A new variation of Bowen's formula for graph directed Markov systems

Mario Roy

doi:10.3934/dcds.2012.32.2533

Article Contents

2012, Volume 32, Issue 7: 2533-2551. Doi: 10.3934/dcds.2012.32.2533

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A new variation of Bowen's formula for graph directed Markov systems

Mario Roy^1,

1.
Glendon College, York University, 2275 Bayview Avenue, Toronto, M4N 3M6

Received: November 30, 2009

Revised: April 30, 2010

Published: June 2012

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Abstract

We introduce a new variation of Bowen's formula for conformal graph directed Markov systems (a.k.a. CGDMSs). This new variation applies to a very large collection of non-irreducible systems and is shown to coincide with the well-known Bowen's formula that holds for all finite or finitely irreducible CGDMSs (cf. [2], [4] and [1]). We further show that the original version of Bowen's formula may not hold even for non-irreducible CGDMSs whose components are IFSs, justifying thereby the introduction of a new variation. This answers two questions that were raised by Ghenciu and Mauldin in [1]. Their third question is also %partially tackled. addressed. Indeed, we prove that Ghenciu and Mauldin's conjecture about the finiteness parameters of the partition functions of the pressure is false even within the class of irreducible systems.

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Mathematics Subject Classification: Primary: 37D35.

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References

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[5]	R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.doi: 10.1090/S0002-9947-1988-0961615-4.