Classical and Dynamical Markov and Lagrange Spectra

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:Contents: Classical Lagrange and Markov SpectraSome Results About the Intermediate Portions of the Classical SpectraContinuity of Hausdorff Dimension Across Classical and Dynamical SpectraBeginning of Dynamical Lagrange and Markov SpectraIntervals and Hall Rays in Dynamical Lagrange and Markov SpectraAppendices:Proof of Hurwitz TheoremProof of Euler's RemarkContinued Fractions, Binary Quadratic Forms, and Markov SpectrumFreiman's Rightmost Gap on Markov SpectrumSoft Bounds on the Hausdorff Dimension of Dynamical Cantor SetsModulus of Continuity of the Dimension Across Classical SpectraClosedness of the Dynamical Spectra Associated to Horseshoes
Readership: Graduate students and researchers in Number Theory and Dynamical Systems.Markov and Lagrange Spectra;Diophantine Approximations;Fractal Geometry;Hyperbolic Dynamical Systems0Key Features:Presents classical results from a modern viewpoint Presents recent research results in a very classical subjectCombines ideas from several fields of Mathematics