Skip to main content
×
×
Home

Escape rates for special flows and their higher order asymptotics

  • FABIAN DREHER (a1) and MARC KESSEBÖHMER (a1)
Abstract

In this paper escape rates and local escape rates for special flows are studied. In a general context the first result is that the escape rate depends monotonically on the ceiling function and fulfils certain scaling, invariance and continuity properties. In the context of metric spaces local escape rates are considered. If the base transformation is ergodic and exhibits an exponential convergence in probability of ergodic sums, then the local escape rate with respect to the flow is just the local escape rate with respect to the base transformation, divided by the integral of the ceiling function. Also, a reformulation with respect to induced pressure is presented. Finally, under additional regularity conditions higher order asymptotics for the local escape rate are established.

Copyright
References
Hide All
[Amb41] Ambrose, W.. Representation of ergodic flows. Ann. of Math. (2) 42(3) (1941), 723739.
[BJP14] Bandtlow, O. F., Jenkinson, O. and Pollicott, M.. Periodic points, escape rates and escape measures. Ergodic Theory, Open Dynamics, and Coherent Structures (Springer Proceedings in Mathematics & Statistics, 70) . Eds. Bahsoun, W., Bose, C. and Froyland, G.. Springer, New York, 2014, pp. 4158.
[BY11] Bunimovich, L. A. and Yurchenko, A.. Where to place a hole to achieve a maximal escape rate. Israel J. Math. 182(1) (2011), 229252.
[Cip15] Cipriano, I.. Entry times, escape rates and smoothness of stationary measures. PhD Thesis, University of Warwick, 2015.
[CKD13] Cristadoro, G., Knight, G. and Degli Esposti, M.. Follow the fugitive: an application of the method of images to open systems. J. Phys. A 46(27) (2013), 272001.
[CMS97] Collet, P., Martnez, S. and Schmitt, B.. The Pianigiani–Yorke measure for topological Markov chains. Israel J. Math. 97(1) (1997), 6170.
[Dre15] Dreher, F.. Über Ausströmraten spezieller Flüsse. PhD Thesis, Universität Bremen, 2015.
[Ell85] Ellis, R. S.. Entropy, Large Deviation, and Statistical Mechanics (Grundlehren der mathematischen Wissenschaften, 271) . Springer, New York, 1985.
[FP12] Ferguson, A. and Pollicott, M.. Escape rates for Gibbs measures. Ergod. Th. & Dynam. Sys. 32(3) (2012), 961988.
[Gur65] Gurevich, B. M.. Construction of increasing partitions for special flows. Theory Probab. Appl. 10(4) (1965), 627645.
[Jac60] Jacobs, K.. Neuere Methoden und Ergebnisse der Ergodentheorie (Ergebnisse der Mathematik und ihrer Grenzgebiete, 29) . Springer, Berlin, 1960.
[JKL14] Jaerisch, J., Kesseböhmer, M. and Lamei, S.. Induced topological pressure for countable state Markov shifts. Stoch. Dyn. 14(2) (2014), 1350016-1–31.
[Kea72] Keane, M.. Strongly mixing g-measures. Invent. Math. 16 (1972), 309324.
[Kes01] Kesseböhmer, M.. Large deviation for weak Gibbs measures and multifractal spectra. Nonlinearity 14(2) (2001), 395409.
[KL09] Keller, G. and Liverani, C.. Rare events, escape rates and quasistationarity: some exact formulae. J. Stat. Phys. 135(3) (2009), 519534.
[KT61] Katz, M. and Thomasian, A. J.. A bound for the law of large numbers for discrete Markov processes. Ann. Math. Statist. 32(1) (1961), 336337.
[Led74] Ledrappier, F.. Principe variationnel et systèmes dynamiques symboliques. Z. Wahrscheinlichkeitsth. Verw. Geb. 30 (1974), 185202.
[Lin89] Lind, D. A.. Perturbations of shifts of finite type. SIAM J. Discrete Math. 2(3) (1989), 350365.
[LMD03] Liverani, C. and Maume-Deschamps, V.. Lasota–Yorke maps with holes: conditionally invariant probability measures and invariant probability measures on the survivor set. Ann. Inst. Henri Poincaré (B) Probab. Stat. 39(3) (2003), 385412.
[MN91] Magnus, J. R. and Neudecker, H.. Matrix Differential Calculus with Applications in Statistics and Econometrics (Wiley Series in Probability and Mathematical Statistics) , Revised edn. John Wiley, Chichester, 1991.
[PY79] Pianigiani, G. and Yorke, J. A.. Expanding maps on sets which are almost invariant. Decay and chaos. Trans. Amer. Math. Soc. 252 (1979), 351366.
[Roc72] Rockafellar, R. T.. Convex Analysis (Princeton Mathematical Series, 28) , Second printing edn. Princeton University Press, Princeton, NJ, 1972.
[Rou12] Rousseau, J.. Recurrence rates for observations of flows. Ergod. Th. & Dynam. Sys. 32(5) (2012), 17271751.
[Wal00] Walters, P.. An Introduction to Ergodic Theory (Graduate Texts in Mathematics, 79) , First softcover printing edn. Springer, New York, 2000.
[Yur98] Yuri, M.. Zeta functions for certain non-hyperbolic systems and topological Markov approximations. Ergod. Th. & Dynam. Sys. 18 (1998), 15891612.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 21 *
Loading metrics...

Abstract views

Total abstract views: 138 *
Loading metrics...

* Views captured on Cambridge Core between 25th September 2017 - 14th August 2018. This data will be updated every 24 hours.