We consider partially hyperbolic diieomorphisms preserving a splitting of the tangent bundle into a strong-unstable subbundle E uu (uniformly expanding) and a subbundle E c , dominated by E uu. We prove that if the central direction E c is mostly contracting for the diieomorphism (negative Lyapunov exponents), then the ergodic Gibbs u-states are the Sinai-Ruelle-Bowen measures, there are nitely...

In this paper and in the forthcoming Part II we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M , possibly having critical points of infinite Morse index and co-index. The idea is to consider an infinite dimensional subbundle or more generally an essential subbundle of the tangent bundle of M , suitably related with the gradient flow of...

We consider dynamical systems generated by partially hyperbolic surface endomorphisms of class Cr with one-dimensional strongly unstable subbundle. As the main result, we prove that such a dynamical system generically admits finitely many ergodic physical measures whose union of basins of attraction has total Lebesgue measure, provided that r ≥ 19.

Take an n-dimensional manifold M . Endow it with a distribution, by which I mean a smooth linear subbundle D ⊂ TM of its tangent bundle TM . So, for x ∈ M , we have a k-plane Dx ⊂ TxM , and by letting x vary we obtain a smoothly varying family of k-planes on M . Put a smoothly varying family g of inner products on each k-plane. The data (M,D, g) is, by definition, a sub-Riemannian geometry. Tak...

Take an n-dimensional manifold M . Endow it with a distribution, by which I mean a smooth linear subbundle D ⊂ TM of its tangent bundle TM . So, for x ∈ M , we have a k-plane Dx ⊂ TxM , and by letting x vary we obtain a smoothly varying family of k-planes on M . Put a smoothly varying family g of inner products on each k-plane. The data (M,D, g) is, by definition, a sub-Riemannian geometry. Tak...

In order to obtain a framework in which both non-holonomic mechanical systems and non-holonomic mechanical systems with symmetry can be described, we introduce in this paper the notion of a Lagrangian system on a subbundle of a Lie algebroid.

Abstract Let $f\,:\,C\,\longrightarrow \,D$ be a nonconstant separable morphism between irreducible smooth projective curves defined over an algebraically closed field. We say that $f$ is genuinely ramified if ${\mathcal O}_D$ the maximal semistable subbundle of $f_*{\mathcal O}_C$ (equivalently, induced homomorphism $f_*\,:\, \pi _1^{\textrm{et}}(C)\,\longrightarrow \, _1^{\textrm{et}}(D)$ éta...

A natural generalization of analytic functions to n-dimensional Euclidean space are quasiregular mappings.(See [2] and [3].) An analogue of Theorem 1.1 for quasiregular mappings in John domains in Euclidean space appeared in [4]. Recently the analytical tools used in the proof of this result have been generalized to Carnot groups. We give an account of some of these advances and obtain an analo...

In recent years the study of the differential geometry of the total space E, of a vector bundle π : R → M , initiated by R.Miron [11], [12] has been developed by many people (see [13] and the references therein). If we take a horizontal complement of the vertical subbundle V E, we can express the geometrical objects defined on E in a more simplified form and new geometric objects can be obtaine...