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Iterated function systems and dynamical systems.
1Department of Mathematics, Concordia University, 7141 Sherbrooke Street West, Montreal, Quebec H4B 1R6, Canada.
Chaos (Woodbury, N.Y.)
|December 1, 1995
Summary
This study explores invariant measures for expanding transformations and their inverse systems. A key finding is that invariant measures are preserved if the transformation is equivalent to a piecewise linear one.
Area of Science:
- Dynamical Systems
- Measure Theory
- Ergodic Theory
Background:
- Invariant measures are fundamental in understanding the long-term behavior of dynamical systems.
- Piecewise expanding transformations and iterated function systems are crucial in various fields, including chaos theory and fractal geometry.
Purpose of the Study:
- To investigate the relationship between invariant measures of a piecewise expanding transformation (tau) and its generated iterated function system (T(tau)).
- To identify the conditions under which an absolutely continuous measure invariant under tau is also invariant under T(tau).
Main Methods:
- Analysis of invariant measures for piecewise expanding maps on compact metric spaces.
- Characterization of transformations that are absolutely continuously conjugated to piecewise linear maps.
- Exploration of measures of maximal entropy and general equilibrium states.
Main Results:
- The core result establishes that a tau-invariant absolutely continuous measure is T(tau) invariant if and only if tau is absolutely continuously conjugated with a piecewise linear transformation.
- This provides a precise criterion for the preservation of invariant measures under the iterated function system.
Conclusions:
- The study clarifies the connection between the properties of a transformation and the invariance of its associated measures.
- The findings contribute to the understanding of invariant measures in dynamical systems, with implications for the study of entropy and equilibrium states.