1 2 Fe b 20 07 A PRIORI BOUNDS FOR SOME INFINITELY

Exact Bounds for Distributed Graph Colouring

We prove exact bounds on the time complexity of distributed graph colouring. If we are given a directed path that is properly coloured with n colours, by prior work it is known that we can find a proper 3-colouring in 12 log ∗(n) ± O(1) communication rounds. We close the gap between upper and lower bounds: we show that for infinitely many n the time complexity is precisely 12 log ∗ n communicat...

Physical Measures for Infinitely Renormalizable Lorenz Maps

A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics. Namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article we construct examples of i...

A Priori Bounds for Some Infinitely Renormalizable Quadratics: I. Bounded Primitive Combinatorics

A Priori Bounds for Some Infinitely Renormalizable Quadratics: Ii. Decorations

Exponential Triples

Using ultrafilter techniques we show that in any partition of N into 2 cells there is one cell containing infinitely many exponential triples, i.e. triples of the kind a, b, ab (with a, b > 1). Also, we will show that any multiplicative IP ∗ set is an “exponential IP set”, the analogue of an IP set with respect to exponentiation. Introduction A well-known theorem by Hindman states that given an...