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We all want to maximize our gains and minimize our losses, but decisions have uncertain outcomes. What if you could choose between an expected return of $1000 with no chance of losing any amount, or an expected return of $5000 with a chance of losing $50,000. Which would you choose? The answer depends upon how risk-averse you are. Many would happily take(More)
The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors. Abstract For piecewise monotone(More)
We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the(More)
– For piecewise expanding one-dimensional maps without periodic turning points we prove that isolated eigenvalues of small (random) perturbations of these maps are close to isolated eigenvalues of the unperturbed system. (Here " eigenvalue " means eigenvalue of the corresponding Perron-Frobenius operator acting on the space of functions of bounded variation(More)
Let X R 2 be a nite union of bounded polytopes and let T : X ! X be piecewise aane and eventually expanding. Then the Perron-Probenius operator L of T is quasicompact as an operator on the space of functions of bounded variation on R 2 and its isolated eigenvalues (including multiplicities) are just the reciprocals of the poles of the dynamical zeta(More)
We construct a mixing continuous piecewise linear map on [−1, 1] with the property that a two-dimensional lattice made of these maps with a linear north and east nearest neighbour coupling admits a phase transition. We also provide a modification of this construction where the local map is an expanding analytic circle map. The basic strategy is borroughed(More)
– We prove stochastic stability of chaotic maps for a general class of Markov random perturbations (including singular ones) satisfying some kind of mixing conditions. One of the consequences of this statement is the proof of Ulam's conjecture about the approximation of the dynamics of a chaotic system by a finite state Markov chain. Conditions under which(More)
In this note we give a computable criterion for a piecewise expanding interval map T to be mixing, which at the same time not only establishes explicit bounds on the spectral gap of the associated Perron Frobenius operator acting on the space of functions of bounded variation, but establishes strict contraction rates for this operator. Of course such a(More)