國立清華大學 數學系

Abstract：
We study the composition relations of rational functions, focusing on two main aspects. First, we establish a stronger version of the Tits alternative for endomorphisms of the projective line, proving that for almost all rational functions f of degree greater than 3, the semigroup generated by f and any fixed morphism g is free. Our approach utilizes dynamical canonical heights and their properties. Second, we develop algorithms to detect composition relations between rational functions. 
We also investigate finitely represented semigroups, inspired by monomial compositions, and prove that such semigroups have finite sets of basic relations of equal size. Our results contribute to the classical theory of function composition initiated by Fatou, Julia, and Ritt, with applications in arithmetic dynamics and graph theory.