Generalized fractal dimensions of invariant measures of full-shift systems over compact and perfect spaces: Generic behavior

• Autores: Carvalho, Silas L., Condori, Alexander
• Formato: artículo
• Fecha de Publicación: 2021
• Institución: Consejo Nacional de Ciencia Tecnología e Innovación
• Repositorio: CONCYTEC-Institucional
• Lenguaje: inglés
• OAI Identifier: oai:repositorio.concytec.gob.pe:20.500.12390/2384
• Enlace del recurso: https://hdl.handle.net/20.500.12390/2384; https://doi.org/10.1515/forum-2020-0023
• Nivel de acceso: acceso abierto
• Materia: invariant measures; correlation dimension; Full-shift over an uncountable alphabet; generalized fractal dimensions; http://purl.org/pe-repo/ocde/ford#2.02.03

In this paper, we show that, for topological dynamical systems with a dense set (in the weak topology) of periodic measures, a typical (in Baire's sense) invariant measure has, for each q>0, zero lower q-generalized fractal dimension. This implies, in particular, that a typical invariant measure has zero upper Hausdorff dimension and zero lower rate of recurrence. Of special interest is the full-shift system (X,T) (where X=Mℤ is endowed with a sub-exponential metric and the alphabet M is a compact and perfect metric space), for which we show that a typical invariant measure has, for each q>1, infinite upper q-correlation dimension. Under the same conditions, we show that a typical invariant measure has, for each s∈(0,1) and each q>1, zero lower s-generalized and infinite upper q-generalized dimensions. © 2021 Walter de Gruyter GmbH, Berlin/Boston 2021.