Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics

Abstract: The aims of this paper are answering several conjectures and questions about multiplier spectrum of rational maps and giving 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 where 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 generalize 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.