Abstract
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.
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References
O. Bandtlow, H.H. Rugh, Entropy continuity for interval maps with holes. Ergod. Theory Dyn. Syst. 1â26 (2017). Published online
H. Bruin, M.F. Demers, M. Todd, Hitting and escaping statistics: mixing, targets, and holes. Preprint (2016)
L. Bunimovich, A. Yurchenko, Where to place a hole to achieve a maximal escape rate. Isr. J. Math. 182, 229â252 (2011)
C. Carminati, G. Tiozzo, The local Hölder exponent for the dimension of invariant subsets of the circle. Ergod. Theory Dyn. Syst. 1â16. Published online: 08 March 2016
P. Collet, S. MartĂnez, B. Schmitt, The Yorke-Pianigiani measure and the asymptotic law on the limit Cantor set of expanding systems. Nonlinearity 7, 1437â1443 (1994)
P. Collet, S. MartĂnez, J. San MartĂn, Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems. Probability and Its Applications (Springer, Berlin, 2013)
M. Demers, Escape rates and physical measures for the infinite horizon Lorentz gas with holes. Dyn. Syst. 28, 393â422 (2013)
M. Demers, M. Todd, Equilibrium states, pressure and escape for multimodal maps with holes. Isr. J. Math. 221(1), 367â424 (2017)
M. Demers, L.-S. Young, Escape rates and conditionally invariant measures. Nonlinearity 19, 377â397 (2006)
M. Demers, H.-K. Zhang, A functional analytic approach to perturbations of the Lorentz gas. Commun. Math. Phys. 324(3), 767â830 (2013)
M. Demers, C. Ianzano, P. Mayer, P. Morfe, E. Yoo, Limiting distributions for countable state topological Markov chains with holes. Discrete Contin. Dyn. Syst. 37, 105â130 (2017)
M. Denker, M. UrbaĆski, Ergodic theory of equilibrium states for rational maps. Nonlinearity 4, 103â134 (1991)
A. Ferguson, M. Pollicott, Escape rates for Gibbs measures. Ergod. Theory Dyn. Syst. 32, 961â988 (2012)
G. Keller, Rare events, exponential hitting times and extremal indices via spectral perturbation. Dyn. Syst. Int. J. 27, 11â27 (2012)
G. Keller, M. Liverani, Stability of the spectrum for transfer operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28, 141â152 (1999)
G. Keller, M. Liverani, Rare events, escape rates and quasi-stationarity: some exact formulae. J. Stat. Phys. 135, 519â534 (2009)
C. Liverani, V. Maume-Deschamps, Lasota-Yorke maps with holes: conditionally invariant probability measures and invariant measures on the survivor set. Ann. Inst. H. PoincarĂ© Probab. Stat. 39, 385â412 (2003)
V. Lucarini, D. Faranda, A.C. Freitas, J.M. Freitas, M. Holland, T. Kuna, M. Nicol, M. Todd, S. Vaienti, Extremes and Recurrence in Dynamical Systems. Pure and Applied Mathematics: A Wiley Series of Texts, Monographs, and Tracts (Wiley, New York, 2016)
D. Mauldin, M. UrbaĆski, Dimensions, measures in infinite iterated function systems. Proc. Lond. Math. Soc. 73, 105â154 (1996)
D. Mauldin, M. UrbaĆski, Gibbs states on the symbolic space over an infinite alphabet. Isr. J. Math. 125, 93â130 (2001)
D. Mauldin, M. UrbaĆski, Graph Directed Markov Systems: Geometry and Dynamics of Limit Sets (Cambridge University Press, Cambridge, 2003)
V. Mayer, M. UrbaĆski, Geometric thermodynamical formalism and real analyticity for meromorphic functions of finite order. Ergod. Theory Dyn. Syst. 28, 915â946 (2008)
V. Mayer, M. UrbaĆski, Thermodynamical Formalism and Multifractal Analysis for Meromorphic Functions of Finite Order, vol. 203. Memoirs of American Mathematical Society, no. 954 (American Mathematical Society, Providence, RI, 2010)
L. Pawelec, M. UrbaĆski, A. Zdunik, Exponential distribution of return times for weakly Markov systems. Preprint (2016). arXiv:1605.06917
G. Pianigiani, Conditionally invariant measures and exponential decay. J. Math. Anal. Appl. 82, 75â88 (1981)
G. Pianigiani, J.A. Yorke, Expanding maps on sets which are almost invariant: decay and chaos. Trans. Am. Math. Soc. 252, 351â366 (1979)
M. Szostakiewicz, M. UrbaĆski, A. Zdunik, Fine inducing and equilibrium measures for rational functions of the Riemann sphere. Isr. J. Math. 210, 399â465 (2015)
M. UrbaĆski, Hausdorff dimension of invariant sets for expanding maps of a circle. Ergod. Theory Dyn. Syst. 6, 295â309 (1986)
M. UrbaĆski, Invariant subsets of expanding mappings of the circle. Ergod. Theory Dyn. Syst. 7, 627â645 (1987)
M. UrbaĆski, The Hausdorff dimension of the set of points with non-dense orbit under a hyperbolic dynamical system. Nonlinearity 4, 385â397 (1991)
M. UrbaĆski, Rational functions with no recurrent critical points. Ergod. Theory Dyn. Syst. 14, 391â414 (1994)
M. UrbaĆski, Geometry and ergodic theory of conformal non-recurrent dynamics. Ergod. Theory Dyn. Syst. 17, 1449â1476 (1997)
Acknowledgements
The authors wish to thank all the referees of this manuscript for their very valuable comments and suggestions that considerably influenced the final exposition and the content of the manuscript. The authors also wish to thank Tushar Das for supplying them with some relevant references and for fruitful conversations (with the second named author) about the topic of this manuscript.
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Pollicott, M., UrbaĆski, M. (2017). Introduction. In: Open Conformal Systems and Perturbations of Transfer Operators. Lecture Notes in Mathematics, vol 2206. Springer, Cham. https://doi.org/10.1007/978-3-319-72179-8_1
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DOI: https://doi.org/10.1007/978-3-319-72179-8_1
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