On Anosov diffeomorphisms with asymptotically conformal periodic data

Boris Kalinin; Victoria Sadovskaya

Home/Archive/arXiv

arXiv · 0709.3790

On Anosov diffeomorphisms with asymptotically conformal periodic data

Boris Kalinin·Victoria Sadovskaya

View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider transitive Anosov diffeomorphisms for which every periodic orbit has only one positive and one negative Lyapunov exponent. We establish various properties of such systems including strong pinching, C^{1+β} smoothness of the Anosov splitting, and C^1 smoothness of measurable invariant conformal structures and distributions. We apply these results to volume preserving diffeomorphisms with two-dimensional stable and unstable distributions and diagonalizable derivatives of the return maps at periodic points. We show that a finite cover of such a diffeomorphism is smoothly conjugate to an Anosov automorphism of a torus. As a corollary we obtain local rigidity for such diffeomorphisms. We also establish a local rigidity result for Anosov diffeomorphisms in dimension three.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Boris Kalinin, Victoria Sadovskaya. 2008-03-29. On Anosov diffeomorphisms with asymptotically conformal periodic data. https://arxiv.org/abs/0709.3790

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

KEEP EXPLORING

Related papers

arXiv2026

Central Limit Theorem for Sequential Dynamical Systems

We present a general approach to establish the Central Limit Theorem with error bounds for sequential dynamical systems. The main tool we develop is the application to this setting of a projective metric on complex cones, following the ideas introduced by Rugh and Dubois. To demonstrate the power of the proposed setting, we apply it to both sequential expanding maps, where similar results are known, and to sequential dispersing billiards, for which no such results are currently known.

math.DS↗

arXiv2026

Reduced-Order Inference with Structure-Preserving Parametrization for Bending and Rotating Systems

Mechanical systems are often characterized only by their response to certain loads known from experiments or simulations. The obtained data can be used for various purposes: system analysis, design of mathematical models, or construction of reduced-order models for further simulations under different loading conditions. The use of data for reduced-order modeling is an important developing research direction, especially when the high-dimensional system operators are unknown and their low-dimensional approximation is required for accurate but fast simulations. Our goal is to obtain the low-dimensional surrogate model from the available input signal and deformation trajectory data, capturing the correct system behavior for the basic deformation cases, namely bending and rotation, which are present in almost every complex mechanical system. In this work, we propose a methodology to infer the system operators by solving a nonlinear unconstrained optimization problem. The methodology is based on the operator inference approach for second-order systems and includes a parametrization of the unknown operators that preserves their symmetric positive definite or skew-symmetric structure. We demonstrate the performance of the novel approach for three numerical examples that are used to simulate basic bending and rotating.

math.DS↗

arXiv2026

Affine mappings of translation surfaces: shrinking targets and Diophantine properties

Let $(X,ω)$ be a translation surface whose Veech group $Γ_ω$ is a lattice. We prove that the generic orbit of the group of affine homeomorphisms of $(X,ω)$ can be used to approximate each point of $X$ with Diophantine precision. The proof utilizes an induced $SL_2(\mathbb{R})$-action on a fiber bundle $Y_ω$ whose base is $SL_2(\mathbb{R})/Γ_ω$ and whose fiber is a quotient of $X$. We observe that this bundle embeds as an $SL_2(\mathbb{R})$-orbit closure in the moduli space of once marked translation surfaces, and hence we may invoke the spectral gap results of Avila and Gouëzel and a quantitative mean ergodic theorem for the $SL_2(\mathbb{R})$-action on the mean-zero, square-integrable functions on $Y_ω$.

math.DS↗