Michael Brin’s research while affiliated with University of Maryland, College Park and other places

A partially hyperbolic diffeomorphism is dynamically coherent if its center, center-stable and center-unstable invariant distributions are integrable, i.e. tangent to foliations. Dynamical coherence is a key assumption in the theory of stable ergodicity. The main result: a partially hyperbolic diffeomorphism f\colon M\to M is dynamically coherent if the strong stable and unstable foliations are quasi-isometric in the universal cover \widetilde{M}, i.e. for any two points in the same leaf, the distance between them in \widetilde{M} is bounded from below by a linear function of the distance along the leaf.

A Euclidean complex X is a simplicial complex whose simplices are (flat) Euclidean simplices. We construct a natural Brownian motion on X and show that if X has nonpositive curvature and satisfies Gromov's hyperbolicity condition, then, with probability one, Brownian motion tends to a random limit on the Gromov boundary. Applying a combination of geometric and probabilistic techniques we describe spaces of harmonic functions on X.

We consider deterministic perturbations $\ddot q^\varepsilon(t)+F'(q^\varepsilon(t))=\varepsilon b(\dot q^\varepsilon(t),q^\varepsilon(t))$ of an oscillator $\ddot q+F'(q)=0$, $q\in{\mathbb R}^1$. Assume that $\lim_{|q|\to\infty}F(q)=\infty$ and that $F'(q)$ has a finite number of nondegenerate zeros. For a generic F, if $\partial b/\partial\dot q<0$ (as in the case of friction), then typical orbits are attracted to points where F has a local minimum. For $0<\varepsilon\ll1$, the equilibrium to which the trajectory is attracted is . To study this randomness, which is caused by the sensitive behavior of trajectories near the saddle points, we consider the graph $\Gamma$ homeomorphic to the space of connected components of the level sets of the Hamiltonian $H(p,q)=p^2/2+F(q)$. We show that, as $\varepsilon\to0$, the slow component of $(p^\epsilon(t/\epsilon),q^\epsilon(t/\epsilon))$ tends to a certain stochastic process on $\Gamma$ which is deterministic inside the edges and branches at the interior vertices into adjacent edges with probabilities which can be calculated through the Hamiltonian H and the perturbation b.

. A Euclidean polyhedron (a simplicial complex whose simplices are Euclidean) of nonpositive curvature (in the sense of Alexandrov) has rank 2 if every finite geodesic segment is a side of a flat rectangle. We prove that if a three-dimensional, geodesically complete, simply connected Euclidean polyhedron X of rank 2 and of nonpositive curvature admits a cocompact and properly discontinuous group of isometries, then X is either a Riemannian product or a thick Euclidean building of type e A 3 or e B 3 . 1. Introduction Let X be a simply connected, complete, geodesic space of nonpositive curvature in the sense of Alexandrov [Bal95]; in other words, X is a Hadamard space. Assume that X admits a properly discontinuous and cocompact group Gamma of isometries. If X is a Riemannian manifold and every geodesic in X bounds a flat strip, then X is either a Riemannian product or a symmetric space of higher rank, see [Bal95]. In the general case we also expect that X belongs to a re...

. We classify geodesically complete, compact 2-dimensional spherical polyhedra X of diameter and injectivity radius ß. If X contains a point whose link has diameter ? ß then either (i) X is the spherical join of the finite set P of points whose link has diameter ? ß with the metric graph E = fx 2 X : d(x; P ) = ß=2g whose diameter is ? ß or (ii) X is a hemispherex, that is, X is obtained by attaching hemispheres to the standard sphere S along great circles so that not all of them pass through the same pair of opposite points in S. If all links of X have diameter ß then either (i) X is a thick spherical building of type A 3 or B 3 , or (ii) X is the spherical join of a finite set with a graph E of diameter ß. In each case the injectivity radius of E is ß. 1. Introduction The diameter rigidity question considered in this paper is motivated by the rank rigidity problem for spaces of nonpositive curvature. The rank of a complete, simply connected space Y of nonpositive curvature is 2 i...

A C 1 -map f of a compact manifold M is a Morse–Smale endomorphism if the nonwandering set of f is finite and hyperbolic and the local stable and global unstable manifolds of periodic points intersect transversally. Morse–Smale endomorphisms appear naturally in the dynamics of the evolution operator on the set of traveling wave solutions for lattice models of unbounded media. The main result of this paper is the openness of the set of Morse–Smale endomorphisms in the space C 1 (M, M) of C 1 -maps of M into itself. The usual order relation on f (given by the intersections of local stable and global unstable manifolds) is used to describe the orbit structure of f and its small C 1 -perturbations.

A 2-dimensional orbihedron of nonpositive curvature is a pair (X, Γ), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and Γ is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X, Γ) which turns out to depend only on Γ and prove that, if X is boundaryless, then either (X, Γ) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense.

We study the structure of certain simply connected 2-dimensional complexes with non-positive curvature. We obtain a precise description of how these complexes behave at infinity and prove an existence theorem which gives an abundance of such complexes. We also investigate the structure of groups which act transitively on the set of vertices of such a complex.

Let M be the universal cover of a compact nonflat surface N of nonpositive curvature. We show that on the average the Brownian motion on M behaves similarly to the Brownian motion on negatively curved manifolds. We use this to prove that harmonic measures on the sphere at infinity have positive Hausdorff dimension and if the geodesic flow on N is ergodic then the harmonic and geodesic measure classes at infinity are singular unless the curvature is constant.