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...w.r.t. ν0. (µ0 is indeed the equilibrium state for logg on the “non-trapped” set Xnt := {x ∈ [0, 1] : T n x ̸∈ ◦ I0 ∀n ≥ 0}. ϕε is also the conditionally invariant density for the “hole” Iε, see e.g. =-=[18]-=-.) We need to check assumptions (A5) and (A6). First note that (3.2) ηε = sup |ν0(P0(ψ1Iε\I0 ‖ψ‖≤1 ))| = |λ0| ∫ sup ψ dν0 ‖ψ‖≤1 ∣ Iε\I0 ∣ ≤ |λ0| ν0(Iε \ I0). In particular, |ν0(P0−Pε)(ϕ0)| = |λ0| ∫ Iε...

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...in a much more simple, direct and compact way than has previously been available. In one dimension, piecewise expanding maps with non-Markov holes have been studied via a variety of approaches, [BC], =-=[LiM]-=-, [D1]; logistic maps with non-Markov holes were studied in [D2]. M n 3. Banach space embeddings We must start with the overdue exact definition of the family of admissible leaves Σ, which is a set of...

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... that the eigenvalue one is common to all the operators. If 3.1-(2) is not assumed, then the operator L" will always have one eigenvalue close to one, but the spectral radius could vary slightly,=-= see [60] for -=-such a situation. INVARIANT MEASURES AND THEIR PROPERTIES 25 Thus the operator L " is nothing else than a matrix. Accordingly its eigenvalues and eigenvectors can be explicitly computed and, if t...

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...Markov restriction on the holes was relaxed, but the results still used strongly the Markov partitions associated with Anosov diffeomorphisms. In low-dimensional settings, efforts to drop the Markov requirements on both the map and the holes have had some success for expanding maps of the interval. A spectral analysis of the transfer operator was performed in [BK] and the stability of the spectrum was established in [KL] for perturbations of expanding maps 2 including small holes. More constructive techniques using bounded variation and contraction mapping arguments have been used in [BC] and [LiM] to prove the existence and properties of a.c.c.i.m. All these results assume that the potential associated with the transfer operator has bounded variation. This brief survey highlights the classes of systems with holes which have been studied to date: expanding maps in one dimension; and in higher dimensions, systems which admit finite Markov partitions. These systems are all uniformly hyperbolic. The purpose of this paper is to develop a method to study dynamical systems with holes which relies on neither finite Markov partitions nor uniform hyperbolicity. The Markov extension is a flexible...

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...d ones. Basically so far the problems studied were on the existence of conditionally invariant measures, their properties, and the existence of the escape rates [4], [6], [9], [10], [11], [13], [14], =-=[27]-=-, [32], [34]. In this paper we address a natural question which, to the best of our knowledge, has not been studied so far. Obviously, if one enlarges a hole then the escape rate of the orbits will in...

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... Markov restriction on the holes was relaxed, but the results still used strongly the Markov partitions associated with Anosov diffeomorphisms. In low-dimensional settings, efforts to drop the Markov requirements on both the map and the holes have had some success for expanding maps of the interval. A spectral analysis of the transfer operator was performed in [BK] and the stability of the spectrum was established in [KL] for perturbations of expanding maps including small holes. More constructive techniques using bounded variation and contraction mapping arguments have been used in [BCh] and [LiM] to prove the existence and properties of conditionally invariant measures. Markov extensions were used in [D] to drop some of the earlier technical requirements and limit only the size of the holes. This brief survey highlights the classes of systems with holes which have been studied to date: expanding maps in one dimension; and in higher dimensions, systems which admit finite Markov partitions. These systems are all uniformly hyperbolic. In this paper, we seek to understand the escape dynamics of a class of open systems which are not uniformly hyperbolic, but which do exhibit exponential re...