Shadowing and Hyperbolicity for Endomorphisms of Locally Compact Groups

We study the shadowing property for continuous endomorphisms of locally compact groups, using the left uniformity.

For Lie groups we obtain a complete infinitesimal characterization: an endomorphism has shadowing if and only if its differential is hyperbolic.

As consequences, positively expansive Lie group endomorphisms are automatically topologically expanding, and for Lie group automorphisms, expansiveness, shadowing, two-sided shadowing and being topologically Anosov are equivalent.

We also show that, for connected semisimple Lie groups, shadowing endomorphisms are precisely nilpotent endomorphisms.

In contrast, for totally disconnected locally compact groups, shadowing is automatic: every continuous endomorphism has shadowing.

The proof uses Willis' tidy-above decomposition for endomorphisms.

This yields, in the totally disconnected case, that topological expansion is equivalent to positive expansiveness and that being topologically Anosov is equivalent to expansiveness.

We also discuss connections with group shifts and derive a compactness consequence for topologically mixing automorphisms.