This workshop, sponsored by AIM and NSF, was devoted to the emerging theory of stability and hyperbolicity of functions. These notions are well known in the univariate setting, where stability means that all zeros lie in the left-half plane and hyperbolicity means that all zeros are real [CM, Post, Fisk, Rah]. The multivariate generalizations go back to 1950s and are being actively explored now...

If X is a geodesic metric space and x1, x2, x3 ∈ X , a geodesic triangle T = {x1, x2, x3} is the union of the three geodesics [x1x2], [x2x3] and [x3x1] in X . The space X is δ-hyperbolic (in the Gromov sense) if any side of T is contained in a δ-neighborhood of the union of the two other sides, for every geodesic triangle T in X . We denote by δ(X) the sharpest hyperbolicity constant of X , i.e...

A new paradigm for numerically approximating trajectories of an ODE is espoused. We ask for a one to one correspondence between trajectories of an ODE and its discrete approximation. The results enable one, in principal, to compute a trajectory of a discrete approximation, and to use this computation to rigorously prove the existence of a trajectory of the ODE near the discrete trajectory. More...

The authors show that for dynamical systems that possess a type of piecewise hyperbolicity in which there is no decrease in the number of stable modes, the global error in a numerical approximation may be obtained as a reasonable magnification of the local error. In particular, under certain conditions the authors prove the existence of a trajectory on an infinite time interval of the given ord...

We provide several equivalent characterizations of Kobayashi hyperbolicity in unbounded convex domains in terms of peak and anti-peak functions at infinity, affine lines, Bergman metric and iteration theory.

We give a set of sufficient and necessary conditions for parabolicity and hyperbolicity of a submanifold with controlled mean curvature in a Riemannian manifold with a pole and with sectional curvatures bounded from above or from below.

We study the concepts of continuous shadowing and continuous inverse shadowing with respect to various classes of admissible pseudo orbits, and characterize hyperbolicity and structural stability using the notion of continuous inverse shadowing.

In this note, we prove the generic Kobayashi volume measure hyperbolicity of singular directed varieties (X,V ), as soon as the canonical sheaf KV of V is big in the sense of Demailly.

This note deals with C. Conley's topological approach to hyperbolic invariant sets for continuous flows. It is based on the notions of isolated invariant sets and Morse decompositions and it leads to the concept of weak hyperbolicity.

We show the hyperbolicity of the Feigenbaum fixed point using the inflexibility of the Feigenbaum tower, the Manẽ-Sad-Sullivan λLemma and the existence of parabolic domains (petals) for semi-attractive fixed points.