ar X iv : m at h / 99 08 08 0 v 1 [ m at h . D S ] 1 6 A ug 1 99 9 Topological Entropy and ε - Entropy for Damped Hyperbolic Equations

ar X iv : m at h / 06 08 72 0 v 1 [ m at h . D S ] 2 9 A ug 2 00 6 TOPOLOGICAL ENTROPY AND PARTIALLY HYPERBOLIC DIFFEOMORPHISMS

We consider partially hyperbolic diffeomorphisms on compact manifolds where the unstable and stable foliations stably carry some unique nontrivial homologies. We prove the following two results: if the center foliation is one dimensional, then the topological entropy is locally a constant; and if the center foliation is two dimensional, then the topological entropy is continuous on the set of a...

ar X iv : m at h / 98 08 06 5 v 1 [ m at h . Q A ] 1 4 A ug 1 99 8 QUANTUM DETERMINANTS AND QUASIDETERMINANTS

ar X iv : h ep - t h / 98 08 08 5 v 2 2 5 A ug 1 99 8 Gravitational Entropy and Global Structure

The underlying reason for the existence of gravitational entropy is traced to the impossibility of foliating topologically non-trivial Euclidean spacetimes with a time function to give a unitary Hamiltonian evolution. In d dimensions the entropy can be expressed in terms of the d − 2 obstructions to foliation, bolts and Misner strings, by a universal formula. We illustrate with a number of exam...

ar X iv : m at h / 99 07 08 5 v 2 [ m at h . G R ] 1 7 D ec 1 99 9 LOOPS AND SEMIDIRECT PRODUCTS

ar X iv : m at h - ph / 9 80 50 19 v 1 2 0 M ay 1 99 8 The Definition and Measurement of the Topological Entropy per Unit Volume in Parabolic PDE ’ s

We define the topological entropy per unit volume in parabolic PDE's such as the complex Ginzburg-Landau equation, and show that it exists, and is bounded by the upper Hausdorff dimension times the maximal expansion rate. We then give a constructive implementation of a bound on the inertial range of such equations. Using this bound, we are able to propose a finite sampling algorithm which allow...