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Explore the construction of invariant measures for geodesic flows on hyperbolic surfaces through this mathematical lecture focusing on thermodynamical formalism and its geometric applications. Learn about Gibbs measures as good invariant measures for geodesic flows, understanding their crucial ergodic properties and theoretical foundations. Discover how these mathematical tools solve important geometric problems, including analyzing the regularity of entropy under metric changes and studying horocyclic flows on abelian covers of compact hyperbolic surfaces. Gain insights into the intersection of dynamical systems, differential geometry, and ergodic theory through rigorous mathematical exposition. This first part of a three-lecture series provides essential background in thermodynamical formalism while demonstrating its powerful applications to geometric questions in hyperbolic geometry.
