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Porosity in Conformal Dynamical Systems

Part of: Classical measure theory Set functions and measures on spaces with additional structure Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  14 April 2021

VASILEIOS CHOUSIONIS  and
MARIUSZ URBAŃSKI
VASILEIOS CHOUSIONIS
Affiliation:
Department of Mathematics, University of Connecticut, 341 Marsfield Road 41009, Stars, CT 06269-3009, U.S.A. e-mail: vasileios.chousionis@uconn.edu
MARIUSZ URBAŃSKI
Affiliation:
Department of Mathematics, University of North Texas, 1155 Union Circle #311430, Denton, 7X76203-5017, U.S.A. e-mail: mariuszurbanski@unt.edu
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Abstract

In this paper we study various aspects of porosities for conformal fractals. We first explore porosity in the general context of infinite graph directed Markov systems (GDMS), and we show that their limit sets are porous in large (in the sense of category and dimension) subsets. We also provide natural geometric and dynamic conditions under which the limit set of a GDMS is upper porous or mean porous. On the other hand, we prove that if the limit set of a GDMS is not porous, then it is not porous almost everywhere. We also revisit porosity for finite graph directed Markov systems, and we provide checkable criteria which guarantee that limit sets have holes of relative size at every scale in a prescribed direction.

We then narrow our focus to systems associated to complex continued fractions with arbitrary alphabet and we provide a novel characterisation of porosity for their limit sets. Moreover, we introduce the notions of upper density and upper box dimension for subsets of Gaussian integers and we explore their connections to porosity. As applications we show that limit sets of complex continued fractions system whose alphabet is co-finite, or even a co-finite subset of the Gaussian primes, are not porous almost everywhere, while they are uniformly upper porous and mean porous almost everywhere.

We finally turn our attention to complex dynamics and we delve into porosity for Julia sets of meromorphic functions. We show that if the Julia set of a tame meromorphic function is not the whole complex plane then it is porous at a dense set of its points and it is almost everywhere mean porous with respect to natural ergodic measures. On the other hand, if the Julia set is not porous then it is not porous almost everywhere. In particular, if the function is elliptic we show that its Julia set is not porous at a dense set of its points.

MSC classification

Primary: 28A75: Length, area, volume, other geometric measure theory
Secondary: 28C10: Set functions and measures on topological groups or semigroups, Haar measures, invariant measures 35R03: Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc.
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 172 , Issue 2 , March 2022 , pp. 303 - 371
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Supported in part by the Simons Foundation Collaboration grant no. 521845.

‡

Supported in part by the Simons Foundation Collaboration grant no. 581668.

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