[Paper Review] On the genericity of positive exponents of conservative skew products with two-dimensional fibers

Davi Obata, Mauricio Poletti|arXiv (Cornell University)|Sep 11, 2018

Mathematical Dynamics and Fractals10 references4 citations

TL;DR

This paper establishes the $C^1$-density and $C^r$-openness of positive Lyapunov exponents for conservative skew products over hyperbolic bases with two-dimensional fibers, using center bunching and perturbation techniques. It proves that in a generic $C^r$-open and $C^1$-dense subset of such systems, the integrated center Lyapunov exponent $L(f) > 0$, implying non-uniform hyperbolicity almost everywhere.

ABSTRACT

In this paper we study the existence of positive Lyapunov exponents for three different types of skew products, whose fibers are compact Riemannian surfaces and the action on the fibers are by volume preserving diffeomorphisms. These three types include skew products with a volume preserving Anosov diffeomorphism on the basis; or with a subshift of finite type on the basis preserving a measure with product structure; or locally constant skew products with Bernoulli shifts on the basis. We prove the $C^1$-density and $C^r$-openess of the existence of positive Lyapunov exponents on a set of positive measure in the space of such skew products.

Motivation & Objective

• To investigate the genericity of positive Lyapunov exponents in conservative skew products with two-dimensional fiber surfaces.
• To establish that positive center Lyapunov exponents are $C^1$-dense and $C^r$-open in the space of $C^r$-partially hyperbolic, volume-preserving skew products with center bunching.
• To extend results beyond Anosov or Bernoulli base dynamics by avoiding reliance on accessibility or full volume-preserving structure.
• To provide a framework applicable to skew products over subshifts of finite type and locally constant cocycles with product measures.

Proposed method

• Uses Oseledets' theorem to define Lyapunov exponents along the center direction and defines the integrated center exponent $L(f) = \int \lambda_c^+(x) \, d\text{leb}(x)$.
• Applies center bunching to ensure existence of linear holonomies, crucial for perturbation arguments.
• Employs $C^1$-perturbations to construct skew products with positive $L(f)$, relying on a pinching property in the fiber dynamics.
• Adapts techniques from [20] on measure convergence and holonomy maps to prove weak$^*$ convergence of invariant measures.
• Uses measurable holonomy maps $\mathcal{H}_k$ and their convergence to establish $\tilde{m}^k \to \tilde{m}$, ensuring regularity of conditional measures.
• Applies a measurable selection argument to show that the disintegration $\tilde{m}_p$ is $\mathbf{B}_0$-measurable, completing the measure-theoretic foundation.

Experimental results

Research questions

• RQ1 Is the set of $C^r$-partially hyperbolic, volume-preserving skew products with positive center Lyapunov exponent $C^1$-dense?
• RQ2 Is the set of such systems with positive $L(f)$ also $C^r$-open?
• RQ3 Can the results be extended to skew products over subshifts of finite type with measures having local product structure?
• RQ4 Can the two-dimensional fiber assumption be removed to allow higher-dimensional fibers?
• RQ5 Is the pinching property $C^r$-dense in the space of such skew products?

Key findings

• The set of $C^r$-partially hyperbolic, volume-preserving skew products with $L(f) > 0$ is $C^1$-dense in the space of such systems with center bunching.
• This same set is $C^r$-open, implying that positive Lyapunov exponents are robust under small $C^r$-perturbations.
• The existence of positive $L(f)$ implies non-uniform hyperbolicity almost everywhere, as $\lambda_c^+(x) > 0$ a.e. for such systems.
• The result holds for three classes: skew products over Anosov diffeomorphisms, subshifts of finite type with product-like measures, and locally constant skew products over Bernoulli shifts.
• The proof does not rely on accessibility or full volume-preserving structure, allowing broader applicability than prior results.
• The authors conjecture that the result extends to higher-dimensional fibers, with at least one positive Lyapunov exponent in the generic case.