Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-29T22:51:57.651Z Has data issue: false hasContentIssue false

Metric invariance entropy and conditionally invariant measures

Published online by Cambridge University Press:  20 October 2016

FRITZ COLONIUS*
Affiliation:
Institut für Mathematik, Universität Augsburg, Augsburg, Germany email fritz.colonius@math.uni-augsburg.de

Abstract

Two notions of metric invariance entropy are constructed with respect to conditionally invariant measures for control systems in discrete time and it is shown that they are invariant under conjugacies.

Type
Original Article
Copyright
© Cambridge University Press, 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adler, R., Konheim, A. and McAndrew, M.. Topological entropy. Trans. Amer. Math. Soc. 114 (1965), 6185.Google Scholar
Collett, P., Martinez, S. and San Martin, J.. Quasi-Stationary Distributions: Markov Chains, Diffusions, and Dynamical Systems. Springer, Berlin, 2013.Google Scholar
Colonius, F., Kawan, C. and Nair, G.. A note on topological feedback entropy and invariance entropy. Systems Control Lett. 62 (2013), 377381.Google Scholar
da Silva, A. and Kawan, C.. Invariance entropy of hyperbolic control sets. Discrete Contin. Dyn. Syst. A 36 (2016), 97136.Google Scholar
Delchamps, D.. Stabilizing a linear system with quantized state feedback. IEEE Trans. Automat. Control 35 (1990), 916924.Google Scholar
Demers, M. F. and Young, L.-S.. Escape rates and conditionally invariant measures. Nonlinearity 19 (2006), 377397.Google Scholar
Downarowicz, T.. Entropy in Dynamical Systems. Cambridge University Press, Cambridge, 2011.Google Scholar
Dunford, N. and Schwartz, J. T.. Linear Operators, Part I: General Theory. Wiley-Interscience, New York, 1977.Google Scholar
Gänssler, P. and Stute, W.. Wahrscheinlichkeitstheorie. Springer, Berlin, 1977.Google Scholar
Kawan, C.. Lower bounds for the strict invariance entropy. Nonlinearity 24 (2011), 19091935.CrossRefGoogle Scholar
Kawan, C.. Invariance Entropy for Deterministic Control Systems. An Introduction (Lecture Notes in Mathematics, 2089) . Springer International, Cham, 2013.Google Scholar
Keller, G. and Liverani, C.. Rare events, escape rates, quasistationarity: some exact formulae. J. Stat. Phys. 135 (2009), 519534.Google Scholar
Kifer, Y.. Ergodic Theory of Random Transformations. Birkhäuser, Boston, 1986.Google Scholar
Mehta, P., Vaidya, U. and Banaszuk, A.. Markov chains, entropy, and fundamental limitations in nonlinear stabilization. IEEE Trans. Automat. Control 53 (2008), 784791.Google Scholar
Nair, G., Evans, R. J., Mareels, I. and Moran, W.. Topological feedback entropy and nonlinear stabilization. IEEE Trans. Automat. Control 49 (2004), 15851597.Google Scholar
Oikhberg, T. and Troitsky, V.. A theorem of Krein revisited. Rocky Mountain J. Math. 35 (2005), 195210.Google Scholar
Pianigiani, G. and Yorke, J. A.. Expanding maps on sets which are almost invariant: decay and chaos. Trans. Amer. Math. Soc. 252 (1979), 351366.Google Scholar
Walters, P.. An Introduction to Ergodic Theory. Springer, New York, 1982.Google Scholar
Zmarrou, H. and Homburg, A. J.. Bifurcations of stationary measures of random diffeomorphisms. Ergod. Th. & Dynam. Sys. 27 (2007), 16511692.Google Scholar