Continuity of Hausdorff  measure for conformal dynamical systems

Tomasz Szarek; Mariusz Urbański; Anna Zdunik

doi:10.3934/dcds.2013.33.4647

Article Contents

2013, Volume 33, Issue 10: 4647-4692. Doi: 10.3934/dcds.2013.33.4647

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Continuity of Hausdorff measure for conformal dynamical systems

1.
Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, 80-952 Gdańsk
2.
Department of Mathematics, University of North Texas, Denton, TX 76203-1430
3.
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa

Received: June 30, 2012

Revised: December 31, 2012

Published: September 2013

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Abstract

Developing the pioneering work of Lars Olsen [14], we deal with the question of continuity of the numerical value of Hausdorff measures of some natural families of conformal dynamical systems endowed with an appropriate natural topology. In particular, we prove such continuity for hyperbolic polynomials from the Mandelbrot set, and more generally for the space of hyperbolic rational functions of a fixed degree. We go beyond hyperbolicity by proving continuity for maps including parabolic rational functions, for example that the parameter $1/4$ is such a continuity point for quadratic polynomials $z\mapsto z^2+c$ for $c\in [0,1/4]$. We prove the continuity of the numerical value of Hausdorff measures also for the spaces of conformal expanding repellers and parabolic ones, more generally for parabolic Walters conformal maps. We also prove some partial continuity results for all conformal Walters maps; these are in general of infinite degree. In order to do this, as one of our tools, we provide a detailed local analysis, uniform with respect to the parameter, of the behavior of conformal maps around parabolic fixed points in any dimension. We also establish continuity of numerical values of Hausdorff measures for some families of infinite $1$-dimensional iterated function systems.

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Mathematics Subject Classification: 37F35, 37F45, 37F10, 37E05.

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