Invariant densities for C¹ maps

• Autorzy: Anthony N. Quas
• Języki publikacji: EN

Abstrakty

EN

We consider the set of $C^1$ expanding maps of the circle which have a unique absolutely continuous invariant probability measure whose density is unbounded, and show that this set is dense in the space of $C^1$ expanding maps with the $C^1$ topology. This is in contrast with results for $C^2$ or $C^{1+ε}$ maps, where the invariant densities can be shown to be continuous.

Słowa kluczowe

EN

cocycle; expanding map; invariant density; absolutely continuous invariant measure

Kategorie tematyczne

• 28D05: Measure-preserving transformations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 120
• Numer: 1
• Strony: 83-88

Daty

wydano

1996

otrzymano

1996-03-04

Twórcy

autor

Anthony N. Quas

• Statistical Laboratory, Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge CB2 1SB, England

Bibliografia

• [1] P. Góra and B. Schmitt, Un exemple de transformation dilatante et $C^1$ par morceaux de l'intervalle, sans probabilité absolument continue invariante, Ergodic Theory Dynam. Systems 9 (1989), 101-113.
• [2] G. R. Grimmett and D. R. Stirzaker, Probability and Random Processes, 2nd ed., Oxford Univ. Press, Oxford, 1992.
• [3] K. Krzyżewski, A remark on expanding mappings, Colloq. Math. 41 (1979), 291-295.
• [4] R. Mañé, Ergodic Theory and Differentiable Dynamics, Springer, New York, 1988.
• [5] A. N. Quas, Non-ergodicity for $C^1$ expanding maps and g-measures, Ergodic Theory Dynam. Systems 16 (1996), 1-13.
• [6] A. N. Quas, A $C^1$ expanding map of the circle which is not weak-mixing, Israel J. Math. 93 (1996), 359-372.