We consider a measurable matrix-valued cocycle A: Z+ X Rd d, driven by a measurepreserving transformation T of a probability space (X,, ), with the integrability condition log+ A (1,) L1 (). We show that for -ae x X, if limn 1 n log A (n, x) v= 0 for all v Rd\{0}, then the trajectory {A (n, x) v} n= 0 is far away from 0 (ie lim supn A (n, x) v> 0) and there is some nonzero v such that lim supn A (n, x) v v. This improves the classical multiplicative ergodic theorem of Oseledec. We here present an application to linear random processes to illustrate the importance.

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For any dynamical system T : X → X of a compact metric space X with g-almost product property and uniform separation property, under the assumptions that the periodic points are dense in X and the periodic measures are dense in the space of invariant measures, we distinguish various periodiclike recurrences and find that they all carry full topological entropy and so do their gap-sets. In particular, this implies that any two kind of periodic-like recurrences are essentially different. Moreover, we coordinate periodic-like recurrences with (ir)regularity and obtain lots of generalized multifractal analyses for all continuous observable functions. These results are suitable for all β-shifts (β > 1), topological mixing subshifts of finite type, topological mixing expanding maps or topological mixing hyperbolic diffeomorphisms, etc. Roughly speaking, we combine many different “eyes” (i.e., observable functions and periodic-like recurrences) to observe the dynamical complexity and obtain a Refined Dynamical Structure for Recurrence Theory and Multi-fractal Analysis.

The modified Camassa-Holm (also called FORQ) equation is one of numerous <i>cousins</i> of the Camassa-Holm equation possessing non-smoth solitons (<i>peakons</i>) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissipative^a) the Sobolev <i>H</i>¹ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the <i>H</i>¹ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations are Hamiltonian and, in fact, Liouville integrable. The latter is proved directly by exploiting the inverse spectral techniques, especially asymptotic analysis of solutions, developed elsewhere [3].

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