Generalized bounded variation and applications to piecewise monotonic transformations. (English) Zbl 0574.28014
Authors 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.
Additional remark: The result has consequences for the study of the one- dimensional Poincaré-map of the Lorenz-flow as pointed out by [S. Wong: J. Lond. Math. Soc., II. Ser. 22, 506-520 (1980; Zbl 0459.28022)] and [C. Robinson, Ergodic Theory Dyn. Syst. 4, 605-611 (1984; Zbl 0558.28011)]. Robinson also attempts to give a much shorter proof of the above result (reducing it to a result of Wong). However, his proof is wrong.
Additional remark: The result has consequences for the study of the one- dimensional Poincaré-map of the Lorenz-flow as pointed out by [S. Wong: J. Lond. Math. Soc., II. Ser. 22, 506-520 (1980; Zbl 0459.28022)] and [C. Robinson, Ergodic Theory Dyn. Syst. 4, 605-611 (1984; Zbl 0558.28011)]. Robinson also attempts to give a much shorter proof of the above result (reducing it to a result of Wong). However, his proof is wrong.
MSC:
28D05 | Measure-preserving transformations |
37A99 | Ergodic theory |
47A35 | Ergodic theory of linear operators |
60F05 | Central limit and other weak theorems |
26A45 | Functions of bounded variation, generalizations |
Keywords:
quasi-compactness of the Perron-Frobenius operator of piecewise monotonic transformations; inverse of the derivative; Hölder continuous; bounded p-variation; one-dimensional Poincaré-map of the Lorenz-flowReferences:
[1] | Bowen, R., Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Math. 470 (1975), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0308.28010 |
[2] | Bowen, R., Bernoulli maps of the interval, Isr. J. Math., 28, 161-168 (1978) · Zbl 0377.28010 |
[3] | Bruneau, M., Variation totale d’une fonction, Lecture Notes in Math. 413 (1974), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0288.26005 |
[4] | Denker, M., Keller, G.: Simulations for invariant measures of dynamical systems. Preprint · Zbl 0581.58027 |
[5] | Dieudonné, J., Treatise on analysis, Vol. II (1970), New York: Academic Press, New York · Zbl 0202.04901 |
[6] | Dunford, N.; Schwartz, J. T., Linear operators, Part I. (1958), New York: Interscience, New York · Zbl 0084.10402 |
[7] | Hofbauer, F.; Keller, G., Ergodic properties of invariant measures for piecewise monotonic transformations, Math. Z., 180, 119-140 (1982) · Zbl 0485.28016 |
[8] | Hofbauer, F.; Keller, G., Equilibrium states for piecewise monotonic transformations, Ergodic Theory Dyn. Syst., 2, 23-43 (1982) · Zbl 0499.28012 |
[9] | Ionescu-Tulcea, C.; Marinescu, G., Théorie ergodique pour des classes d’opérations non complètement continues, Ann. Math. (2), 52, 140-147 (1950) · Zbl 0040.06502 |
[10] | Lasota, A.; Yorke, J. A., On the existence of invariant measures for piecewise monotonic transformations, Trans. Am. Math. Soc., 186, 481-488 (1973) · Zbl 0298.28015 |
[11] | Oxtoby, J. C., Measure and category, Graduate texts in Math. Vol.2 (1971), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0217.09201 |
[12] | Rychlik, M., Bounded variation and invariant measures, Studia math., 76, 69-80 (1983) · Zbl 0575.28011 |
[13] | Schaefer, H. H., Topological Vector Spaces (1966), New York: Mac Millan, New York · Zbl 0141.30503 |
[14] | Suquet, P.: Fonctions à variation bornée sur un ouvert de ℝN. Sém. d’Analyse Convexe, Montpellier 1978, Exposé N∘4. |
[15] | Wong, S., Some metric properties of piecewise monotonic mappings of the unit interval, Trans. Am. Math. Soc., 246, 493-500 (1978) · Zbl 0401.28011 |
[16] | Wong, S., Hölder continuous derivatives and ergodic theory, J. Lond. Math. Soc., 22, 506-520 (1980) · Zbl 0459.28022 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.