A birational mapping with a strange attractor: Post critical set and covariant curves

TL;DR

This paper studies specific two-dimensional birational transformations, exploring their post critical sets, covariant curves, and the existence of strange attractors, including cases with infinite post critical sets and unbounded attractors.

Contribution

It links the nature of post critical sets to covariant curves and preserved forms, and characterizes strange attractors in certain birational maps.

Findings

Infinite post critical sets correspond to algebraic covariant curves.

Some mappings have no algebraic covariant curves or preserved forms.

Strange attractors are identified, including an unbounded one.

Abstract

We consider some two-dimensional birational transformations. One of them is a birational deformation of the H\'enon map. For some of these birational mappings, the post critical set (i.e. the iterates of the critical set) is infinite and we show that this gives straightforwardly the algebraic covariant curves of the transformation when they exist. These covariant curves are used to build the preserved meromorphic two-form. One may have also an infinite post critical set yielding a covariant curve which is not algebraic (transcendent). For two of the birational mappings considered, the post critical set is not infinite and we claim that there is no algebraic covariant curve and no preserved meromorphic two-form. For these two mappings with non infinite post critical sets, attracting sets occur and we show that they pass the usual tests (Lyapunov exponents and the fractal dimension) for being strange attractors. The strange attractor of one of these two mappings is unbounded.