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arxiv: 2508.06233 · v4 · submitted 2025-08-08 · 🧮 math.DS

Mostly nonuniformly sectional expanding systems

Vitor Ara\'ujo , Luciana Salgado This is my paper

Pith reviewed 2026-05-19 00:31 UTC · model grok-4.3

classification 🧮 math.DS
keywords mostly nonuniformly sectional expandingMNUSEsingular flowsphysical measuresSRB measuressectional hyperbolicityLorenz-like attractorsattracting sets
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The pith

A mostly nonuniformly sectional expanding property unifies several hyperbolicity notions and guarantees physical measures for singular attractors of any finite codimension.

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

This paper introduces mostly nonuniformly sectional expanding systems for singular flows. The new notion includes sectional hyperbolicity, asymptotically sectional hyperbolicity, and multisingular hyperbolicity. It provides sufficient conditions for the existence of physical or SRB measures on asymptotically sectionally hyperbolic attracting sets regardless of codimension. Examples are given of higher-dimensional contracting Lorenz-like attractors and mixed singularity cases that satisfy the criteria and possess full ergodic basins. The work also constructs higher codimensional nonuniformly sectional expanding attractors.

Core claim

We introduce the notion of mostly nonuniform sectional expanding (MNUSE) for singular flows which encompasses the notions of sectional hyperbolicity, asymptotically sectional and multisingular hyperbolicity. We exhibit an example of a vector field of class C^r, r > 1, whose flow exhibits a nonuniformly sectional hyperbolic set satisfying MNUSE, which is neither sectional hyperbolic nor asymptotically sectional hyperbolic. We obtain sufficient conditions for the existence of physical/SRB measures for asymptotically sectionally hyperbolic attracting sets with any finite co-dimension, extending the co-dimension two case, and provide examples of such attractors with non-sectional hyperbolic or混合

What carries the argument

The mostly nonuniformly sectional expanding (MNUSE) property, which controls nonuniform expansion along sectional directions for singular flows and enables proofs of physical measures even when standard sectional hyperbolicity fails.

If this is right

  • Physical or SRB measures exist for asymptotically sectionally hyperbolic attracting sets of any finite codimension.
  • Higher-dimensional versions of contracting Lorenz-like and Rovella-like attractors admit physical measures with full ergodic basin.
  • Attractors containing both sectional-hyperbolic and non-sectional equilibria in a transitive set still support SRB measures.
  • Non-uniformly sectional expanding attractors can be realized with central direction dimension greater than two.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The MNUSE framework may extend to classify attractors arising in models of fluid flow or plasma dynamics with multiple singularities.
  • Similar conditions could be tested for existence of physical measures in systems with time-dependent or stochastic perturbations.
  • The mixed-singularity examples suggest that statistical properties remain robust when sectional and non-sectional behaviors coexist in one orbit closure.

Load-bearing premise

The vector field is C^r with r greater than 1 and there exists a nonuniformly sectional hyperbolic set that satisfies the MNUSE property but is neither sectional hyperbolic nor asymptotically sectional hyperbolic.

What would settle it

Construct an asymptotically sectionally hyperbolic attracting set of codimension three that meets all stated conditions yet possesses no physical measure with full ergodic basin.

Figures

Figures reproduced from arXiv: 2508.06233 by Luciana Salgado, Vitor Ara\'ujo.

Figure 1
Figure 1. Figure 1: P t x : Nx → NXtx by P t x = OXtx ◦ DXt(x) [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Local stable and unstable manifolds near σ0, σ1 and σ2, and the ellipsoid E on the left hand side; the trapping bi-torus U on the right hand side. eigenvalue with positive real part. Hence, the two-dimensional unstable manifolds Wu (σ1) and Wu (σ2) are contained in the attracting set [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The Lorenz attracting set including the geometric Lorenz at￾tractor and the pair of hyperbolic saddle-type non-Lorenz like singularities σ1, σ2 with complex expanding eigenvalue. This attracting set A = Λ ∪ Wu (σ1) ∪ W2 (σ2) is connected, since these unstable man￾ifold accumulate inside the geometric Lorenz attractor, and clearly sectional-hyperbolic (and singular-hyperbolic) since the hyperbolic splitting… view at source ↗
Figure 4
Figure 4. Figure 4: Construction of a sectional-hyperbolic with no Lorenz-like singularity. After this, consider a vector field X2 on a solid torus M2 presenting two hyperbolic singularities σ1 (a repelling one) and σ2 (a saddle with one-dimensional stable direction). Note that X2 is transverse to the boundary ∂M2 pointing outwardly. Moreover, there is a meridian curve C ′ in ∂M2 which is the boundary of a disk contained in t… view at source ↗
Figure 5
Figure 5. Figure 5: Lorenz one-dimensional transformation with repelling fixed points at the extremes of the interval on the left; and the geometric Lorenz construction with this map as the quotient over the contracting invariant foliation on the cross-section S, with two corresponding periodic saddle-type periodic orbits O(p±). As usual in the geometric Lorenz construction, we assume that in the cube I 3 the flow is linear G… view at source ↗
read the original abstract

We introduce the notion of \emph{mostly nonuniform sectional expanding} (MNUSE) for singular flows which encompasses the notions of sectional hyperbolicity, asymptotically sectional and multisingular hyperbolicity. We exhibit an example of a vector field of class $C^r, r > 1$, whose flow exhibits a nonuniformly sectional hyperbolic set satisfying MNUSE, which is neither sectional hyperbolic nor asymptotically sectional hyperbolic. We obtain sufficient conditions for the existence of physical/SRB measures for asymptotically sectionally hyperbolic attracting sets with any finite co-dimension, extending the co-dimension two case. We provide examples of such attractors, either with non-sectional hyperbolic equilibria, or with sectional-hyperbolic equilibria of mixed type, i.e., with a Lorenz-like singularity together with a Rovella-like singularity in a transitive set. These are higher-dimensional versions of contracting Lorenz-like attractors (also known as Rovella-like attractors) to which we apply our criteria to obtain a physical/SRB measure with full ergodic basin. We also adapt the previous examples to obtain higher co-dimensional (i.e. with central direction of dimension greater than $2$) non-uniformly sectional expanding attractors.

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

0 major / 2 minor

Summary. The paper introduces the notion of mostly nonuniformly sectional expanding (MNUSE) for singular flows, which generalizes sectional hyperbolicity, asymptotically sectional hyperbolicity, and multisingular hyperbolicity. It exhibits a concrete example of a C^r (r>1) vector field whose flow has a nonuniformly sectional hyperbolic set satisfying MNUSE but failing to be sectional or asymptotically sectional hyperbolic. Sufficient conditions are derived for the existence of physical/SRB measures on asymptotically sectionally hyperbolic attracting sets of arbitrary finite co-dimension, extending the co-dimension-two case. Examples of such attractors (with non-sectional or mixed-type equilibria, including higher co-dimensional central bundles) are constructed and the criteria applied to obtain SRB measures with full ergodic basins.

Significance. If the example verification and the extension of the SRB existence arguments hold, the work provides a strictly broader framework than prior notions for establishing physical measures in singular flows with central bundles of dimension greater than two. The explicit C^r construction demonstrating the proper inclusion of MNUSE and the adaptation of nonuniform expansion estimates to higher co-dimensions constitute a concrete advance in the ergodic theory of singular attractors.

minor comments (2)
  1. [Theorem 5.2] In the statement of the main existence theorem for SRB measures, the precise dependence on the MNUSE constants (e.g., the lower bound on the expansion rate along the central bundle) should be made explicit, as it is used in the adapted proof from the co-dimension-two case.
  2. [Figure 7] The figures illustrating the phase portrait of the higher co-dimensional example would benefit from clearer labeling of the central directions and the location of the mixed-type equilibria.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report, so we have no points to address point-by-point at this stage. We will prepare a revised version incorporating any minor editorial or technical suggestions that may arise during the process.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines the new MNUSE property explicitly as a generalization that properly contains sectional hyperbolicity and asymptotically sectional hyperbolicity, then constructs a concrete C^r vector field example (r>1) whose flow satisfies MNUSE while violating the stricter older notions. Sufficient conditions for SRB/physical measures on asymptotically sectionally hyperbolic attractors of arbitrary finite co-dimension are obtained by adapting co-dimension-two arguments to the higher-dimensional central bundle case, with explicit verification on the provided examples (including mixed Lorenz/Rovella singularities). No load-bearing step reduces by definition or by self-citation to the target result itself; all key verifications and extensions are carried out directly in the manuscript without hidden uniformity assumptions or circular appeals.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on standard background results in smooth dynamical systems (existence and uniqueness of flows for C^r vector fields, r>1) together with the specific geometric constructions of the examples; no free parameters or invented entities are visible from the abstract.

axioms (1)
  • standard math Vector fields of class C^r (r>1) generate well-defined flows on the manifold
    Invoked implicitly when stating the existence of the flow and the nonuniformly sectional hyperbolic set.

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