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arxiv: 2604.10913 · v1 · submitted 2026-04-13 · 🧮 math.DS

Non-existence of Lyapunov exponents in the Newhouse domain

Shin Kiriki , Xiaolong Li , Yushi Nakano , Teruhiko Soma This is my paper

Pith reviewed 2026-05-10 16:24 UTC · model grok-4.3

classification 🧮 math.DS
keywords lyapunovexponentsnewhousediffeomorphismsdomainhomoclinicmathcalnon-existence
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The pith

In the Newhouse domain of C^r surface diffeomorphisms, a dense set of maps has no Lyapunov exponents on open sets of points and directions.

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

The paper establishes that non-existence of Lyapunov exponents is a persistent feature within the Newhouse domain. For any C^r diffeomorphism with r at least 2 that belongs to this domain, there is a dense collection of nearby maps where, for some open region of the surface and some open cone of directions, the Lyapunov exponents do not exist at any point in that region. This shows that even with robust homoclinic tangencies, the lack of well-defined expansion rates is common rather than exceptional. A sympathetic reader would care because it challenges assumptions about typical behavior in chaotic dynamical systems on surfaces.

Core claim

Within the Newhouse domain of C^r surface diffeomorphisms (r in [2, infinity)), there exists a dense subset D such that for any f in D, Lyapunov exponents fail to exist for all points in some open set U and all nonzero tangent vectors in some open cone V in R^2. This is achieved by constructing diffeomorphisms with specific oscillatory return times near a homoclinic tangency, incorporating Newhouse theory and results on Lyapunov irregularity.

What carries the argument

Diffeomorphisms with specific oscillatory return times near a homoclinic tangency, constructed via Newhouse theory and Lyapunov irregularity results.

Load-bearing premise

That diffeomorphisms with the required oscillatory return times near homoclinic tangencies can be constructed inside the Newhouse domain while ensuring non-existence of Lyapunov exponents on open sets.

What would settle it

Exhibiting one diffeomorphism inside the Newhouse domain for which Lyapunov exponents exist everywhere on every open set of points and every open cone of directions, or proving that no such dense subset D exists.

Figures

Figures reproduced from arXiv: 2604.10913 by Shin Kiriki, Teruhiko Soma, Xiaolong Li, Yushi Nakano.

Figure 1
Figure 1. Figure 1: The modified rectangle lemma Claim 3.4. For every non-negative integer m, we have f nk+m(ℓ u k+m) ⊂ 1 2 Uk,m+1 [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
read the original abstract

We show that within the Newhouse domain of $C^r$ surface diffeomorphisms ($r \in [2,\infty )$), there exists a dense subset $\mathcal D$ such that for any $f \in \mathcal D$, Lyapunov exponents fail to exist for all points in some open set $U$ and all nonzero tangent vectors in some open cone $V \subset \mathbb{R}^2$. This demonstrates that the non-existence of Lyapunov exponents is a persistent phenomenon in the setting of robust homoclinic tangencies. The proof relies on constructing diffeomorphisms exhibiting specific oscillatory return times near a homoclinic tangency, incorporating techniques from Newhouse theory and recent results on Lyapunov irregularity, alongside several refinements and new arguments.

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

2 major / 2 minor

Summary. The manuscript claims that within the Newhouse domain of C^r surface diffeomorphisms (r ∈ [2, ∞)), there exists a dense subset D such that for any f ∈ D, the Lyapunov exponent lim (1/n) log ||Df^n(x)v|| fails to exist for every x in some open set U and every nonzero v in some open cone V ⊂ R^2. The proof proceeds by constructing diffeomorphisms with specific oscillatory return times near a persistent homoclinic tangency, combining Newhouse-type perturbations with refinements from recent Lyapunov irregularity results to achieve the required non-existence on open sets.

Significance. If the central construction succeeds, the result would establish that non-existence of Lyapunov exponents is a robust, open-set phenomenon inside the Newhouse domain, strengthening earlier findings that were limited to Cantor sets or measure-zero sets. This would be a notable contribution to the study of irregular dynamics for surface diffeomorphisms with homoclinic tangencies, provided the uniformity over open U is rigorously controlled while remaining in the C^r-open Newhouse domain.

major comments (2)
  1. [§3] §3 (Oscillatory return constructions): The claim that refined Newhouse perturbations produce oscillatory return times yielding a positive limsup/liminf gap uniformly for all points of an open set U (and all nonzero vectors in an open cone V) is load-bearing for the open-set statement. The sketch does not yet supply explicit estimates showing that the return-map distortion remains controlled across a whole neighborhood without exiting the Newhouse domain or reducing the oscillation to a Cantor subset.
  2. [§4] §4 (Density argument): The proof that the set D is dense in the Newhouse domain while simultaneously guaranteeing an open U and cone V for each f ∈ D requires a clearer inductive or approximation scheme. It is unclear how the local oscillatory construction can be performed densely without destroying the openness of U or the C^r-robustness of the tangency.
minor comments (2)
  1. [Notation] The notation for the open cone V ⊂ R^2 should be accompanied by a brief geometric description or reference to a figure illustrating its opening angle relative to the tangency direction.
  2. [Introduction] A short paragraph comparing the new oscillatory-return construction with the standard Newhouse Cantor-set constructions would help readers assess the precise novelty of the refinements.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the points where the technical control needs to be made fully explicit. Both major comments concern the uniformity of the construction over open sets while remaining inside the C^r-open Newhouse domain; we believe these can be resolved by adding detailed estimates and a clearer approximation scheme. We outline the responses below and will incorporate the necessary expansions in the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (Oscillatory return constructions): The claim that refined Newhouse perturbations produce oscillatory return times yielding a positive limsup/liminf gap uniformly for all points of an open set U (and all nonzero vectors in an open cone V) is load-bearing for the open-set statement. The sketch does not yet supply explicit estimates showing that the return-map distortion remains controlled across a whole neighborhood without exiting the Newhouse domain or reducing the oscillation to a Cantor subset.

    Authors: We agree that explicit distortion bounds are required for the open-set claim. In the revision we will insert a new subsection in §3 that supplies these estimates. The construction proceeds by first realizing a persistent homoclinic tangency (which exists densely by Newhouse) and then applying a localized C^r-small perturbation supported in a small ball B around the tangency point. The return times to B are made to oscillate between two intervals I_1 and I_2 with a fixed gap by composing a sequence of Newhouse-type bumps whose C^r norms are controlled uniformly on B. Because the perturbation is localized and the Newhouse domain is C^r-open, the resulting map remains inside the domain. Distortion of the return map on the whole open set U = B minus a small neighborhood of the stable/unstable manifolds is bounded by the usual C^r estimates for surface diffeomorphisms (using the fact that the derivative along the orbit stays bounded away from zero on the chosen cone V, which is taken to be a small open cone around the unstable direction at the saddle). Consequently the limsup/liminf gap for (1/n)log||Df^n(x)v|| is uniform on U × V and does not collapse to a Cantor set. We will also verify that the same gap persists under further small perturbations, preserving openness of U. revision: yes

  2. Referee: [§4] §4 (Density argument): The proof that the set D is dense in the Newhouse domain while simultaneously guaranteeing an open U and cone V for each f ∈ D requires a clearer inductive or approximation scheme. It is unclear how the local oscillatory construction can be performed densely without destroying the openness of U or the C^r-robustness of the tangency.

    Authors: We will rewrite the density argument in §4 to make the approximation scheme explicit. Let g be any map in the Newhouse domain. By the density of homoclinic tangencies inside the Newhouse domain, there exists a C^r-small perturbation g_1 of g that possesses a persistent homoclinic tangency at some point p. We then apply the local oscillatory construction of §3 inside a sufficiently small ball B around p; the size of B and the C^r-norm of the perturbation are chosen so that (i) the perturbed map f remains inside the Newhouse domain (by openness), (ii) the tangency at p persists, and (iii) the open set U is taken to be a slightly smaller open disk inside B on which the return-time oscillation is uniform. Because the perturbation is supported in B and arbitrarily small, f can be made arbitrarily close to g. The cone V is chosen once and for all as a small open cone around the unstable eigenvector at the saddle; its openness is preserved under the C^1-small perturbation. No further induction is needed beyond the two-step approximation (first realize the tangency, then add the oscillation locally); the same local construction works at every scale because the Newhouse domain is open. We will add a diagram and a quantitative statement of the C^r-closeness to clarify that openness of U and V is never sacrificed. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external Newhouse theory plus independent new constructions

full rationale

The paper constructs a dense set D inside the Newhouse domain by perturbing near persistent homoclinic tangencies to produce specific oscillatory return times, then invokes established Newhouse results and recent Lyapunov irregularity theorems (external to this work) together with stated new refinements. No equation reduces to a fitted parameter renamed as prediction, no self-definitional loop appears, and no load-bearing uniqueness theorem is imported solely from the authors' prior work. The central claim therefore rests on externally supported constructions rather than on re-labeling its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard background from smooth dynamical systems and the definition of the Newhouse domain; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of C^r diffeomorphisms on compact surfaces
    Invoked throughout the construction and openness arguments.
  • domain assumption Existence and robustness properties of the Newhouse domain
    The ambient space in which the dense subset is constructed.

pith-pipeline@v0.9.0 · 5422 in / 1298 out tokens · 49326 ms · 2026-05-10T16:24:08.359915+00:00 · methodology

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