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

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July 2012, 32(7): 2533-2551. doi: 10.3934/dcds.2012.32.2533

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

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

Received December 2009 Revised May 2010 Published March 2012

Abstract
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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.

Keywords: thermodynamic formalism, pressure, Bowen's formula, Hausdorff dimension., Iterated function systems, graph directed Markov systems.

Mathematics Subject Classification: Primary: 37D3.

Citation: Mario Roy. A new variation of Bowen's formula for graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2533-2551. doi: 10.3934/dcds.2012.32.2533

References:

[1]	A. Ghenciu and R. D. Mauldin, Conformal graph directed Markov systems, preprint, arXiv:0711.1182.
[2]	R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73 (1996), 105-154. doi: 10.1112/plms/s3-73.1.105.
[3]	R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995-5025. doi: 10.1090/S0002-9947-99-02268-0.
[4]	R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets," Cambridge Tracts in Mathematics, 148, Cambridge University Press, Cambridge, 2003.
[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.

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References:

[1]	A. Ghenciu and R. D. Mauldin, Conformal graph directed Markov systems, preprint, arXiv:0711.1182.
[2]	R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 73 (1996), 105-154. doi: 10.1112/plms/s3-73.1.105.
[3]	R. D. Mauldin and M. Urbański, Conformal iterated function systems with applications to the geometry of continued fractions, Trans. Amer. Math. Soc., 351 (1999), 4995-5025. doi: 10.1090/S0002-9947-99-02268-0.
[4]	R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets," Cambridge Tracts in Mathematics, 148, Cambridge University Press, Cambridge, 2003.
[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.

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