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Transfer operator for piecewise affine approximations of interval maps. (English) Zbl 0827.58014

A transformation of the unit interval is called Markov if there exist disjoint open intervals \(I_1, I_2, \dots, I_\ell\), the union of whose closures is [0,1] such that the restriction of \(f\) to each \(I_j\) is monotone and continuous, and such that the closure of each \(f(I_j)\) is the closure of a union of intervals \(I_k\). When \(f\) is a Markov transformation, one may study the dynamical system generated by the iterations of \(f\) using symbolic dynamics and transfer operators as in equilibrium statistical mechanics. However, the statistical properties of the orbits of \(f\) can only be described fully in terms of Markov chains if the restriction of the Markov map to each interval \(I_j\) is affine. Thus, it is desirable to approximate \(f\) with a sequence of piecewise affine Markov maps.
The authors solve this problem constructively. They obtain a sequence of finite Markov stochastic matrices whose normalized eigenvectors with eigenvalue 1 approach the stationary probability density of \(f\) exponentially fast in the uniform norm.

MSC:

37E99 Low-dimensional dynamical systems
82B31 Stochastic methods applied to problems in equilibrium statistical mechanics
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