Thermodynamical approach to the Markoff-Hurwitz equation
Ergodic Theory and Dynamical Systems Seminar
24th May 2018, 3:00 pm – 4:00 pm
Howard House, 4th Floor Seminar Room
I'll first introduce the Markoff-Hurwitz equation and explain how it plays a fundamental role in different areas of mathematics.The main result I'll discuss is a true asymptotic formula for the number of real points in a fixed orbit of the automorphism group of the Markoff-Hurwitz variety with bounded maximal entry. In particular this establishes an asymptotic formula for the number of integer solutions to the Markoff-Hurwitz equation of bounded height. Our results are new when the equation has at least 4 variables, where the previous best result was by Baragar (1998) that gave a polylogarithmic growth rate with a mysterious noninteger exponent of growth. We brought new methods to this problem using the thermodynamical formalism and symbolic dynamics, and as a result, obtained a new characterization of the mysterious exponent of growth in terms of `conformal measures' on projective space supported on fractal sets. By incorporating work of Huang and Norbury, our result for the case of exactly 4 variables allows one to count the number of one sided simple closed curves, by their lengths, on Fuchsian 3 times punctured projective planes. This yields a twisted analog of the counting results of Mirzakhani for simple closed curves that involves a noninteger exponent of growth. This is based on joint work with Alex Gamburd and Ryan Ronan.