Absolutely Continuous Invariant Measures for Piecewise Expanding Chaotic Transformations in R^n with Summable Oscillations of Derivative

Abdusslam Osman Beitalmal (1)
(1) Department of Mathematics, Faculty of Science, Sebha University, Libya


This paper presents a detailed investigation of absolutely continuous invariant measures (ACIMs) for piecewise expanding chaotic transformations in , with particular attention paid to the case where the derivative has summable oscillations. ACIMs are important objects in the study of dynamical systems, as they provide a way to understand the long-term behavior of trajectories and the statistical properties of the system. The paper covers a range of important topics related to ACIMs, including the boundedness condition, distortion condition, localization condition, and Schmitt's condition. It also discusses the Perron-Frobenius operator, which plays a critical role in the existence and properties of ACIMs. The main result of the paper is the proof that the Perron-Frobenius operator is constrictive, which implies the existence of a finite number of ergodic ACIMs that satisfy Schmitt's condition and a condition dependent on the defining partition. This finding has significant implications for the understanding of complex systems and the advancement of research in this field. The paper also discusses the relationship between ACIMs and dynamical systems, highlighting the role of ACIMs in ergodic theory. Overall, this paper provides a valuable reference for researchers interested in the study of ACIMs and their significance in the analysis of dynamical systems and ergodic theory.

Full text article

Generated from XML file


Adl-Zarabi, Kourosh and Proppe, Harald (1997). Existence of many ergodic absolutely continuous invariant measures for piecewise-expanding C2 chaotic transformations in R2 on a fixed number of partitions. Journal of statistical physics Volume 89, pages 537-548 Springer. DOI: https://doi.org/10.1007/BF02765534

Baker, Richard, (1991). Lebesgue measure” on R∞.Proceedings of the Amer-ican Mathematical Society. Vol 113/4,pages 1023-1029. DOI: https://doi.org/10.1090/S0002-9939-1991-1062827-X

Candeloro, D (1987). Misure invariante per transformazioni in piu dimen- sioni.Atti Sem. Mat. Fis. Univ. Modena Vol 35. Pages 33-42

G´ora, Pawel, (1994). Properties of invariant measures for piecewise expanding one-dimensional transformations with summable oscillations of derivative. Ergodic Theory and Dynamical Systems. Vol 14, page 475-492.Cambridge University Press. DOI: https://doi.org/10.1017/S0143385700007987

Gora, Pawel and Boyarsky, Abraham (1989). Absolutely continuous invariant measures for piecewise expanding C 2 transformations in R N . Israel Journal of Mathematics Vol 67, pages 272-286. Springer DOI: https://doi.org/10.1007/BF02764946

Jab-lon´ski, M, (1983). On invariant measures for piecewise C2-transformations of the n-dimensional cube, Annales Polonici Mathematici Vol 2/43, pages 185-195. DOI: https://doi.org/10.4064/ap-43-2-185-195

Keller, G (1979). Propriet´es ergodiques des endomorphismes dilatants, C 2 par morceaux, des r´egions born´ees du plan. These Universite de Rwnnes 1979.

Osman, Abdusslam (1996). Smoothness of invariant densities for certain classes of dynamical systems. Concordia University.

Rychlik, Marek Ryszard (1983). Invariant measures and the variational principle for Lozi mappings.University of California, Berkeley.

Schmitt, Bernard (1986). Contribution `a l’´etude de syst`emes dynamiques unidimensionnels en th´eorie ergodique. School Dijon 1986.

Sprecher, David A (1993). A universal mapping for Kolmogorov’s superposition theorem. Neural networks Vol 6/8 pages 1089-1094. DOI: https://doi.org/10.1016/S0893-6080(09)80020-8


Abdusslam Osman Beitalmal
abd.beitalmal@sebhau.edu.ly (Primary Contact)
Beitalmal , A. O. (2023). Absolutely Continuous Invariant Measures for Piecewise Expanding Chaotic Transformations in R^n with Summable Oscillations of Derivative. Journal of Pure & Applied Sciences, 22(2), 73–82. https://doi.org/10.51984/jopas.v22i2.2342

Article Details

No Related Submission Found