Abstract:
Oseledets regularity functions quantify the deviation of the growth associated with a dynamical system along its Lyapunov bundles from the corresponding uniform exponential growth. Precise degree of regularity of these functions is unknown. We show that for every invariant Lyapunov bundle of a volume preserving Anosov flow on a closed smooth Riemannian manifold, the corresponding Oseledets regularity functions are in $L^p(m)$, for some $p > 0$, where $m$ is the probability measure defined by the volume form. We prove an analogous result for essentially bounded cocycles over volume preserving Anosov flows.

Abstract:
Previously Chicone, Latushkin and Montgomery-Smith [\textbf{Comm. Math. Phys. \textbf{173},(1995)}] have investigated the spectrum of the dynamo operator for an ideally conducting fluid. More recently, Tang and Boozer [{\textbf{Phys. Plasmas (2000)}}], have investigated the anisotropies in magnetic field dynamo evolution, from finite-time, Lyapunov exponents, giving rise to a Riemann metric tensor, in the Alfven twist in magnetic flux tubes (MFTs). In this paper one investigate the role of Perelman Ricci flows constraints in twisted magnetic flux tubes, where the Lyapunov eigenvalue spectra for the Ricci tensor associated with the Ricci flow equation in MFTs leads to a finite-time Lyapunov exponential stretching along the toroidal direction of the tube and a contraction along the radial direction of the tube. It is shown that in the case of MFTs, the sectional Ricci curvature of the flow, is negative as happens in geodesic flows of Anosov type. Ricci flows constraints in MFTs substitute the Thiffeault and Boozer [\textbf{Chaos}(2001)] have vanishing of Riemann curvature constraint on the Lyapunov exponential stretching of chaotic flows. Gauss curvature of the twisted MFT is also computed and the contraints on a negative Gauss curvature are obtained.

Abstract:
Recently Tang and Boozer [{\textbf{Phys. Plasmas (2000)}}], have investigated the anisotropies in magnetic field dynamo evolution, from local Lyapunov exponents, giving rise to a metric tensor, in the Alfven twist in magnetic flux tubes (MFTs). Thiffeault and Boozer [\textbf{Chaos}(2001)] have investigated the how the vanishing of Riemann curvature constrained the Lyapunov exponential stretching of chaotic flows. In this paper, Tang-Boozer-Thiffeault differential geometric framework is used to investigate effects of twisted magnetic flux tube filled with helical chaotic flows on the Riemann curvature tensor. When Frenet torsion is positive, the Riemann curvature is unstable, while the negative torsion induces an stability when time $t\to{\infty}$. This enhances the dynamo action inside the MFTs. The Riemann metric, depends on the radial random flows along the poloidal and toroidal directions. The Anosov flows has been applied by Arnold, Zeldovich, Ruzmaikin and Sokoloff [\textbf{JETP (1982)}] to build a uniformly stretched dynamo flow solution, based on Arnold's Cat Map. It is easy to show that when the random radial flow vanishes, the magnetic field vanishes, since the exponential Lyapunov stretches vanishes. This is an example of the application of the Vishik's anti-fast dynamo theorem in the magnetic flux tubes. Geodesic flows of both Arnold and twisted MFT dynamos are investigated. It is shown that a constant random radial flow can be obtained from the geodesic equation. Throughout the paper one assumes, the reasonable plasma astrophysical hypothesis of the weak torsion. Pseudo-Anosov dynamo flows and maps have also been addressed by Gilbert [\textbf{Proc Roy Soc A London (1993)}

Abstract:
Exponential decay of correlations for $\Co^{(4)}$ Contact Anosov flows is established. This implies, in particular, exponential decay of correlations for all smooth geodesic flows in strictly negative curvature.

Abstract:
Despite the invertible setting, Anosov endomorphisms may have infinitely many unstable directions. Here we prove, under transitivity assumption, that an Anosov endomorphism on a closed manifold $M,$ is either special (that is, every $x \in M$ has only one unstable direction) or for a typical point in $M$ there are infinitely many unstable directions. Other result of this work is the semi rigidity of the unstable Lyapunov exponent of a $C^{1+\alpha}$ codimension one Anosov endomorphism and $C^1$ close to a linear endomorphism of $\mathbb{T}^n$ for $(n \geq 2).$ In the appendix we give a proof for ergodicity of $C^{1+\alpha}, \alpha > 0,$ conservative Anosov endomorphism.

Abstract:
We study the relation between Lyapunov function and exponential trichotomy for the linear equation on time scales. Furthermore, as an application of these results, we give the roughness of exponential trichotomy on time scales. 1. Introduction Exponential trichotomy is important for center manifolds theorems and bifurcation theorems. When people analyze the asymptotic behavior of dynamical systems, exponential trichotomy is a powerful tool. The conception of trichotomy was first introduced by Sacker and Sell [1]. They described SS-trichotomy for linear differential systems by linear skew-product flows. Furthermore, Elaydi and Hájek [2, 3] gave the notions of exponential trichotomy for differential systems and for nonlinear differential systems, respectively. These notions are stronger notions than SS-trichotomy. In 1991, Papaschinopoulos [4] discussed the exponential trichotomy for linear difference equations. And in 1999 Hong and his partners [5, 6] studied the relationship between exponential trichotomy and the ergodic solutions of linear differential and difference equations with ergodic perturbations. Recently, Barreira and Valls [7, 8] gave the conception of nonuniform exponential trichotomy. From their papers, we can see that the exponential trichotomy studied before is just a special case of the nonuniform exponential trichotomy. For more information about exponential trichotomy we refer the reader to papers [9–14]. Many phenomena in nature cannot be entirely described by discrete system, or by continuous system, such as insect population model, the large population in the summer, the number increases, a continuous function can be shown. And in the winter the insects freeze to death or all sleep, their number reduces to zero, until the eggs hatch in the next spring, the number increases again, this process is a jump process. Therefore, this population model is a discontinuous jump function, it cannot be expressed by a single differential equation, or by a single difference equations. Is it possible to use a unified framework to represent the above population model? In 1988, Hilger [15] first introduced the theory of time scales. From then on, there are numerous works on this area (see [16–23]). Time scales provide a method to unify and generalize theories of continuous and discrete dynamical systems. In this paper, motivated by [8], we study the exponential trichotomy on time scales. We firstly introduce -Lyapunov function on time scales. Then we study the relationship between exponential trichotomy and -Lyapunov function on time scales. We

Abstract:
In 1977, Keane and Smorodinsky showed that there exists a finitary homomorphism from any finite-alphabet Bernoulli process to any other finite-alphabet Bernoulli process of strictly lower entropy. In 1996, Serafin proved the existence of a finitary homomorphism with finite expected coding length. In this paper, we construct such a homomorphism in which the coding length has exponential tails. Our construction is source-universal, in the sense that it does not use any information on the source distribution other than the alphabet size and a bound on the entropy gap between the source and target distributions. We also indicate how our methods can be extended to prove a source-specific version of the result for Markov chains.

Abstract:
We consider a perturbation of the Anosov-type system, which leads to the appearance of a hierarchical set of islands-around-islands. We demonstrate by simulation that the boundaries of the islands are sticky to trajectories. This phenomenon leads to the distribution of Poincare recurrences with power-like tails in contrast to the exponential distribution in the Anosov-type systems.

Abstract:
We generalize the definition of quantum Anosov properties and the related Lyapunov exponents to the case of quantum systems driven by a classical flow, i.e. skew-product systems. We show that the skew Anosov properties can be interpreted as regular Anosov properties in an enlarged Hilbert space, in the framework of a generalized Floquet theory. This extension allows us to describe the hyperbolicity properties of almost-periodic quantum parametric oscillators and we show that their upper Lyapunov exponents are positive and equal to the Lyapunov exponent of the corresponding classical parametric oscillators. As second example, we show that the configurational quantum cat system satisfies quantum Anosov properties.

Abstract:
In this paper we mainly address the problem of disintegration of Lebesgue measure and measure of maximal entropy along the central foliation of (conservative) Derived from Anosov (DA) diffeomorphisms. We prove that for accessible DA diffeomorphisms of $\mathbb{T}^3$, atomic disintegration has the peculiarity of being mono-atomic (one atom per leaf). We further provide open and non-empty condition for the existence of atomic disintegration. Finally, we prove some new relations between Lyapunov exponents of DA diffeomorphisms and their linearization.