The book intends to give a modern presentation of the classical Markov and Lagrange spectrum, which are fundamental objects from the theory of Diophantine approximations and of their several generalizations related to Dynamical Systems and Differential Geometry. Besides presenting many classical results, the book includes several topics of recent research on the subject, connecting several fields of Mathematics - Number Theory, Dynamical Systems and Fractal Geometry.It includes topics as:

This book is a remarkable contribution to the literature on dynamical systems and geometry. It consists of a selection of work in current research on Teichmuller dynamics, a field that has continued to develop rapidly in the past decades. After a comprehensive introduction, the author investigates the dynamics of the Teichmuller flow, presenting several self-contained chapters, each addressing a different aspect on the subject. The author includes innovative expositions, all the while solving open problems, constructing examples, and supplementing with illustrations. This book is a rare find in the field with its guidance and support for readers through the complex content of moduli spaces and Teichmuller Theory. The author is an internationally recognized expert in dynamical systems with a talent to explain topics that is rarely found in the field. He has created a text that would benefit specialists in, not only dynamical systems and geometry, but also Lie theory and number theory.