Counting Periodic Orbits of One-Dimensional Maps - Lecture 1

Explore the fascinating world of periodic orbit counting in one-dimensional maps through this lecture by Juan Rivera-Letelier from the University of Rochester. Delve into the connections between mathematics, number theory, and thermodynamic formalism as part of the Simons Semester on Dynamics. Begin with a review of Milnor and Thurston's work on interval maps, examining unweighted periodic orbit counting and the Artin-Mazur zeta function's relationship to topological entropy. Discover open problems surrounding the rationality of the Artin-Mazur zeta function and learn about Olivares-Vinales' ongoing research. Investigate weighted periodic orbit counting for interval and complex rational maps, focusing on Ruelle's weighted version of the Artin-Mazur zeta function. Analyze the pressure function and other thermodynamic formalism concepts through the study of Ruelle's zeta function's analytic properties. Highlight the prime orbit theorem established by Parry and Pollicott for uniformly hyperbolic maps and its extension to non-uniformly hyperbolic maps in dimension 1 by Zhiqiang Li and the lecturer. Compare the prime orbit theorem's statement and proof to the prime number theorem in number theory, and examine the induced scheme developed by Feliks Przytycki and the lecturer for non-uniformly hyperbolic cases.