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The spectra of Poincaré recurrences for two classes of dynamical systems are obtained in the framework of the Carathéodory construction. One class contains systems which are topologically conjugate to subshifts with the specification property, the other consists of minimal multipermutative symbolic systems. The spectra are shown to be solutions of a non-homogeneous Bowen equation, and their relationship with multifractal spectra of Lyapunov exponents is exposed.
Suppose that $f$ is a surface diffeomorphism with a hyperbolic fixed point $\mathcal O$ and this fixed point has a transversal homoclinic orbit. It is well known that in a vicinity of this type of homoclinic there are hyperbolic invariants sets. We introduce smooth invariants for the homoclinic orbit which we call the multipliers. As an application, we study the influence of the multipliers on numerical invariants of the hyperbolic invariant sets as the vicinity becomes small.
We introduce pointwise dimensions and spectra associated with Poincaré recurrences. These quantities are then calculated for any ergodic measure of positive entropy on a weakly specified subshift. We show that they satisfy a relation comparable to Young's formula for the Hausdorff dimension of measures invariant under surface diffeomorphisms. A key-result in establishing these formula is to prove that the Poincaré recurrence for a 'typical' cylinder is asymptotically its length. Examples are provided which show that this is not true for some systems with zero entropy. Similar results are obtained for special flows and we get a formula relating spectra for measures of the base to the ones of the flow.
We prove, under some general assumptions, that master-slave synchronization implies generalized synchronization, that is we show the existence and continuity of the functional dependence between the “slave” coordinates and the “master” ones. Then, we prove that this function may be Lipschitz continuous and even less “smooth”, that is only Hölder continuous, depending on the coupling strength. We go beyond the above mentioned assumptions by coupling two identical maps of the interval that are neither continuous nor invertible to prove ‘almost-everywhere’ synchronization instead of global synchronization. Then we relate the Hausdorff dimension and the dimension for Poincaré recurrence of the attractor of master and slave systems. We provide some examples illustrating these results.
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