general estimates in a family of interval maps by computing the metric mean dimension in a way similar to the Misiurewicz formula for the entropy, which in particular shows that our bounds are sharp. As an application, we determine the metric mean dimension of the classical Weierstrass functions. Of independent interest, we develop a dynamical analogue of the Minkowski-Bouligand dimension for subshifts on Ahlfors regular alphabets, which also provides an entropy formula in terms of the size of the set of admissible words, generalizing the classical result for subshifts on finite alphabets.

math.DS↗

arXiv2024

Metric mean dimension via subshifts of compact type

We investigate the metric mean dimension of subshifts of compact type. We prove that the metric mean dimensions of a continuous map and its inverse limit coincide, generalizing Bowen's entropy formula. Building upon this result, we extend the notion of metric mean dimension to discontinuous maps in terms of suitable subshifts. As an application, we show that the metric mean dimension of the Gauss map and that of induced maps of the Manneville-Pomeau family is equal to the box dimension of the corresponding set of discontinuity points, which also coincides with a critical parameter of the pressure operator associated to the geometric potential.

math.DS↗