Measure rigidity for regular polynomial endomorphisms and applications

Abstract

We prove measure-rigidity results for regular polynomial endomorphisms of~$\mathbb{C}^k$. For maps of the same degree whose leading homogeneous parts differ by an invertible linear map on the target, equality of equilibrium measures is equivalent to postcomposition by an affine symmetry of the Julia set. We then show that, for $f$ in a nonempty Zariski open subset of the parameter space, $\mu_g=\mu_f$ implies $g=f$, with no hypothesis relating the leading terms; in dimension two this holds across all degrees, with $g$ an iterate of~$f$. We also establish finiteness results for maps with a prescribed equilibrium measure under essentially necessary hypotheses. Our approach rests on the stable manifold structure near the hyperplane at infinity. Equality of equilibrium measures yields a Zariski-dense family of common local stable-manifold germs, whose intrinsic first-order jets lead to a divisibility incidence problem on a Grassmannian. We resolve this problem by analyzing the power-map fiber and applying a generic stabilizer argument. In dimension two, applications include a characterization in terms of preperiodic sets, a Tits-type alternative, results on iterated centralizers, and an arithmetic non-density theorem.