Article history: Received 21 August 2007 Accepted 12 September 2008 Available online 22 October 2008 Submitted by R.A. Brualdi AMS classification: 15A60 65F99

Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set A we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph...

On extrinsic geometry of unit normal vector fields of Riemannian hyperfoliations. Abstract We consider a unit normal vector field of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in ...

Consider a one parameter family of diffeomorphisms fε such that f0 is an Anosov element in a standard abelian Anosov action having sufficiently strong mixing properties. Let νε be any u-Gibbs state for fε. We prove (Theorem 1) that if A is a C ∞ function then the map A → νε(A) is differentiable at ε = 0. This implies (Corollary 1) that the difference of Birkhoff averages of the perturbed and un...

An almost contact metric structure is parametrized by a section σ of an associated homogeneous fibre bundle, and conditions for σ to be a harmonic section, and a harmonic map, are studied. These involve the characteristic vector field ξ, and the almost complex structure in the contact subbundle. Several examples are given where the harmonic section equations for σ reduce to those for ξ, regarde...

For almost all values of x 2 R, the classical theta sum S N (x) = N ?1=2 N X n=1 e 2i n 2 x exhibits an extremely irregular behaviour, as N tends to innnity. This limit is investigated by exploiting a connection to the ergodic properties of ows on the unit tangent bundle of a surface of constant negative curvature.

We establish that, for every hyperbolic orbifold of type (2, q,∞) and for every orbifold of type (2, 3, 4g+2), the geodesic flow on the unit tangent bundle is left-handed. This implies that the link formed by every collection of periodic orbits (i) bounds a Birkhoff section for the geodesic flow, and (ii) is a fibered link. We also prove similar results for the torus with any flat metric. Besid...

Studying the space of Lagrangian submanifolds is a fundamental problem in symplectic topology. Lagrangian spheres appear naturally in the Leftschetz pencil picture of symplectic manifolds. In this paper we demonstrate the uniqueness up to Hamiltonian isotopy of the Lagrangian spheres in some 4-dimensional Stein symplectic manifolds. The most important example is the cotangent bundle of the 2-sp...

To reduce the computational complexity of classification systems using tangent distance, Hastie et al. (HSS) developed an algorithm to devise rich models for representing large subsets of the data which computes automatically the "best" associated tangent subspace. Schwenk & Milgram proposed a discriminant modular classification system (Diabolo) based on several autoassociative multilayer perce...