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The focus of this book is on open conformal dynamical systems corresponding to the escape of a point through an open Euclidean ball. The ultimate goal is to understand the asymptotic behavior of the escape rate as the radius of the ball tends to zero. In the case of hyperbolic conformal systems this has been addressed by various authors. The conformal maps considered in this book are far more general, and the analysis correspondingly more involved.

The asymptotic existence of escape rates is proved and they are calculated in the context of (finite or infinite) countable alphabets, uniformly contracting conformal graph-directed Markov systems, and in particular, conformal countable alphabet iterated function systems. These results have direct applications to interval maps, rational functions and meromorphic maps.

Towards this goal the authors develop, on a purely symbolic level, a theory of singular perturbations of Perron--Frobenius (transfer) operators associated with countable alphabet subshifts of finite type and Hölder continuous summable potentials. This leads to a fairly full account of the structure of the corresponding open dynamical systems and their associated surviving sets.



Chapter 1. Introduction

We give an introduction to the main ideas, techniques, and results in this book. We discuss their relations with the relevant previous research on open systems.
Mark Pollicott, Mariusz Urbański

Chapter 2. Singular Perturbations of Classical Original Perron–Frobenius Operators on Countable Alphabet Symbol Spaces

In this chapter we first recall, from Mauldin and Urbański (Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets. Cambridge University Press, Cambridge, 2003), the basics of thermodynamic formalism on a countable alphabet symbol space. We present means the concept and various properties of Perron–Frobenius (transfer) operators. Then we define a non-standard Banach space on which such operators also act. We endow it with two non-standard norms. Using Keller–Liverani Perturbation Theorem from Appendix A, we then deal at length with singular perturbations of Perron–Frobenius operators from a “strong” norm to a “weak” one. We prove the Fundamental Perturbative Result, Proposition 2.4.2. This result, fundamental for all what follows in the manuscript, enables us in particular, at the end of the chapter, to calculate asymptotic behavior of leading eigenvalues of perturbed operators. This is the key starting point for our further results on the behavior of escape rates.
Mark Pollicott, Mariusz Urbański

Chapter 3. Symbol Escape Rates and the Survivor Set K(U n )

This chapter is still of purely symbolic character. We consider open symbolic dynamical systems generated by carefully chosen, with the intent of applying them in further chapters to conformal dynamical systems, open sets, denoted by U n . So, as a matter of fact, we deal with a sequence of holes U n . In the first section we define escape rates, prove their existence for the open systems generated by the holes U n (equality of the lower and upper escape rates), and relate them to the eigenvalues of the perturbed operators \({\mathcal {L}}_n\). We also prove the existence and restricted uniqueness of conditional invariant measures absolutely continuous with respect to the original equilibrium state μ φ , on \(U_n^c\), the complements of the holes U n . The majority of this chapter is though devoted to study the shift-invariant survivor set K(U n ) and the symbolic dynamics it generates. For all n ≥ 1 large enough we prove the Variational Principle for the dynamical system σ : K(U n ) → K(U n ) and the potential \(\varphi |{ }_{K(U_n)}\). This is not trivial for at least two reasons. Firstly, the sets K(U n ) need not be compact and so, the classical version of the Variational Principle (Bowen, Equilibrium States and the Ergodic Theory for Anosov Diffeomorphisms, Springer, Berlin, 1975; Ruelle, Thermodynamic Formalism. Addison-Wesley, Reading, 1978; Walters, An Introduction to Ergodic Theory. Springer, Berlin, 1982; Parry and Pollicott, Asterisque 268, 1990, or Przytycki and Urbański, Conformal Fractals Ergodic Theory Methods. Cambridge University Press, Cambridge, 2010 for ex.) does not apply. Secondly, since we do not know whether the subshifts σ : K(U n ) → K(U n ) are of finite type, nor even whether these are topologically mixing, the theory of Mauldin and Urbański (Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets. Cambridge University Press, Cambridge, 2003) does not apply either. Our primary tool is then the original and perturbed Perron–Frobenius operator, particularly the spectral decomposition of the Fundamental Perturbative Result, Proposition 2.​4.​2. We also prove the existence and uniqueness of equilibrium states for the “surviving” dynamical system σ : K(U n ) → K(U n ) with the potential \(\varphi |{ }_{K(U_n)}\). It turns out to be the functional μ n of Corollary 2.​4.​5, which we are able to extend to an ordinary Borel probability measure on K(U n ). Assuming the standard normalization, we show that the negative of the corresponding topological pressure on the survivor set K(U n ) is equal to the escape rates generated by the holes U n . Finally, we prove strong stochastic properties of this dynamical system, such as exponential decay of correlations and an Almost Sure Invariance Principle. The appropriate versions of the Central Limit Theorem and the Law of Iterated Logarithm then follow.
Mark Pollicott, Mariusz Urbański

Chapter 4. Escape Rates for Conformal GDMSs and IFSs

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).
Mark Pollicott, Mariusz Urbański

Chapter 5. Applications: Escape Rates for Multimodal Maps and One-Dimensional Complex Dynamics

In this chapter we apply our previous results to study escape rates for multimodal maps of an interval and one-dimensional complex dynamics. The former includes Topological Collet–Eckmann and subexpanding maps while the latter includes all rational functions of the Riemann sphere and a large class of transcendental meromorphic functions on the complex plane. The tool which makes such applications possible is the first return map defined on a sufficiently “good” set, ex. a nice set in the context of conformal dynamics. In the first three sections, under appropriate hypotheses that include an appropriate form of large deviations, we relate the escape rates, both for equilibrium measures and Hausdorff dimension, of the original systems and the induced ones. This is done in a fairly general abstract setting. In the last three sections, making use of the first three ones, and the previous chapter, we prove the existence of asymptotic escape rates generated by Euclidean balls for both, equilibrium measures μ φ and Hausdorff dimension, for the above mentioned 1-dimensional systems. We furthermore provide closed formulas for the values of these rates.
Mark Pollicott, Mariusz Urbański


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