This book investigates the intricate relationship between ergodic theory and the behavior of differentiable maps on manifolds, specifically focusing on equivariant diffeomorphisms. It delves into the properties of dynamical systems where the maps preserve certain symmetries, exploring concepts such as invariant measures, recurrence, and the statistical behavior of orbits. The work examines how the equivariant nature of these diffeomorphisms constrains their ergodic properties, often leading to more structured and predictable dynamics compared to general diffeomorphisms. The analysis is grounded in advanced mathematical techniques from differential geometry and measure theory.

The book is recognized as a specialized monograph within the field of dynamical systems and ergodic theory. Its focus on equivariant diffeomorphisms suggests a rigorous mathematical treatment aimed at researchers and advanced graduate students. The subject matter indicates a dense, technical text that builds upon foundational concepts in topology, differential geometry, and measure theory. The work is likely to be valued for its contribution to understanding the interplay between symmetry and chaotic behavior in mathematical models.