A Minimum Principle for Lyapunov Exponents and a Higher-dimensional Version of a Theorem of Mañé
نویسندگان
چکیده
We consider compact invariant sets Λ for C maps in arbitrary dimension. We prove that if Λ contains no critical points then there exists an invariant probability measure with a Lyapunov exponent λ which is the minimum of all Lyapunov exponents for all invariant measures supported on Λ. We apply this result to prove that Λ is uniformly expanding if every invariant probability measure supported on Λ is hyperbolic repelling. This generalizes a well known theorem of Mañé to the higher-dimensional setting.
منابع مشابه
Three-axis optimal control of satellite attitude based on Ponteryagin maximum principle
A long time ago, since the launch of the first artificial satellite in 1957, controling attitude of satellites has been considered by the designers and engineers of aerospace industry. Considering the importance of this issue various methods of control in response to this need have been presented and analyzed until now. In this paper, we propose and analyze a three-axis optimal control on the s...
متن کاملEntropy, Smooth Ergodic Theory, and Rigidity of Group Actions
This is a preliminary version of lecture notes based on a 6 hour course given at the workshop Dynamics Beyond Uniform Hyperbolicity held in Provo, Utah June 2017. The goal of this course is two-fold. First we will present a number of tools and results in the smooth ergodic theory of actions of higher-rank abelian groups including Lyapunov exponents, metric entropy, and the relationship between ...
متن کاملModel Based Method for Determining the Minimum Embedding Dimension from Solar Activity Chaotic Time Series
Predicting future behavior of chaotic time series system is a challenging area in the literature of nonlinear systems. The prediction's accuracy of chaotic time series is extremely dependent on the model and the learning algorithm. On the other hand the cyclic solar activity as one of the natural chaotic systems has significant effects on earth, climate, satellites and space missions. Several m...
متن کاملAn Lp-Lq-version Of Morgan's Theorem For The Generalized Fourier Transform Associated with a Dunkl Type Operator
The aim of this paper is to prove new quantitative uncertainty principle for the generalized Fourier transform connected with a Dunkl type operator on the real line. More precisely we prove An Lp-Lq-version of Morgan's theorem.
متن کاملExtension of Higher Order Derivatives of Lyapunov Functions in Stability Analysis of Nonlinear Systems
The Lyapunov stability method is the most popular and applicable stability analysis tool of nonlinear dynamic systems. However, there are some bottlenecks in the Lyapunov method, such as need for negative definiteness of the Lyapunov function derivative in the direction of the system’s solutions. In this paper, we develop a new theorem to dispense the need for negative definite-ness of Lyapunov...
متن کامل