Entropy formula and continuity of entropy for piecewise expanding maps

  • José F. Alves

    Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
  • Antonio Pumariño

    Departamento de Matemáticas, Facultad de Ciencias de la Universidad de Oviedo, Federico García Lorca, 18, 33007 Oviedo, Spain
Entropy formula and continuity of entropy for piecewise expanding maps cover

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Abstract

We consider some classes of piecewise expanding maps in finite dimensional spaces having invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures and, using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to a classical one-dimensional family of tent maps and a family of two-dimensional maps which arises as the limit of return maps when a homoclinic tangency is unfolded by a family of three dimensional diffeomorphisms.

Cite this article

José F. Alves, Antonio Pumariño, Entropy formula and continuity of entropy for piecewise expanding maps. Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 1, pp. 91–108

DOI 10.1016/J.ANIHPC.2020.06.003