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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

Mario Roy 1,

1. 

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

Received  December 2009 Revised  May 2010 Published  March 2012

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 & Continuous Dynamical Systems - A, 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, ().   Google Scholar

[2]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems,, Proc. London Math. Soc. (3), 73 (1996), 105.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar

[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.  doi: 10.1090/S0002-9947-99-02268-0.  Google Scholar

[4]

R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,", Cambridge Tracts in Mathematics, 148 (2003).   Google Scholar

[5]

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

show all references

References:
[1]

A. Ghenciu and R. D. Mauldin, Conformal graph directed Markov systems,, preprint, ().   Google Scholar

[2]

R. D. Mauldin and M. Urbański, Dimensions and measures in infinite iterated function systems,, Proc. London Math. Soc. (3), 73 (1996), 105.  doi: 10.1112/plms/s3-73.1.105.  Google Scholar

[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.  doi: 10.1090/S0002-9947-99-02268-0.  Google Scholar

[4]

R. D. Mauldin and M. Urbański, "Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets,", Cambridge Tracts in Mathematics, 148 (2003).   Google Scholar

[5]

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

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