Dissertations, Theses, and Capstone Projects

Date of Degree

9-2017

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Alex Gamburd

Committee Members

Alex Gamburd

Jozef Dodziuk

Melvyn Nathanson

Subject Categories

Dynamical Systems | Number Theory

Keywords

Diophantine equation, symbolic dynamics, transfer operator, Fourier analysis, strong approximation

Abstract

The Markoff equation is a Diophantine equation in 3 variables first studied in Markoff's celebrated work on indefinite binary quadratic forms. We study the growth of solutions to an n variable generalization of the Markoff equation, which we refer to as the Markoff-Hurwitz equation. We prove explicit asymptotic formulas counting solutions to this generalized equation with and without a congruence restriction. After normalizing and linearizing the equation, we show that all but finitely many solutions appear in the orbit of a certain semigroup of maps acting on finitely many root solutions. We then pass to an accelerated subsemigroup of maps for which the dynamical system is uniformly contracting and show that both asymptotic formulas are consequences of a general asymptotic formula for a vector-valued counting function related to this accelerated dynamical system. This general formula is obtained using methods from symbolic dynamics, following a technique due to Lalley.

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