Curvature computations for a two-component Camassa-Holm equation with vorticity

In the present paper, a two-component Camassa-Holm (2CH) system with vorticity is studied as a geodesic flow on a suitable Lie group. The paper aims at presenting various details of the geometric formalism and a major result is the computation of the sectional curvature K of the underlying configuration manifold. As a further result, we show that there are directions for which K is strictly positive and bounded away from zero.     

Trivial and simple spectrum for SL(d, ℝ) cocycles with free base and fiber dynamics

Let AC_D(M,SL(d,R)) denote the pairs (f,A) so that f ∈ A ⊂ Diff¹(M) is a C¹-Anosov transitive diffeomorphisms and A is an SL(d,R) cocycle dominated with respect to f. We prove that open and densely in AC_D(M,SL(d,R)), in appropriate topologies, the pair (f,A) has simple spectrum with respect to the unique maximal entropy measure μ_f. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in AutLeb(M) × L^p(M,SL(d,R)).     

Diffeomorphisms with <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-shadowing property

We give a characterization of structurally stable diffeomorphisms by making use of the notion of L ^p -shadowing property. More precisely, we prove that the set of structurally stable diffeomorphisms coincides with the C ¹-interior of the set of diffeomorphisms having L ^p -shadowing property.     

The barycenter property for robust and generic diffeomorphisms

Let f:M~d→M~d(d≥2) be a diffeomorphism on a compact C~∞ manifold on M.If a diffeomorphism f belongs to the C~1-interior of the set of all diffeomorphisms having the barycenter property,then f is Ω-stable.Moreover,if a generic diffeomorphism f has the barycenter property,then f is Ω-stable.We also apply our results to volume preserving diffeomorphisms.     

On the well-posedness of the Holm-Staley b-family of equations

In this paper we consider the Holm-Staley b-family of equations in the Sobolev spaces H^s(?) for s>3/2. Using a geometric approach we show that, for any value of the parameter b, the corresponding solution map,u(0)?u(T ), is nowhere locally uniformly continuous.     

Quasi-hyperbolic semigroups

We introduce the notion of quasi-hyperbolic operators and C₀-semigroups. Examples include the push-forward operator associated with a quasi-Anosov diffeomorphism or flow. A quasi-hyperbolic operator can be characterised by a simple spectral property or as the restriction of a hyperbolic operator to an invariant subspace. There is a corresponding spectral property for the generator of a C₀-semigroup, and it characterises quasi-hyperbolicity on Hilbert spaces but not on other Banach spaces. We exhibit some weaker properties which are implied by the spectral property.     

On lengths of filling closed geodesics of hyperbolic punctured Riemann spheres

Let S be a Riemann sphere with n ≥ 4 points deleted. In this article we investigate certain filling closed geodesics of S and give quantitative common lower bounds for the hyperbolic lengths of those geodesics with respect to any hyperbolic structure on S (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enumeration of Mapping Classes for the Torus

We enumerate periodic, reducible, and Anosov elements of the mapping class group of the torus, and determine their growth function. Then we prove that almost all elements of the mapping class group of the torus are Anosov.     

Asymptotic Expansions for Closed Orbits in Homology Classes

In this paper, we study the behaviour of the counting function associated to the closed geodesics lying in a prescribed homology class on a compact negatively curved manifold. Our main result is an asymptotic expansion. We also obtain results in the wider context of periodic orbits of Anosov flows.