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

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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.
    Mathematics Subject Classification: Primary: 37D35.

    Citation:

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  • [1]

    A. Ghenciu and R. D. MauldinConformal 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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