We prove several rigidity results on multiplier spectrum and length spectrum. For example, we show that for every non-exceptional rational map \(f:\mathbb {P}^1(\mathbb {C})\rightarrow \mathbb {P}^1(\mathbb {C})\) of degree \(d\ge 2\) , the \(\mathbb {Q}\) -vector space generated by all the (finite) characteristic exponents of periodic points of f has infinite dimension. This answers a stronger version of a question of Levy and Tucker. Our result can also be seen as a generalization of recent results of Ji-Xie and of Huguin which proved Milnor’s conjecture about rational maps having integer multipliers. We also get a characterization of postcritically finite maps by using their length spectra. Finally as an application of our result, we get a new proof of the Zariski-dense orbit conjecture for endomorphisms on \((\mathbb {P}^1)^N, N\ge 1\) .
