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Explicit formulas for the average density of conformal repellers

Published online by Cambridge University Press:  04 July 2006

LUIS BARREIRA
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (e-mail: barreira@math.ist.utl.pt, vsaraiva@math.ist.utl.pthttp://www.math.ist.utl.pt/˜barreira/)
VITOR SARAIVA
Affiliation:
Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (e-mail: barreira@math.ist.utl.pt, vsaraiva@math.ist.utl.pthttp://www.math.ist.utl.pt/˜barreira/)

Abstract

We obtain explicit formulas for the average density, also called order-two density, of an arbitrary conformal repeller of a $C^{1+\epsilon}$ transformation. We note that unlike with the pointwise densities, the average densities exist almost everywhere on each repeller, and are in fact constant almost everywhere. Thus, they are natural parameters to describe the geometric structure of the corresponding invariant sets, and as such it is of interest to obtain explicit formulas for their almost constant value. We obtain simultaneously formulas without using symbolic dynamics, and formulas based on the symbolic dynamics generated by a given Markov partition. The proofs strongly depend on the use of appropriate Markov partitions, and are also based on methods of Falconer and of Zähle.

Type
Research Article
Copyright
2006 Cambridge University Press

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