scholarly journals On symmetric logarithm and some old examples in smooth ergodic theory

2003 ◽  
Vol 180 (3) ◽  
pp. 241-255 ◽  
Author(s):  
K. Frączek ◽  
M. Lemańczyk
Keyword(s):  
Ergodic Theory ◽  
Smooth Ergodic Theory

Smooth Ergodic Theory for Endomorphisms

2009 ◽  
Author(s):  
Min Qian ◽  
Jian-Sheng Xie ◽  
Shu Zhu
Keyword(s):  
Ergodic Theory ◽  
Smooth Ergodic Theory

Vanishing transverse entropy in smooth ergodic theory

2010 ◽  
Vol 31 (4) ◽  
pp. 1229-1235 ◽  
Author(s):  
FRANÇOIS LEDRAPPIER ◽  
JIAN-SHENG XIE
Keyword(s):  
Ergodic Theory ◽  
Measure Preserving Transformation ◽  
Smooth Ergodic Theory

AbstractFor a measure-preserving transformation, the entropy being zero means that there is no increasingσ-algebra. In this note, we prove that a similar phenomenon occurs forC2diffeomorphisms when considering the increment between the partial entropies associated with different exponents.

A new entropy formula of Ledrappier–Young type for linear toral dynamics

2012 ◽  
Vol 34 (3) ◽  
pp. 1037-1054
Author(s):  
JIAN-SHENG XIE
Keyword(s):  
Dynamical System ◽  
Ergodic Theory ◽  
Linear Dynamical System ◽  
Linear Dynamics ◽  
Entropy Formula ◽  
System A ◽  
Smooth Ergodic Theory

AbstractA detailed Ledrappier–Young theory is presented for linear toral dynamics. First, a proof is given for a simplified definition for local entropies which holds in more general settings besides the current linear dynamics. Then it is shown that the transverse dimensions can be defined directly via the Smale structure of the linear dynamical system. A new entropy formula of Ledrappier–Young type is obtained. The conjecture of Ledrappier and Xie [Vanishing transverse entropy in smooth ergodic theory. Ergod. Th. & Dynam. Sys. 31(4) (2011), 1229–1235] is also discussed for such linear dynamics.

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