Dynamical Objects for Cohomologically Expanding Maps

TL;DR

This paper develops a method to construct unique invariant currents and measures for non-invertible smooth maps on compact manifolds by leveraging sheaf cohomology and expansion rates, unifying smooth and holomorphic cases.

Contribution

It introduces a cohomological framework to produce and analyze invariant currents and measures, ensuring their uniqueness under certain growth conditions, applicable in both smooth and holomorphic contexts.

Findings

Constructed unique invariant currents representing cohomological classes.

Established conditions for sheaf cohomology to match de Rham cohomology.

Applied the main theorem to both smooth and holomorphic maps.

Abstract

The goal of this paper is to construct invariant dynamical objects for a (not necessarily invertible) smooth self map of a compact manifold. We prove a result that takes advantage of differences in rates of expansion in the terms of a sheaf cohomological long exact sequence to create unique lifts of finite dimensional invariant subspaces of one term of the sequence to invariant subspaces of the preceding term. This allows us to take invariant cohomological classes and under the right circumstances construct unique currents of a given type, including unique measures of a given type, that represent those classes and are invariant under pullback. A dynamically interesting self map may have a plethora of invariant measures, so the uniquess of the constructed currents is important. It means that if local growth is not too big compared to the growth rate of the cohomological class then the expanding cohomological class gives sufficient "marching orders" to the system to prohibit the formation of any other such invariant current of the same type (say from some local dynamical subsystem). Because we use subsheaves of the sheaf of currents we give conditions under which a subsheaf will have the same cohomology as the sheaf containing it. Using a smoothing argument this allows us to show that the sheaf cohomology of the currents under consideration can be canonically identified with the deRham cohomology groups. Our main theorem can be applied in both the smooth and holomorphic setting.