An Introduction to the Kolmogorov–Bernoulli Equivalence

1. Front Matter

   Pages i-xiv

   PDF

2. Introduction

   • Gabriel Ponce, Régis Varão

   Pages 1-7

3. Preliminaries in Ergodic Theory

   • Gabriel Ponce, Régis Varão

   Pages 9-32

4. Kolmogorov–Bernoulli Equivalence for Ergodic Automorphisms of \(\mathbb T^2\)

   • Gabriel Ponce, Régis Varão

   Pages 33-56

5. Hyperbolic Structures and the Kolmogorov–Bernoulli Equivalence

   • Gabriel Ponce, Régis Varão

   Pages 57-90

6. State of the Art

   • Gabriel Ponce, Régis Varão

   Pages 91-116

7. Back Matter

   Pages 117-119

   PDF

This book offers an introduction to a classical problem in ergodic theory and smooth dynamics, namely, the Kolmogorov–Bernoulli (non)equivalence problem, and presents recent results in this field. Starting with a crash course on ergodic theory, it uses the class of ergodic automorphisms of the two tori as a toy model to explain the main ideas and technicalities arising in the aforementioned problem. The level of generality then increases step by step, extending the results to the class of uniformly hyperbolic diffeomorphisms, and concludes with a survey of more recent results in the area concerning, for example, the class of partially hyperbolic diffeomorphisms. It is hoped that with this type of presentation, nonspecialists and young researchers in dynamical systems may be encouraged to pursue problems in this area.

• Kolmogorov systems
• Bernoulli systems
• ergodic theory
• isomorphism problem
• disintegration of measures
• 37A35
• 37C40
• 37D30

“All the proofs given in the book are complete, the list of references is adequate and the language is very clear, making the reading captivating.” (Sergey Gennadievich Kryzhevich, Mathematical Reviews, May, 2021)

• IMECC, University of Campinas - UNICAMP, Campinas, Brazil

  Gabriel Ponce, Régis Varão

Gabriel Ponce is an Associate Professor at the Institute of Mathematics, Statistics and Scientific Computing (IMECC) at University of Campinas. He holds a Bachelor's degree in Mathematics (2009) and a PhD in Dynamical Systems (2014), both from the University of São Paulo (USP), under the supervision of Ali Tahzibi. He was also a visiting PhD student at Penn State University, USA, where he worked with Federico Rodriguez Hertz, followed by a period of postdoctoral study at the University of Campinas (UNICAMP). His main field of research is Dynamical Systems and Ergodic Theory.

Régis Varão holds a Bachelor's degree in Mathematics from the University of Campinas (Unicamp) and received his Master's degree (2008) and PhD (2012) from IMPA, under the supervision of Marcelo Viana.  He also engaged in postdoctoral at the University of São Paulo (USP) to work with Prof. Ali Tahzibi and at the University of Chicago to work with Prof. Amie Wilkinson. In 2014 he joined theInstitute of Mathematics, Statistics and Scientific Computing (IMECC) at University of Campinas as an Associate Professor. He received an honorary mention for the best PhD thesis in mathematics in Brazil (Gutierrez Prize). His main research area is Dynamical Systems and Ergodic Theory.

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