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Star flows: a characterization via Lyapunov functions

Luciana Silva Salgado

Abstract

In this work, it is presented a characterization of star property for a $C^1$ vector field based on Lyapunov functions. It is also obtained conditions to strong homogeneity for singular sets by using the notion of infinitesimal Lyapunov functions. As an application, we obtain some results related to singular hyperbolic sets for flows.

Star flows: a characterization via Lyapunov functions

Abstract

In this work, it is presented a characterization of star property for a vector field based on Lyapunov functions. It is also obtained conditions to strong homogeneity for singular sets by using the notion of infinitesimal Lyapunov functions. As an application, we obtain some results related to singular hyperbolic sets for flows.

Paper Structure

This paper contains 13 sections, 21 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Some definitions and auxiliary results
  3. Fields of quadratic forms
  4. Positive and negative cones
  5. $\EuScript{J}$-separated linear maps
  6. $\EuScript{J}$-symmetrical matrixes and $\EuScript{J}$-selfadjoint operators
  7. Lyapunov exponents
  8. Some applications
  9. Some results about partial and sectional hyperbolicity from $\EuScript{J}$-separation
  10. Proof of Corollaries \ref{['mcor:2-sec-exp-J-monot']} and \ref{['thm:lyap-est-sec-hyp']}
  11. Proof of Theorems \ref{['cor:star-flow']}, \ref{['mthm:strong-homog-equiv']} and Corollary \ref{['mcor:strong-homog']}
  12. Extended Linear Poicaré Flow
  13. Proof of Theorem \ref{['mthm:strong-homog-to-quad']}

Key Result

Theorem \oldthetheorem

ArSal2012 Let $X_t$ be a flow defined on a positive invariant subset $U \subset M$, $A_t(x)$ a cocycle over $X_t$ on $U$ and $D(x)=\lim_{t\to0}(A_t(x)-Id)/t$ its infinitesimal generator. Then, for all $v\in E_x$ and $x\in U$, where and $D(x)^*$ denotes the adjoint of the linear map $D(x):E_x\to E_x$ with respect to the adapted inner product at $x$.

Figures (2)

  • Figure 1: Geometric Lorenz attractor is an example of singular star flow
  • Figure 2: Lorenz equations simulated by Octave, it is an example of singular star system

Theorems & Definitions (36)

  • Definition 1
  • Remark \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Sun's Theorem
  • Definition 2
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • ...and 26 more