• Corpus ID: 211259208

Rotation numbers of perturbations of smooth dynamics

@article{Gourmelon2020RotationNO,
  title={Rotation numbers of perturbations of smooth dynamics},
  author={Nicolas Gourmelon},
  journal={arXiv: Dynamical Systems},
  year={2020}
}
  • N. Gourmelon
  • Published 17 February 2020
  • Mathematics
  • arXiv: Dynamical Systems
The small perturbations of a linear cocycle have a relative rotation number associated to each pair formed of an invariant measure of the base dynamics and a $2$-dimensional bundle of the finest dominated splitting (provided that some orientation is preserved). The properties of that relative rotation number allow some small steps towards dichotomies between complex eigenvalues and dominated splittings in higher dimensions and higher regularity. 
3 Citations

SIMPLICITY OF LYAPUNOV SPECTRUM FOR GEODESIC FLOWS IN NEGATIVE CURVATURE

We study the Lyapunov spectrum of the geodesic flow of negatively curved 1/4-pinched Riemannian manifolds. We show that the space of metrics with simple Lyapunov spectrum with respect to the

Parameter Spaces of Locally Constant Cocycles

This article concerns the locus of all locally constant $\mathrm{SL}(2,\mathbb{R})$-valued cocycles that are uniformly hyperbolic, called the hyperbolic locus. Using the theory of semigroups of

Simplicity of the Lyapunov spectrum for classes of Anosov flows

<jats:p>For <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" xlink:href="S0143385722000268_inline1.png" />

References

SHOWING 1-10 OF 27 REFERENCES

Perturbations of the derivative along periodic orbits

We show that a periodic orbit of large period of a diffeomorphism or flow either admits a dominated splitting of a prescribed strength or can be turned into a sink or a source by a $C^1$-small

Density of positive Lyapunov exponents for symplectic cocycles

  • Disheng Xu
  • Mathematics
    Journal of the European Mathematical Society
  • 2019
We prove that Sp(2d;R), HSp(2d) and pseudo unitary cocycles with at least one non-zero Lyapunov exponent are dense in all usual regularity classes for non periodic dynamical systems. For

Lyapunov exponents with multiplicity 1 for deterministic products of matrices

We exhibit an explicit criterion for the simplicity of the Lyapunov spectrum of linear cocycles, either locally constant or dominated, over hyperbolic (Axiom A) transformations. This criterion is

Adapted metrics for dominated splittings

  • N. Gourmelon
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2007
Abstract A Riemannian metric is adapted to a hyperbolic set of a diffeomorphism if, in this metric, the expansion/contraction of the unstable/stable directions is seen after only one iteration. A

Some characterizations of domination

We show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates. Then we consider sets Σ in $${{\rm GL}(d, \mathbb R)}$$

A structural stability theorem

precise definition of structural stability given in ? 5 is somewhat stronger; it demands that the conjugacy 9q can be found within an arbitrary CO neighborhood of the identity when g is sufficiently

Internal perturbations of homoclinic classes: non-domination, cycles, and self-replication

Abstract Conditions are provided under which lack of domination of a homoclinic class yields robust heterodimensional cycles. Moreover, so-called viral homoclinic classes are studied. Viral classes

Towards a Classification for Quasiperiodically Forced Circle Homeomorphisms

Poincaré's classification of the dynamics of homeomorphisms of the circle is one of the earliest, but still one of the most elegant, classification results in dynamical systems. Here we generalize

Monotonic cocycles

We develop a “local theory” of multidimensional quasiperiodic $${\mathrm {SL}}(2,{\mathbb R})$$SL(2,R) cocycles which are not homotopic to a constant. It describes a $$C^1$$C1-open neighborhood of

Necessary conditions for stability of diffeomorphisms

S. Smale has recently given sufficient conditions for a diffeomorphism to be Q-stable and conjectured the converse of his theorem. The purpose of this paper is to give some limited results in the