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.

Autorentext

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 the Institute 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.

Inhalt
Introduction.- Preliminaries in Ergodic Theory.- KolmogorovBernoulli Equivalence for Ergodic Automorphism of T².- Hyperbolic Structures and the KolmogorovBernoulli Equivalence.- State of the Art.- Index.