On automorphic measures, Lyapunov exponents and instability of rational maps

Abstract

To construct obstructions to the stability of rational maps with non-summable critical points in their Julia sets, we introduce automorphic measures with complex eigenvalues for rational maps on the Riemann sphere. In particular, these measures extend the classical notions of quasi-invariant and conformal measures by allowing the respective Radon--Nikodym derivative to be complex-valued and proportional to a multiplicative cocycle. \[ j_{(s,t)}(R) = |R'|^{s} \left(\frac{|R'|}{R'}\right)^{t}, \] which plays the role of a generalized automorphy factor in the sense of group actions. The existence of such measures reveals a close connection between geometric and dynamical properties of rational maps. We show that the existence of certain automorphic measures, particularly unimodular measures and their associated vector fields, implies instability of the corresponding rational map. Specifically, for a weakly dissipative rational map admitting a $( -1, 1)$-unimodular measure, there exists an integer $q \ge 1$ such that the map is $q$-unstable. This result generalizes earlier instability criteria involving pseudoconformal measures and connects the presence of such measures to the failure of structural stability. Furthermore, we establish ergodic and combinatorial conditions ensuring the existence of unimodular measures or vector fields -- most notably, through bounded recurrence, bounded velocity of arguments, and relations with the Milnor--Thurston kneading theory. These criteria provide a unified framework linking automorphic measures, Lyapunov spectra, and the geometric deformation spaces of rational maps.