Dynamically incoherent surface endomorphisms

TL;DR

This paper constructs explicit examples of partially hyperbolic endomorphisms on the 2-torus that are dynamically incoherent, showing branching of center curves in a way not seen in invertible systems.

Contribution

It introduces new examples of non-invertible, dynamically incoherent partially hyperbolic maps on the torus with novel branching behavior.

Findings

Existence of dynamically incoherent partially hyperbolic endomorphisms on $\

Branching of center curves occurs along countably many circles.

These examples demonstrate a new form of coherence not observed in invertible systems.

Abstract

We explicitly construct a dynamically incoherent partially hyperbolic endomorphisms of $\mathbb{T}^2$ in the homotopy class of any linear expanding map with integer eigenvalues. These examples exhibit branching of centre curves along countably many circles, and thus exhibit a form of coherence that has not been observed for invertible systems.