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doctoral thesis

Contribution to the ergodic theory of piecewise monotone continuous maps

Faller, Bastien
2008

This thesis is devoted to the ergodic theory of the piecewise monotone continuous maps of the interval. The coding is a classical approach for these maps. Thanks to the coding, we get a symbolic dynamical system which is almost isomorphic to the initial dynamical system. The principle of the coding is very similar to the one of expansion of real numbers. We first define the coding in a perspective similar to the one of the expansions of real numbers; this perspective was already adopted by Rényi and Parry in their papers about the expansions of numbers. Then we present the theory of Hofbauer about the links between the ergodic properties of a piecewise monotone continuous map of the interval and the corresponding symbolic dynamical system. We prove that there is a bijection between the sets of measures of maximal entropy of these two dynamical systems. We apply these results to the study of two families of maps: first the maps Tα,β(x) := βx + α mod 1, then the maps we will call generalized β-transformations. For the family of maps Tα,β, we describe in detail the family of symbolic dynamical systems obtained by the coding. Then we turn to the question of normality of the orbits for the maps Tα,β. Finally we study the generalized β-transformations: we prove that most of them have a a unique measure of maximal entropy, then we also study the normality of the orbits for theses maps.

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Type
doctoral thesis
DOI
10.5075/epfl-thesis-4232
Author(s)
Faller, Bastien
Advisors
Pfister, Charles-Edouard  
Date Issued

2008

Publisher

EPFL

Publisher place

Lausanne

Thesis number

4232

Total of pages

107

Subjects

maps of the interval

•

symbolic dynamic

•

expansion of numbers

•

topological entropy

•

Markov diagram

•

normal numbers

•

applications de l'intervalle

•

dynamique symbolique

•

développement des nombres

•

entropie topologique

•

diagramme de Markov

•

nombres normaux

EPFL units
GR-PF  
Faculty
SB  
School
IACS  
Doctoral School
EDMA  
Available on Infoscience
September 25, 2008
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/29673
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