Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2112.11388

Strong Fast Invertibility and Lyapunov Exponents for Linear Systems

Florian Noethen
View original source ↗Read source document ↗Export metadata ↓

Abstract

In 2019 Anthony Quas, Philippe Thieullen and Mohamed Zarrabi introduced the concept of strong fast invertibility for linear cocycles. It relates the growth of volumes between different initial times and, together with a condition on singular value gaps, yields the existence of a dominated splitting of the dynamics. The properties of this splitting largely coincide with those for systems with stable Lyapunov exponents. In this article, we take a closer look at strongly fast invertible systems with bounded coefficients. By linking the dimensions at which a system admits strong fast invertibility to the multiplicities of Lyapunov exponents, we are able to give a full characterization of regular strongly fast invertible systems similar to that of systems with stable Lyapunov exponents. In particular, we show that the stability of Lyapunov exponents implies strong fast invertibility (even in the absence of regularity). Central to our arguments are certain induced systems on spaces of exterior products that represent the evolution of volumes. Finally, we derive convergence results for the computation of Lyapunov exponents via Benettin's algorithm using perturbation theory. While the stronger assumption of stable Lyapunov exponents clearly leaves more freedom on how to choose stepsizes, we derive conditions for the stepsizes with which convergence can be ensured even if a system is only strongly fast invertible.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Florian Noethen. 2021-12-21. Strong Fast Invertibility and Lyapunov Exponents for Linear Systems. https://doi.org/10.14232/ejqtde.2025.1.33

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Admissible Fourier Lengths, KAM Reducibility, and Spectral Applications

We develop a perturbative KAM reducibility theory for one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles based on an admissible Fourier length $\ell$. The regularity relevant to the iteration is measured by positive adapted Fourier width rather than ordinary smoothness in the Euclidean length $|n|$. The same length governs Fourier decay, truncation and resonance scales, and the arithmetic condition controlling the small divisors. This framework contains the classical analytic and Gevrey settings, while non-monotone choices of $\ell$ allow classical nowhere differentiable Weierstrass-type perturbations and continuous perturbations outside every positive H\"older class. As spectral applications, we obtain purely absolutely continuous spectrum for every phase and $1/2$-H\"older continuity of the integrated density of states for the associated quasiperiodic Schr\"odinger operators. The Aubry dual has pure point spectrum for Lebesgue almost every dual phase, with eigenfunctions exponentially localized in the metric induced by $\ell$. We also construct nowhere differentiable quasiperiodic potentials with purely absolutely continuous Cantor spectrum.

math.DS↗
arXiv2026

Dynamics inside the attracting basins of some skew products

Polynomial skew products in $\mathbb{C}^2$ are maps of the form $F(z,w)=(P(z),Q(z,w))$, where $P$ and $Q$ are polynomials. Their local dynamics have been widely investigated. In this paper, we study the global dynamics inside Fatou components of some skew products. We consider all the inverse images in a Fatou component of a given point and use the Kobayashi metric to measure the distance between points. In the cases we consider, there are always arbitrarily large Kobayashi balls in the complement of these inverse sets.

math.DS↗
arXiv2026

Ergodicity of dynamical systems without uniqueness of orbits

Recently, there has been considerable interest in the study of non-deterministic dynamical systems. To analyze the chaotic behavior of such systems from a measure-theoretic viewpoint, it is desirable to consider ergodicity. However, the classical definition of ergodicity involves invariant sets, whose definition is not unique for non-deterministic dynamical systems. Thus, we are led to the question of which invariance yields an interesting definition of ergodicity. Here, we propose a definition based on the strong backward invariance and show that analogs of classical results hold. We also consider implications of the Birkhoff ergodic theorem for systems without uniqueness of orbits.

math.DS↗