Homoclinic orbits, multiplier spectrum and rigidity theorems in complex dynamics

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