Thermodynamical Formalism and Markoff-Hurwitz Equations

Explore the fascinating intersection of number theory, dynamical systems, and fractals in this 51-minute lecture by Michael Magee. Delve into the world of Markoff-Hurwitz equations, beginning with a simple question about the relationship between squares and products of integers. Discover how transfer operators and thermodynamical formalism can be applied to count integer solutions to these Diophantine equations. Uncover new connections between Diophantine geometry and fractals, with a focus on the intriguing 'Rauzy gasket'. Learn how this fractal appears in diverse areas of mathematics, including triply periodic surfaces, dynamics of circle maps, higher-dimensional generalizations of continued fractions, Teichmuller theory, and Diophantine geometry. Gain insights from this joint work with Alex Gamburd and Ryan Ronan, presented as part of the Hausdorff Trimester Program "Dynamics: Topology and Numbers" conference on transfer operators in number theory and quantum chaos.

Michael Magee: Thermodynamical formalism and Markoff-Hurwitz equations