Shadowing and Hyperbolicity for Endomorphisms of Locally Compact Groups

Abstract

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. This connects the result with smooth hyperbolic dynamics. On compact connected Lie groups, the maps with nonsingular differential covered by the theorem are Anosov endomorphisms in the classical non-invertible sense; for automorphisms of compact Lie groups, classical Anosov dynamics, topological Anosov dynamics, expansiveness and shadowing are equivalent. The theorem also applies to singular endomorphisms, which lie outside the usual theory of Anosov endomorphisms. As further consequences, positively expansive Lie group endomorphisms are automatically topologically expanding, and shadowing endomorphisms of connected semisimple Lie groups are precisely the nilpotent ones. In the totally disconnected setting, Aoki had already proved automatic shadowing for automorphisms of compact metrizable groups. We extend this to arbitrary totally disconnected locally compact groups and to non-invertible endomorphisms: every continuous endomorphism has shadowing. The proof uses Willis' tidy-above decomposition. We also discuss group shifts and show that an automorphism with a dense orbit can occur only on a compact group; in that case the automorphism is topologically mixing.