Lyapunov exponents, dual Lyapunov exponents, and multifractal analysis.

Multifractal Analysis for Lyapunov Exponents on Nonconformal Repellers

For nonconformal repellers satisfying a certain cone condition, we establish a version of multifractal analysis for the topological entropy of the level sets of the Lyapunov exponents. Due to the nonconformality, the Lyapunov exponents are averages of nonadditive sequences of potentials, and thus one cannot use Birkhoff’s ergodic theorem neither the classical thermodynamic formalism. We use ins...

The Lyapunov Spectrum for Conformal Expanding Maps and Axiom-A Surface Diffeomorphisms

Lyapunov exponents measure the exponential rate of divergence of infinitesimally close orbits of a smooth dynamical system. These exponents are intimately related to the global stochastic behavior of the system and are fundamental invariants of a smooth dynamical system. In [EP], Eckmann and Procaccia suggested an analysis of Lyapunov exponents for chaotic dynamical systems. This suggestion was...

On a General Concept of Multifractality : Multifractal Spectra for Dimensions , Entropies , and Lyapunov Exponents

To appear in Israel Journal of Mathematics LYAPUNOV EXPONENTS FOR PRODUCTS OF MATRICES AND MULTIFRACTAL ANALYSIS. PART II: NON-NEGATIVE MATRICES

We continue the study in [11, 14] on the upper Lyapunov exponents for products of matrices. Here the matrix function M(x) is only assumed to be non-negative. In general, the variational formula about Lyapunov exponents we obtained in part I does not hold in this setting. Anyway we focus our interest on a special case where M(x) takes finite values M1, . . ., Mm. In this case we prove the variat...

Nonhyperbolic Step Skew-products: Entropy Spectrum of Lyapunov Exponents

We study the fiber Lyapunov exponents of step skew-product maps over a complete shift of N , N ≥ 2, symbols and with C1 diffeomorphisms of the circle as fiber maps. The systems we study are transitive and genuinely nonhyperbolic, exhibiting simultaneously ergodic measures with positive, negative, and zero exponents. We derive a multifractal analysis for the topological entropy of the level sets...