However, it turns out to be H�lder continuous, with an exponent depending on the constant of hyperbolicity of A.
12.
Lanford later gave a shorter proof using the Leray-Schauder fixed point theorem but establishing only the fixed point without the hyperbolicity.
13.
For surfaces, hyperbolicity of the nonwandering set implies the density of periodic points, but this is no longer true in higher dimensions.
14.
Although the definition of hyperbolicity is fundamentally a qualitative one, there are precise criteria that depend on the particular kind of differential equation under consideration.
15.
The definition of the coned off Cayley graph can be generalized to the case of a collection of subgroups and yields the corresponding notion of relative hyperbolicity.
16.
Such groups are in fact virtually free ( i . e . contain a finitely generated free subgroup of finite index ), which gives another proof of their hyperbolicity.
17.
In mathematical physics, "'global hyperbolicity "'is a certain condition on the causal structure of a spacetime manifold ( that is, a Lorentzian manifold ).
18.
It allows to read off certain geometric properties of the latter, in particular it explains the empirical fact that while Teichm�ller space itself is not hyperbolic it retains certain features of hyperbolicity.
19.
Since the curvature is multiplied by \ lambda ^ {-1 } we see that in this example " the more ( negatively ) curved the space is, the more hyperbolic it is ( measured by its hyperbolicity constant \ delta ) ".
20.
With respect to curvature and hyperbolicity it should be noted however that while curvature is a property that is essentially local, hyperbolicity is a large-scale property which does not see local ( i . e . happening in a bounded region ) metric phenomena.
