## 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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### References:

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