Year: 2023
Type: preprint
Abstract: We prove an equidistribution result for iterated preimages of curves by a large class of rational maps $f:\mathbb{CP}^2\dashrightarrow\mathbb{CP}^2$ that cannot be birationally conjugated to algebraically stable maps. The maps, which include recent examples with transcendental first dynamical degree, are distinguished by the fact that they have constant Jacobian determinant relative to the natural holomorphic two form on the algebraic torus. Under the additional hypothesis that $f$ has "small topological degree'' we also prove an equidistribution result for iterated forward images of curves. To prove our results we systematically develop the idea of a positive closed $(1,1)$ current and its cohomology class on the inverse limit of all toric surfaces. This, in turn, relies upon a careful study of positive closed $(1,1)$ currents on individual toric surfaces. This framework may be useful in other contexts.
Source: arXiv (Cornell University)
Authors: Diller, Jeffrey, Roeder, Roland
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4362598916