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Erschienen in:
2017 | OriginalPaper | Buchkapitel
2. Singular Perturbations of Classical Original Perron–Frobenius Operators on Countable Alphabet Symbol Spaces
verfasst von : Mark Pollicott, Mariusz Urbański
Erschienen in: Open Conformal Systems and Perturbations of Transfer Operators
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Abstract
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.