Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-16T17:01:16.705Z Has data issue: false hasContentIssue false
  • English
  • Français

The Perron-Frobenius operator in spaces of smooth functions on an interval

Published online by Cambridge University Press:  19 September 2008

B. Szewc
B. Szewc
Affiliation:
SGGW-Warsaw Agricultural University, Institute of Applied Mathematics and Statistics, Warsaw, Poland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The densities of invariant measures for Misiurewicz maps and Lasota-Yorke maps of class Cr are of class Cr-1 on certain intervals (forming the partition of an interval in case of Misiurewicz maps). For these maps the Perron-Frobenius operator has an unambiguous decomposition into the sum of projections onto eigenspaces (multiplied by the eigenvalues) and a remainder operator. The remainder operator has spectral radius less than one in certain spaces of smooth functions.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 4 , Issue 4 , December 1984 , pp. 613 - 643
Copyright
Copyright © Cambridge University Press 1984

References

REFERENCES

[1]Hofbauer, F. & Keller, G.. Ergodic properties of invariant measures for piecewise monotonic transformations. Math. Z. 180 (1982), 119–140.Google Scholar
[2]Hofbauer, F. & Keller, G.. Equilibrium states for piecewise monotonic transformations. Erg. Th. & Dynam. Sys. 2 (1982), 23–43.Google Scholar
[3]Kowalski, Z. S.. Invariant measure of piecewise monotonic transformation has a positive lower bound on its support. Bull. Acad. Polon. Sci., Ser. Sci. Math. 27 (1979), 53–57.Google Scholar
[4]Krzyzewski, K.. Some results on expanding mappings. Asterisque 50 (1977), 205–218.Google Scholar
[5]Lasota, A. & Yorke, J.. On the existence of invariant measures for piecewise monotonic transformations. Trans. Amer. Math. Soc. 186 (1973), 481–488.Google Scholar
[6]Misiurewicz, M.. Absolutely continuous measures for certain maps of an interval. Inst. Hautes Etudes Sci. Publ. Math. 53 (1981), 17–51.Google Scholar
[7]Ognev, A. J.. Metric properties of some class of maps of an interval. Mat. Zametki 30 No 5 (1981) 723–736. (In Russian.)Google Scholar
[8]Rychlik, M.. Bounded variation and invariant measures. Preprint, to appear in Studia Mathematica.Google Scholar
[9]Sacksteder, R.. The measures invariant under an expanding map. In Springer Lecture Notes in Maths 392 (1974).Google Scholar
[10]Szlenk, W.. Some dynamical properties of certain differentiable mappings of an interval, Part l-Boletin de la Sociedad Mathematica Mexicana, 24 No 2 (1979), 57–82; Part II, Ergodic Theory and Dynamical Systems II (Proceedings Special Year, Maryland 1979–80). Birkhäuser.Google Scholar
[11]Ziemian, K.. Thesis, Warsaw University.Google Scholar