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