Thermodynamical approach to the Markoff-Hurwitz equation

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 count 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. Due to work of Huang and Norbury, the count for the case of exactly 4 variables matches up with counting the number of one sided simple closed curves on Fuchsian/quasi-Fuchsian 3 times punctured projective planes, so our result can be viewed as an analog for nonorientable surfaces of the counting results of Mirzakhani. This is joint work with Alex Gamburd and Ryan Ronan.