## On the rate of convergence to equilibrium in one-dimensional systems.(English)Zbl 0576.58016

Author’s summary: We determine the essential spectral radius of the Perron-Frobenius-operator for piecewise expanding transformations considered as an operator on the space of functions of bounded variation and relate the speed of convergence of equilibrium in such one- dimensional systems to the greatest eigenvalues of generalized Perron- Frobenius-operators of the transformations (operators which yield singular invariant measures).
Since this paper appeared, the isolated eigenvalues of such Perron- Frobenius operators have been characterized for several classes of transformations by the poles of the corresponding Ruelle zeta-functions [see F. Hofbauer and the author, J. Reine Angew. Math. 352, 100-113 (1984; Zbl 0533.28011); ”Markov extensions, zeta-functions, and Fredholm theory for piecewise invertible dynamical systems”, Univ. Heidelberg (preprint)].

### MSC:

 37A99 Ergodic theory 28D05 Measure-preserving transformations 47A10 Spectrum, resolvent 37D99 Dynamical systems with hyperbolic behavior 37A30 Ergodic theorems, spectral theory, Markov operators

Zbl 0533.28011
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### References:

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