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We present necessary and sufficient conditions in the spirit of Rolewicz under which all Lyapunov exponents of a given linear cocycle are either positive or negative. As a consequence, we formulate new conditions for the existence of the so-called tempered exponential dichotomies. We consider cocycles over both maps and flows. [ABSTRACT FROM AUTHOR]

We consider a finite-dimensional deterministic dynamical system with the global attractor 𝒜 which supports a unique ergodic probability measureP. The measurePcan be considered as the uniform long-term mean of the trajectories staying in a bounded domainDcontaining 𝒜. We perturb the dynamical system by a multiplicative heavy tailed Lévy noise of small intensity ϵ > 0 and solve the asymptotic first exit time and location problem fromDin the limit of ϵ↘0. In contrast to the case of Gaussian perturbations, the exit time has an algebraic exit rate as a function of ϵ, just as in the case when 𝒜 is a stable fixed point studied earlier in [9, 14, 19, 26]. As an example, we study the first exit problem from a neighborhood of the stable limit cycle for the Van der Pol oscillator perturbed by multiplicative α-stable Lévy noise. [ABSTRACT FROM PUBLISHER]

It has been shown in Gaidashev and Johnson [D. Gaidashev and T. Johnson, Dynamics of the universal area-preserving map associated with period doubling: stable sets, J. Mod. Dyn. 3(4) (2009), pp. 555–587.] and Gaidashev et al. [D. Gaidashev, T. Johnson, and M. Martens, Rigidity for infinitely renormalizable area-preserving maps, in preparation.] that infinitely renormalizable area-preserving maps admit invariant Cantor sets with a maximal Lyapunov exponent equal to zero. Furthermore, the dynamics on these Cantor sets for any two infinitely renormalizable maps is conjugated by a transformation that extends to a differentiable function whose derivative is Hölder continuous of exponent α > 0. In this article we investigate numerically the specific value of α. We also present numerical evidence that the normalized derivative cocycle with the base dynamics in the Cantor set is ergodic. Finally, we compute renormalization eigenvalues to a high accuracy to support a conjecture that the renormalization spectrum is real. [ABSTRACT FROM PUBLISHER]