Summary
We prove the quasi-compactness of the Perron-Frobenius operator of piecewise monotonic transformations when the inverse of the derivative is Hölder-continuous or, more generally, of bounded p-variation.
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Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. 470. Berlin-Heidelberg-New York: Springer 1975
Bowen, R.: Bernoulli maps of the interval. Isr. J. Math. 28, 161–168 (1978)
Bruneau, M.: Variation totale d'une fonction. Lecture Notes in Math. 413. Berlin-Heidelberg-New York: Springer 1974
Denker, M., Keller, G.: Simulations for invariant measures of dynamical systems. Preprint
Dieudonné, J.: Treatise on analysis, Vol. II. New York: Academic Press 1970
Dunford, N., Schwartz, J.T.: Linear operators, Part I. New York: Interscience 1958
Hofbauer, F., Keller, G.: Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180, 119–140 (1982)
Hofbauer, F., Keller, G.: Equilibrium states for piecewise monotonic transformations. Ergodic Theory Dyn. Syst. 2, 23–43 (1982)
Ionescu-Tulcea, C., Marinescu, G.: Théorie ergodique pour des classes d'opérations non complètement continues. Ann. Math. (2) 52, 140–147 (1950)
Lasota, A., Yorke, J.A.: On the existence of invariant measures for piecewise monotonic transformations. Trans. Am. Math. Soc. 186, 481–488 (1973)
Oxtoby, J.C.: Measure and category. Graduate texts in Math. Vol.2. Berlin-Heidelberg-New York: Springer 1971
Rychlik, M.: Bounded variation and invariant measures. Studia math. 76, 69–80 (1983)
Schaefer, H.H.: Topological Vector Spaces. New York: Mac Millan 1966
Suquet, P.: Fonctions à variation bornée sur un ouvert de ℝN. Sém. d'Analyse Convexe, Montpellier 1978, Exposé N∘4.
Wong, S.: Some metric properties of piecewise monotonic mappings of the unit interval. Trans. Am. Math. Soc. 246, 493–500 (1978)
Wong, S.: Hölder continuous derivatives and ergodic theory. J. Lond. Math. Soc. 22, 506–520 (1980)
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This work has been supported by the Deutsche Forschungsgemeinschaft
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Keller, G. Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrscheinlichkeitstheorie verw Gebiete 69, 461–478 (1985). https://doi.org/10.1007/BF00532744
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DOI: https://doi.org/10.1007/BF00532744