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Expanding maps of an interval with holes. (English) Zbl 1022.37024

Let \(T\) be an expanding map on [0,1], and let \(H\) be a finite union of pairwise disjoint open intervals. Define \(\Lambda =[0,1]\setminus (\bigcup_{n=0}^{\infty}T^{-n}H)\) (this is interpretet as the set of all \(x\) whose orbits never “fall” in the “holes” \(H\)), and consider the dynamics of \(T|_{\Lambda}\). A measure \(\mu\) is called conditionally invariant, if there exists a constant \(\lambda >0\) such that \(\mu (T^{-1}A)=\lambda \mu (A)\) holds for every Borel set \(A\subseteq [0,1]\). The authors assume that for each \(x\in [0,1]\) there is a \(y\in [0,1]\setminus H\) with \(Ty=x\), and another technical condition (these conditions are called genericity conditions on the holes by the authors). Moreover, they assume the existence of a mixing invariant measure whose density is bounded away from \(0\).
It is proved that there exists a unique absolutely continuous (with respect to the Lebesgue measure) conditionally invariant measure \(\mu\). Furthermore for a large class of probability measures it is proved, that starting with \(\nu_{0}=\nu\) (\(\nu\) belongs to the above mentioned class) and setting \(\nu_{n}(A)= \frac{1}{\nu_{n-1}([0,1])}\nu_{n-1}(T^{-1}(A))\) one obtains \(\nu_{n}\to\mu\). This generalizes a result of G. Pianigiani and J. A. Yorke [Trans. Am. Math. Soc. 252, 351-366 (1979; Zbl 0417.28010)], where an analogous result was obtained, if the map satisfies additionally the Markov property. Although these results could be derived using perturbation theory of the Perron-Frobenius operator developped e.g. in G. Keller and C. Liverani [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 28, 141-152 (1999; Zbl 0956.37003)] the authors choose a more elementary approach which is close to the classical proofs of the existence of absolutely continuous invariant measures for expanding maps without holes.
Finally, an invariant measure \(\overline{\mu}\) on \(\Lambda\) is constructed from \(\mu\).
Reviewer: Peter Raith (Wien)

MSC:

37E05 Dynamical systems involving maps of the interval
37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
37A05 Dynamical aspects of measure-preserving transformations
37A30 Ergodic theorems, spectral theory, Markov operators
47A35 Ergodic theory of linear operators
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