Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics

TL;DR

This paper advances understanding of complex dynamics by providing new proofs and generalizations of rigidity theorems related to multiplier and length spectra of rational maps, using tools from both complex and non-archimedean dynamics.

Contribution

It offers a new proof of McMullen's theorem without quasiconformal maps, generalizes the theorem to length spectra, and proves conjectures characterizing exceptional rational maps.

Findings

The length spectrum determines the conjugacy class of rational maps, excluding flexible Lattès maps.

A new proof of Zdunik's rigidity theorem is provided.

Characterization of exceptional rational maps based on multipliers and Lyapunov exponents.

Abstract

The aims of this paper are to answer several conjectures and questions about multiplier spectrum of rational maps and to give new proofs of several rigidity theorems in complex dynamics, by combining tools from complex and non-archimedean dynamics. A remarkable theorem due to McMullen asserts that aside from the flexible Latt\`es family, the multiplier spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. The proof relies on Thurston's rigidity theorem for post-critically finite maps, in which Teichm\"uller theory is an essential tool. We will give a new proof of McMullen's theorem without using quasiconformal maps or Teichm\"uller theory. We show that aside from the flexible Latt\`es family, the length spectrum of periodic points determines the conjugacy class of rational maps up to finitely many choices. This generalizes the aforementioned McMullen's theorem. We will also prove a rigidity theorem for marked length spectrum. Similar ideas also yield a simple proof of a rigidity theorem due to Zdunik. We show that a rational map is exceptional if and only if one of the following holds (i) the multipliers of periodic points are contained in the integer ring of an imaginary quadratic field; (ii) all but finitely many periodic points have the same Lyapunov exponent. This solves two conjectures of Milnor.