Mostly Nonuniform Sectional Expanding (MNUSE)

• MNUSE is a weak-hyperbolicity condition in singular flows where two-dimensional volumes in the center-unstable direction expand exponentially on average on a positive Lebesgue measure set.
• It extends traditional sectional hyperbolicity by accommodating nonuniform expansion and slow recurrence near singularities, thereby encompassing ASH and multisingular hyperbolicity.
• Proof techniques combine partial hyperbolicity, nonuniform expansion estimates via the linear Poincaré flow, and Pesin theory to establish the existence and uniqueness of physical SRB measures.

Searching arXiv for the cited papers to ground the article in current records. Mostly Nonuniformly Sectional Expanding (MNUSE) denotes a weak-hyperbolicity condition for singular flows in which two-dimensional volumes inside a center-unstable direction expand exponentially on average for a positive-Lebesgue-measure set of initial conditions, rather than uniformly for all points and times. In the recent literature, MNUSE has been formulated for partially hyperbolic attracting sets with singularities, related to sectional hyperbolicity, asymptotically sectional hyperbolicity, and multisingular hyperbolicity, and used to derive existence, finiteness, and, in transitive settings, uniqueness of physical/SRB measures under $C^2$ hypotheses (Araújo et al., 8 Aug 2025).

1. Formal setting and core definition

Let $M$ be a compact Riemannian manifold, $X\in C^1(M)$ or $G\in C^r(M)$, and $\phi_t$ the generated flow. The ambient object is typically a forward-invariant set $U\subset M$ with maximal invariant set

$\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$

or, in trapping-region form,

$\Lambda=\bigcap_{t\ge0}\phi_t(U).$

The basic structural assumption is a partially hyperbolic splitting

$T_\Lambda M = E^s \oplus E^{cu},$

with $E^s$ uniformly contracting and $M$0 dominating $M$1. One standard formulation requires a constant $M$2 such that for all $M$3 and $M$4,

$M$5

For flows, the neutral flow direction lies in $M$6, so domination already enforces the partial-hyperbolic asymmetry between $M$7 and $M$8 (Araujo et al., 24 Nov 2025).

In the 2025 formulation, one fixes $M$9, sets $X\in C^1(M)$0, extends $X\in C^1(M)$1 continuously to $X\in C^1(M)$2, and says that $X\in C^1(M)$3 is mostly nonuniformly sectional expanding if there exist a positive-Lebesgue-measure set $X\in C^1(M)$4 and $X\in C^1(M)$5 such that for every $X\in C^1(M)$6,

$X\in C^1(M)$7

This is an averaged two-dimensional co-volume expansion condition along $X\in C^1(M)$8: on a positive-volume set of initial conditions, the average two-dimensional Jacobian of $X\in C^1(M)$9 along $G\in C^r(M)$0 grows exponentially in forward time (Araújo et al., 8 Aug 2025).

A related description emphasizes that the adjective “mostly” refers only to the size of the set of initial conditions: the sectional-expansion property is not required everywhere, but on a large set in the measure-theoretic sense, namely a set of positive Lebesgue measure (Araujo et al., 24 Nov 2025).

2. Continuous-time, discrete-time, and singularity control

An earlier formulation for partially hyperbolic attracting sets expresses sectional expansion directly in continuous time. On a positive-volume set, one requires

$G\in C^r(M)$1

Equivalently, one may work with the time-one map $G\in C^r(M)$2 and the linear Poincaré flow $G\in C^r(M)$3 on the normal bundle $G\in C^r(M)$4, demanding

$G\in C^r(M)$5

where $G\in C^r(M)$6. In this framework, MNUSE means partial hyperbolicity on $G\in C^r(M)$7, nonuniform sectional expansion on a positive-Lebesgue-measure subset, and slow recurrence to equilibria (Araujo et al., 2022).

The slow-recurrence condition is designed to control visits near singularities. It is stated as follows: for every $G\in C^r(M)$8 there exists $G\in C^r(M)$9 such that for Lebesgue-almost every $\phi_t$0,

$\phi_t$1

where $\phi_t$2 is the $\phi_t$3-truncated distance to the singular set. In codimension two, the same paper shows that one can weaken the requirement and ask only for expansion along a sequence of times, formulated through $\phi_t$4 (Araujo et al., 2022).

These formulations all focus on two-plane growth inside $\phi_t$5. In the language of Lyapunov theory, they encode positivity of sectional Lyapunov exponents in every two-dimensional direction in $\phi_t$6, at least on a positive-volume subset. This distinguishes MNUSE from uniform sectional-hyperbolic conditions, which impose deterministic lower bounds for all points and all times rather than asymptotic or averaged negativity of inverse exterior-square norms.

3. Relation to sectional, asymptotical, and multisingular hyperbolicities

MNUSE was introduced as a notion for singular flows that contains sectional hyperbolicity and also encompasses asymptotically sectional and multisingular hyperbolicities (Araújo et al., 8 Aug 2025). The nearby notions are standardly organized by the way two-dimensional expansion is imposed.

Notion	Defining feature
Sectional hyperbolicity (SH)	Uniform lower bound $\phi_t$7 for every $\phi_t$8, $\phi_t$9, and every two-plane $U\subset M$0
Asymptotically sectional hyperbolicity (ASH)	$U\subset M$1 for every two-plane $U\subset M$2 and every orbit avoiding stable manifolds of singularities
Multisingular hyperbolicity (MSH)	Dominated splitting of the linear Poincaré flow $U\subset M$3, uniform contraction/expansion outside neighborhoods of singularities, plus Lorenz-type eigenvalue conditions
MNUSE	Average two-dimensional expansion on a positive-Lebesgue-measure set

For $U\subset M$4 vector fields, one inclusion theorem asserts

$U\subset M$5

The same work also states that there are explicit examples of MNUSE attracting sets which are not SH, not $U\subset M$6-SH, not ASH, and not MSH, so the inclusions are strict (Araújo et al., 8 Aug 2025).

A related precursor appears in the study of partially hyperbolic flows with two-dimensional center. There the condition is phrased as “Gibbs sectional expanding”: for every Gibbs $U\subset M$7-state $U\subset M$8 supported on the attractor and every two-plane $U\subset M$9,

$\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$0

with $\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$1. This is not identical to the positive-volume definition of MNUSE, but it occupies the same conceptual space: nonuniform positivity of sectional growth sufficient to recover SRB/physical measures in partially hyperbolic flow settings (Mi et al., 2020).

4. Physical and SRB measures

A central motivation for MNUSE is the production of physical/SRB measures on singular attractors. One theorem for $\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$2 vector fields states that a partially hyperbolic attracting set $\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$3 is mostly asymptotically sectional expanding on a positive-volume set $\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$4 if and only if there exists an ergodic hyperbolic physical/SRB measure $\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$5 for the flow with

$\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$6

Moreover, there are at most finitely many such ergodic physical/SRB measures $\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$7, each satisfying the Pesin formula

$\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$8

so that they are $\Lambda=\bigcap_{t\ge0}\overline{\phi_t(U)}$9-Gibbs states, and their basins cover $\Lambda=\bigcap_{t\ge0}\phi_t(U).$0 up to zero volume (Araujo et al., 2022).

A later theorem gives a complementary existence statement for MNUSE in the $\Lambda=\bigcap_{t\ge0}\phi_t(U).$1 setting. If $\Lambda=\bigcap_{t\ge0}\phi_t(U).$2 admits a forward-invariant compact $\Lambda=\bigcap_{t\ge0}\phi_t(U).$3 whose maximal invariant set $\Lambda=\bigcap_{t\ge0}\phi_t(U).$4 is MNUSE on a positive-volume set $\Lambda=\bigcap_{t\ge0}\phi_t(U).$5, and if either $\Lambda=\bigcap_{t\ge0}\phi_t(U).$6 or the no-negative-central-exponents condition

$\Lambda=\bigcap_{t\ge0}\phi_t(U).$7

holds, then $\Lambda=\bigcap_{t\ge0}\phi_t(U).$8 supports an ergodic hyperbolic physical/SRB measure. If $\Lambda=\bigcap_{t\ge0}\phi_t(U).$9 is transitive, this SRB measure is unique (Araújo et al., 8 Aug 2025).

The precursor results for partially hyperbolic attractors with $T_\Lambda M = E^s \oplus E^{cu},$0 are also relevant. In that setting, a $T_\Lambda M = E^s \oplus E^{cu},$1 vector field already guarantees the existence of at least one SRB measure, while the $T_\Lambda M = E^s \oplus E^{cu},$2 Gibbs sectional expanding hypothesis yields only finitely many SRB/physical measures and a full-Lebesgue-measure cover of the topological basin by their basins; transitivity implies uniqueness (Mi et al., 2020). Taken together, these results place MNUSE within a broader strategy for converting nonuniform sectional expansion into measure-theoretic predictability.

5. Mechanisms of proof

The proof strategy for MNUSE-based SRB existence passes through the linear Poincaré flow and nonuniform expansion estimates transverse to the flow direction. In one formulation, the forward-time averaged estimate (MNUE) on a positive-volume set yields a discrete-time nonuniform expansion condition

$T_\Lambda M = E^s \oplus E^{cu},$3

where $T_\Lambda M = E^s \oplus E^{cu},$4 is the linear Poincaré flow in the direction normal to $T_\Lambda M = E^s \oplus E^{cu},$5. This produces positive Lyapunov exponents on $T_\Lambda M = E^s \oplus E^{cu},$6 (Araújo et al., 8 Aug 2025).

Once partial hyperbolicity is combined with such nonuniform expansion, the argument enters the standard Pesin-theoretic regime. Oseledets’ theorem yields measurably varying unstable directions, local unstable manifolds are constructed, absolute continuity of conditional measures along unstable leaves is established, and the Ledrappier–Young or Katok formula is used to produce an invariant ergodic probability measure $T_\Lambda M = E^s \oplus E^{cu},$7 satisfying

$T_\Lambda M = E^s \oplus E^{cu},$8

with basin of positive Lebesgue measure. In transitive settings, uniqueness follows by excluding coexistence of two ergodic SRB limits on a single dense orbit (Araújo et al., 8 Aug 2025).

The flow-specific difficulties had already been highlighted in the earlier partially hyperbolic theory. Those arguments work with Poincaré sections and the linear Poincaré flow to remove the neutral flow direction, use Katok shadowing in a flow version, and exploit Pliss-type arguments to produce local center-unstable disks in transverse sections. The resulting geometric control is then lifted back to the flow and saturated by strong stable leaves to obtain uniform positive-volume pieces of basins; finiteness follows because each physical measure must claim at least a fixed amount of volume inside the basin (Mi et al., 2020).

In settings with singularities, slow recurrence plays an additional technical role. It controls the contribution of trajectories visiting neighborhoods of equilibria and allows generalized Pesin-entropy estimates to remain effective even when regular orbits accumulate on singularities. This is the mechanism behind the equivalence between nonuniform sectional expansion plus slow recurrence and the existence of physical/SRB measures in the earlier continuous-time theory (Araujo et al., 2022).

6. Examples, strict inclusions, and higher-codimensional developments

Several classical families lie inside the MNUSE framework. These include Anosov or uniformly hyperbolic flows, singular-hyperbolic attractors of Lorenz type, multidimensional Lorenz attractors, sectional-hyperbolic attractors in higher $T_\Lambda M = E^s \oplus E^{cu},$9, and contracting Lorenz (Rovella) attractors. In these examples, MNUSE either holds trivially from uniform sectional expansion or follows from nonuniform expansion plus slow recurrence, yielding physical/SRB measures by the general theory (Araujo et al., 2022).

A key strict-inclusion example is a three-dimensional geometric Lorenz-type attractor whose Poincaré return on a cross-section $E^s$0 is a skew product

$E^s$1

where $E^s$2 is the Manneville–Pomeau–type map

$E^s$3

This map is ergodic with respect to Lebesgue and has indifferent fixed points at $E^s$4. The corresponding flow has two periodic orbits $E^s$5 that are null Lyapunov along one direction, so the attractor is not uniformly sectional-hyperbolic and not ASH. Nevertheless,

$E^s$6

where $E^s$7 is the SRB-type measure of the suspension, and the attractor is MNUSE but not $E^s$8-SH nor ASH (Araújo et al., 8 Aug 2025).

The higher-codimensional extension is developed for asymptotically sectional hyperbolic attracting sets of any finite codimension. Sufficient conditions are obtained for the existence of physical/SRB measures for asymptotically sectionally hyperbolic attracting sets with any finite co-dimension, extending the co-dimension two case. The examples include attractors with non-sectional hyperbolic equilibria and attractors with sectional-hyperbolic equilibria of mixed type, namely a Lorenz-like singularity together with a Rovella-like singularity in a transitive set. The paper also gives higher-dimensional versions of contracting Lorenz-like attractors to which the criteria apply, obtaining a physical/SRB measure with full ergodic basin, and adapts these constructions to produce higher co-dimensional non-uniformly sectional expanding attractors and asymptotical $E^s$9-sectional hyperbolic attractors which are not non-uniformly $M$00-expanding for any finite $M$01 (Araujo et al., 24 Nov 2025).

A plausible implication is that MNUSE should be viewed not as a single rigid condition but as a measure-theoretic expansion principle that organizes a spectrum of weak-hyperbolicity classes for singular flows. This reading is consistent with the open problems explicitly identified in the literature: removing the $M$02 hypothesis, weakening domination to continuity or Hölder continuity of the splitting, proving statistical stability or continuity of the SRB measure under $M$03 perturbations, and extending the theory within higher-dimensional $M$04-sectional frameworks (Araújo et al., 8 Aug 2025).