Markov extensions for dynamical systems with holes: an application to expanding maps of the interval. (English) Zbl 1079.37031
A dynamical system with holes is one for which trajectories falling into a sizable set cannot be continued. From the statistical point of view, an invariant measure cannot exist and one looks for conditionally invariant measures for which the loss of measure falling into the “holes” is balanced with a normalization. The paper establishes the existence of absolutely continuous conditionally invariant measures for a class of interval maps with holes, otherwise similar to the classical Lasota-Yorke maps without holes. The proof proceeds by constructing an induced “tower with holes” and the related Markov extension of the original system.
This is a powerful method in the classical case and one might expect it to be widely applicable in the case with holes as well, and indeed the author announces results in a forthcoming paper for the logistic map with holes. Given the high level of both the paper itself and the journal in which it was published, I was amazed to see the name of the “Perron-Frobenius” operator misspelled.
This is a powerful method in the classical case and one might expect it to be widely applicable in the case with holes as well, and indeed the author announces results in a forthcoming paper for the logistic map with holes. Given the high level of both the paper itself and the journal in which it was published, I was amazed to see the name of the “Perron-Frobenius” operator misspelled.
Reviewer: Grzegorz Swiatek (University Park)
MSC:
| 37E05 | Dynamical systems involving maps of the interval |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 37F15 | Expanding holomorphic maps; hyperbolicity; structural stability of holomorphic dynamical systems |
| 37D20 | Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) |
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