ar X iv : m at h / 05 05 36 8 v 3 [ m at h . PR ] 6 A pr 2 00 6 SLE Coordinate Changes

ar X iv : m at h / 05 05 36 8 v 1 [ m at h . PR ] 1 7 M ay 2 00 5 SLE coordinate changes

The purpose of this note is to describe a framework which unifies radial, chordal and dipolar SLE. When the definition of SLE(κ; ρ) is extended to the setting where the force points can be in the interior of the domain, radial SLE(κ) becomes chordal SLE(κ; ρ), with ρ = κ − 6, and vice versa. We also write down the martingales describing the Radon-Nykodim derivative of SLE(κ; ρ1, . . . , ρn) wit...

ar X iv : m at h / 05 04 08 2 v 4 [ m at h . D G ] 1 5 A pr 2 00 6 COMPLETE PROJECTIVE CONNECTIONS

The first examples of complete projective connections are uncovered: on surfaces, normal projective connections whose geodesics are all closed and embedded are complete. On manifolds of any dimension, normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature are complete.

ar X iv : m at h / 05 05 01 1 v 4 [ m at h . PR ] 5 J un 2 00 6 EXCHANGEABLE , GIBBS AND EQUILIBRIUM MEASURES FOR MARKOV

We study a class of strongly irreducible, multidimensional, topological Markov shifts, comparing two notions of “symmetric measure”: exchangeability and the Gibbs (or conformal) property. We show that equilibrium measures for such shifts (unique and weak Bernoulli in the one dimensional case) exhibit a variety of spectral properties.

ar X iv : m at h / 05 01 26 2 v 3 [ m at h . O A ] 2 6 A pr 2 00 5 Minimal Homeomorphisms and Approximate Conjugacy in Measure ∗

Let X be an infinite compact metric space with finite covering dimension. Let α, β : X → X be two minimal homeomorphisms. Suppose that the range of K0-groups of both crossed products are dense in the space of real affine continuous functions. We show that α and β are approximately conjugate uniformly in measure if and only if they have affine homeomorphic invariant probability measure spaces.

ar X iv : m at h / 05 04 37 3 v 1 [ m at h . Q A ] 1 9 A pr 2 00 5 Lax Operator for the Quantised Orthosymplectic