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arxiv: 2606.27647 · v1 · pith:VIINS4YDnew · submitted 2026-06-26 · 🧮 math.GR · math.DS

Shadowing and Hyperbolicity for Endomorphisms of Locally Compact Groups

Dekui Peng This is my paper

Pith reviewed 2026-06-29 00:18 UTC · model grok-4.3

classification 🧮 math.GR math.DS
keywords shadowing propertylocally compact groupsLie groupsendomorphismshyperbolicitytotally disconnected groupsexpansive mapstopologically Anosov
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The pith

Endomorphisms of Lie groups have the shadowing property if and only if their differentials are hyperbolic.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that for continuous endomorphisms of Lie groups, the shadowing property holds exactly when the derivative at the identity is a hyperbolic linear map. This gives an infinitesimal test for a global dynamical property. In contrast, every continuous endomorphism of a totally disconnected locally compact group has shadowing. The results also equate several notions of expansiveness and hyperbolicity for automorphisms in these classes of groups.

Core claim

An endomorphism of a Lie group has shadowing if and only if its differential at the identity is hyperbolic. For totally disconnected locally compact groups every continuous endomorphism has shadowing, proved using the tidy-above decomposition.

What carries the argument

The shadowing property defined with respect to the left uniformity on the group, characterized infinitesimally by hyperbolicity of the differential for Lie groups.

Load-bearing premise

That the left uniformity on the group makes the shadowing property characterizable by the differential at the identity for Lie groups, and that the tidy decomposition applies to endomorphisms.

What would settle it

A continuous endomorphism of a Lie group whose differential at the identity is not hyperbolic yet satisfies the shadowing property, or a continuous endomorphism of a totally disconnected locally compact group that lacks shadowing.

read the original 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. 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper studies the shadowing property for continuous endomorphisms of locally compact groups with respect to the left uniformity. For Lie groups it claims a complete infinitesimal characterization: shadowing holds if and only if the differential at the identity is hyperbolic. Consequences include that positively expansive Lie endomorphisms are topologically expanding and that expansiveness, shadowing, two-sided shadowing and being topologically Anosov are equivalent for Lie automorphisms; for connected semisimple Lie groups, shadowing endomorphisms are precisely the nilpotent ones. For totally disconnected locally compact groups the paper asserts that shadowing is automatic for every continuous endomorphism, proved via an extension of Willis' tidy-above decomposition; this yields equivalences between topological expansion and positive expansiveness, and between being topologically Anosov and expansiveness. Connections to group shifts and a compactness result for topologically mixing automorphisms are also discussed.

Significance. If the derivations hold, the work supplies a clean infinitesimal criterion for Lie groups that links differential hyperbolicity directly to a uniform dynamical property, together with automatic shadowing in the tdlc setting. The resulting equivalences among expansiveness notions and the compactness consequence for mixing automorphisms are concrete advances at the interface of topological dynamics and the structure theory of locally compact groups. The manuscript employs standard tools (Willis decomposition, Lie algebra differentials) without introducing free parameters or ad-hoc entities.

major comments (1)
  1. [tdlc results / proof of automatic shadowing] The tdlc claim that every continuous endomorphism has shadowing (abstract and the section developing the tdlc results) rests on applying Willis' tidy-above decomposition directly to endomorphisms. Standard Willis theory constructs tidy subgroups for automorphisms via the scale function; for a general endomorphism φ the image φ(G) may be proper and the preimage structure differs, so the decomposition must be shown to produce the uniform contraction/expansion estimates needed for shadowing. This justification is load-bearing for the automatic-shadowing conclusion and the subsequent equivalences.
minor comments (2)
  1. [Introduction] The introduction should recall the precise definition of the left uniformity on a locally compact group and how it induces the shadowing notion, to make the setup self-contained.
  2. When citing Willis' results, include specific theorem or proposition numbers rather than the general reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the positive assessment of the paper's significance and contributions. We address the single major comment below.

read point-by-point responses

  1. Referee: [tdlc results / proof of automatic shadowing] The tdlc claim that every continuous endomorphism has shadowing (abstract and the section developing the tdlc results) rests on applying Willis' tidy-above decomposition directly to endomorphisms. Standard Willis theory constructs tidy subgroups for automorphisms via the scale function; for a general endomorphism φ the image φ(G) may be proper and the preimage structure differs, so the decomposition must be shown to produce the uniform contraction/expansion estimates needed for shadowing. This justification is load-bearing for the automatic-shadowing conclusion and the subsequent equivalences.

    Authors: The manuscript extends Willis' tidy-above decomposition to continuous endomorphisms rather than applying the automorphism case verbatim. In the tdlc section we construct the relevant tidy subgroups by working with the closed image φ(G) and the preimage filtration induced by iterates of φ, adapting the scale function and the tidy-above property to this setting. We then verify directly that the resulting decomposition yields uniform contraction estimates on the contracting component and uniform expansion estimates on the expanding component (with respect to the left uniformity). These estimates are precisely what is needed to establish the shadowing property by the standard argument for uniform dynamical systems. The construction accounts for the possibility that φ(G) is proper and does not rely on surjectivity. We believe this supplies the required justification; however, if the referee finds the exposition of the adaptation insufficiently explicit, we are prepared to expand the relevant proofs and add a dedicated lemma isolating the endomorphism case. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations rely on external standard tools

full rationale

The paper derives the Lie-group characterization (shadowing iff differential hyperbolic) from the left uniformity and standard differential structure on Lie groups, without reducing to self-definition or fitted parameters. The tdlc claim (automatic shadowing for all continuous endomorphisms) is presented as a direct consequence of applying Willis' tidy-above decomposition, which is external prior work with no author overlap. No self-citation chains, ansatzes smuggled via citation, or renamings of known results appear as load-bearing steps. The central claims remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper relies on established mathematical frameworks and prior results without introducing new free parameters or postulated entities.

axioms (3)
  • standard math Properties of the left uniformity on locally compact groups
    Used to define the shadowing property.
  • domain assumption Willis' tidy-above decomposition for endomorphisms of tdlc groups
    Key tool for proving shadowing in the totally disconnected case.
  • standard math Standard differential calculus on Lie groups
    For the infinitesimal characterization.

pith-pipeline@v0.9.1-grok · 5687 in / 1294 out tokens · 39935 ms · 2026-06-29T00:18:40.347171+00:00 · methodology

discussion (0)

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Reference graph

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