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2017 | OriginalPaper | Buchkapitel

4. Escape Rates for Conformal GDMSs and IFSs

verfasst von : Mark Pollicott, Mariusz Urbański

Erschienen in: Open Conformal Systems and Perturbations of Transfer Operators

Verlag: Springer International Publishing

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Abstract

This is the first chapter where we go beyond symbolic dynamics dealing with open conformal dynamical systems generated by Euclidean open balls. More precisely, we study the survivor set and the escape rates along with their asymptotics in the case of conformal countable alphabet (strictly contracting) Graph Directed Markov Systems (GDMS)s and Iterated Function Systems (IFS)s of Mauldin and Urbański, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets, Cambridge University Press, Cambridge, 2003. We prove the asymptotic existence of escape rates generated by Euclidean balls for both, equilibrium measures μ φ and Hausdorff dimension. As a matter of fact, we provide closed formulas for the values of these rates. From these results we deduce their analogues for parabolic GDMSs and IFSs. Our approach to proving results on escape rates for conformal graph directed Markov systems and conformal iterated function systems is based on the symbolic dynamics, more precisely, the symbolic thermodynamic formalism, developed in the preceding sections. In order to deal with asymptotics of Hausdorff dimension, we consider the geometric potentials \(t\zeta (\omega )= t\log |\varphi ^{\prime }_{\omega _0}(\pi (\sigma (\omega )))|\) and, essentially staying on the symbolic level, undertake a very technical task of calculating the asymptotics \(\lambda _n^{\prime }(t)\) and \(\lambda _n^{\prime \prime }(t)\) of the first and second derivatives of the leading eigenvalues of the corresponding perturbed Perron–Frobenius operators \({\mathbb {L}}_n\). We also use heavily thin annuli properties of appropriate equilibrium measures, which is possible thanks to the progress done in Pawelec et al. (Exponential distribution of return times for weakly Markov systems. Preprint 2016, arXiv:1605.06917).

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Fußnoten
1
As the relevant proof in [34] shows, the Geometric Condition (e) is not needed at all for this theorem; comp. Remark 4.1.12.
 
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Metadaten
Titel
Escape Rates for Conformal GDMSs and IFSs
verfasst von
Mark Pollicott
Mariusz Urbański
Copyright-Jahr
2017
DOI
https://doi.org/10.1007/978-3-319-72179-8_4