Structurally Stable Grassmann Transformations

Transformations of Grassmann spaces

This is a current version of a part of the book “Transformations of Grassmann spaces”. We study transformations of Grassmann spaces preserving certain geometrical constructions (related with buildings). The next part will be devoted to Grassmann spaces associated with polar spaces.

Transformations on the Product of Grassmann Spaces

Let Gk denote the set of all k-dimensional subspaces of an n-dimensional vector space. We recall that two elements of Gk are called adjacent if their intersection has dimension k − 1. The set Gk is point set of a partial linear space, namely a Grassmann space for 1 < k < n − 1 (see Section 5) and a projective space for k ∈ {1, n − 1}. Two adjacent subspaces are—in the language of partial linear...

Transformations of Grassmann Spaces I (preliminary Version)

2 Grassmann spaces 13 2.1 Adjacency relation . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Base subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Principles of duality for projective spaces . . . . . . . . . . . . 18 2.4 Mappings induced by strong embeddings . . . . . . . . . . . . 19 2.5 Adjacency preserving transformations . . . . . . . . . . . . . . 21 2.6 Trans...

Structurally stable heteroclinic cycles

This paper describes a previously undocumented phenomenon in dynamical systems theory; namely, the occurrence of heteroclinic cycles that are structurally stable within the space of C vector fields equivariant with respect to a symmetry group. In the space X(M) of C vector fields on a manifold M, there is a residual set of vector fields having no trajectories joining saddle points with stable m...

Tying Hairs for Structurally Stable Exponentials