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Mostly nonuniformly sectional expanding systems

Vitor Araújo, Luciana Salgado

TL;DR

The paper introduces mostly nonuniformly sectional expanding systems (MNUSE) for singular flows, unifying SH, ASH, and MSH while exhibiting that MNUSE properly contains these classes. It proves that SH, ASH, and MSH attractors are MNUSE and provides explicit $C^r$ examples showing MNUSE can occur without SH, ASH, or MSH, clarifying the landscape of hyperbolicity in singular dynamics. Under smoothness and central-bundle Lyapunov-exponent conditions, attracting MNUSE sets carry a hyperbolic physical/SRB measure, with uniqueness in the transitive case, linking geometric structure to statistical behavior. The work further develops the theory by detailing the necessary definitions (partial hyperbolicity, singular/ASH, MSH, NUSH, and hyperbolic measures) and supplying constructive examples (Lorenz-type, Rovella-type) to delineate the boundaries between MNUSE and the traditional hyperbolicity notions, with implications for robustness and measure-theoretic properties of singular flows.

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 $C^1$ nonuniformly sectional hyperbolic set satisfying MNUSE, which is neither sectional hyperbolic nor asymptotically sectional hyperbolic. Moreover, under some smoothness assumptions together with, either the dimension or the sign of Lyapunov exponents along the central subbundle, we show that attracting MNUSE sets support a physical/SRB measure. This measure is unique if the dynamics is transitive.

Mostly nonuniformly sectional expanding systems

TL;DR

The paper introduces mostly nonuniformly sectional expanding systems (MNUSE) for singular flows, unifying SH, ASH, and MSH while exhibiting that MNUSE properly contains these classes. It proves that SH, ASH, and MSH attractors are MNUSE and provides explicit examples showing MNUSE can occur without SH, ASH, or MSH, clarifying the landscape of hyperbolicity in singular dynamics. Under smoothness and central-bundle Lyapunov-exponent conditions, attracting MNUSE sets carry a hyperbolic physical/SRB measure, with uniqueness in the transitive case, linking geometric structure to statistical behavior. The work further develops the theory by detailing the necessary definitions (partial hyperbolicity, singular/ASH, MSH, NUSH, and hyperbolic measures) and supplying constructive examples (Lorenz-type, Rovella-type) to delineate the boundaries between MNUSE and the traditional hyperbolicity notions, with implications for robustness and measure-theoretic properties of singular flows.

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 nonuniformly sectional hyperbolic set satisfying MNUSE, which is neither sectional hyperbolic nor asymptotically sectional hyperbolic. Moreover, under some smoothness assumptions together with, either the dimension or the sign of Lyapunov exponents along the central subbundle, we show that attracting MNUSE sets support a physical/SRB measure. This measure is unique if the dynamics is transitive.

Paper Structure

This paper contains 15 sections, 8 theorems, 14 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Statement of the results
  3. Conjectures and organization of the text
  4. Preliminary definitions and auxiliary results
  5. Singular hyperbolic structures
  6. Partial hyperbolic attracting sets for vector fields
  7. Singular/asymptotical sectional-hyperbolicity and uniform hyperbolicity
  8. Multi-singular hyperbolicity
  9. Weak versus uniform hyperbolicity - Hyperbolic Lemma
  10. Nonuniform sectional hyperbolicity
  11. Hyperbolic physical/SRB measures
  12. Proof of Theorem \ref{['mthm:SH-ASH-NUSH']}
  13. Proof of Theorem \ref{['mthm:nonmsh']}
  14. Proof of Theorem \ref{['mthm:nonsec']}
  15. Proof of Theorem \ref{['mthm:SRB']}

Key Result

Theorem A

Let $\Lambda \subset M$ be a compact invariant subset for a $C^2$ vector field. Then the following classes of attracting sets are mostly nonuniformly sectional hyperbolic (MNUSE) sets:

Figures (5)

  • Figure 1: A representation of the Linear Poincaré flow $P^t_x$ of a vector $v\in T_xM$ with $x\in M\setminus\operatorname{Sing}(X)$ and the orthogonal projection $O_{X_t x}:T_{X_tx}M\to N_{X_tx}$, with $N_{X_tx}=X(X_t x)^\perp$.
  • Figure 2: Local stable and unstable manifolds near $\sigma_0, \sigma_1$ and $\sigma_2$, and the ellipsoid $E$ on the left hand side; the trapping bi-torus $U$ on the right hand side.
  • Figure 3: The Lorenz attracting set including the geometric Lorenz attractor and the pair of hyperbolic saddle-type non-Lorenz like singularities $\sigma_1,\sigma_2$ with complex expanding eigenvalue.
  • Figure 4: Construction of a sectional-hyperbolic with no Lorenz-like singularity.
  • Figure 5: Lorenz one-dimensional transformation $f$ 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 ${\mathcal{O}}(p_{\pm})$.

Theorems & Definitions (25)

  • Theorem A
  • Remark 2.1: ASH and MSH attractors
  • Theorem B
  • Theorem C
  • Theorem D
  • Example 2.2: Robustly transitive set not admitting a dominated splitting
  • Remark 3.1: adapted metric
  • Definition 3.2
  • Definition 3.3
  • proof
  • ...and 15 more