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The Physical Measure and SRB Measure in Partially Hyperbolic Attractors

Author: MiZeYa
Tutor: CaoYongLuo
School: Suzhou University
Course: Applied Mathematics
Keywords: partially hyperbolic Lyapunov exponent physical measure SRBmeasure u-Gibbs measure
CLC: O19
Type: Master's thesis
Year: 2013
Downloads: 6
Quote: 0
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In dynamical systems, particularly in the non-conservative dynamical systems, the existence and finiteness of physical measure is an important content of our research. In this paper, we mainly consider the partially hyperbolic attrctor A. Firstly, we prove that if for Lebesgue almost every point x in the topological basin B(A) of A. Then there exsits physical measure and SRB measure in the system. Secondly, we show that if for Lebesgue almost every point x in the topological basin B(A) of A. Then there exsits finitely many physical measures, they are all SRB measues, and the union of these basins of physical measures is a full Lebesgue subset of B(A).Because of the special function of u-Gibbs measue in the constrction of physical measure, we make some research about it in this paper, we prove the finiteness of the u-Gibbs measue which satisfy that the central Lyapunov exponent is less than some negative constant for almost everywhere. And in the case of one dimensional central direction, we construct an example such that there exists countable infinite ergodic u-Gibbs measures for which the central Lyapunov exponent is less than zero but not some negtive number almost everywhere, this example illustrate the above property better. In the case of two dimensional central direction, through an example we construct shows that even though there are finitely many (large than1) ergodic u-Gibbs measures in the system cant’t guarantee the existence of physical measue.

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