Energy, equilibrium measure and entropy for toric surface maps
Jeffrey Diller, Roland Roeder
公開日: 2025/9/4
Abstract
We consider the ergodic theory of plane rational maps that preserve the natural holomorphic volume form on the algebraic torus. Specifically we construct natural invariant probability measures for a large class of such maps by intersecting the equilibrium currents we constructed in our previous work [DR]. We show further that these measures are mixing and that each admits an underlying geometric product structure. The main result of [DDG3] then implies that the topological entropy of each map covered by our results is the log of its first dynamical degree. In light of examples presented in [BDJ], this implies in particular that the entropy of a rational map can equal the log of a transcendental number.