  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.
