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

  • Silas L. Carvalho ORCID logo and Alexander Condori ORCID logo EMAIL logo
From the journal Forum Mathematicum

Abstract

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=M 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.

MSC 2010: 37A05; 28D05; 37A50

Communicated by Siegfried Echterhoff


Award Identifier / Grant number: 001/17/CEX-APQ-00352-17

Funding statement: Silas L. Carvalho was partially supported by FAPEMIG (a Brazilian government agency; Universal Project 001/17/CEX-APQ-00352-17). Alexander Condori was partially supported by CIENCIACTIVA C.G. 176-2015.

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Received: 2020-01-28
Revised: 2020-11-11
Published Online: 2021-01-15
Published in Print: 2021-03-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

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