Complex Bounds for Multimodal Maps: Bounded Combinatorics

Real Bounds, Ergodicity and Negative Schwarzian for Multimodal Maps

Over the last 20 years, many of the most spectacular results in the field of dynamical systems dealt specifically with interval and circle maps (or perturbations and complex extensions of such maps). Primarily, this is because in the one-dimensional case, much better distortion control can be obtained than for general dynamical systems. However, many of these spectacular results were obtained s...

Light paths in large polyhedral maps with prescribed minimum degree

Let k be an integer and M be a closed 2-manifold with Euler characteristic χ(M) ≤ 0. We prove that each polyhedral map G on M with minimum degree δ and large number of vertices contains a k-path P , a path on k vertices, such that: (i) for δ ≥ 4 every vertex of P has, in G, degree bounded from above by 6k − 12, k ≥ 8 (It is also shown that this bound is tight for k even and that for k odd this ...

Renormalization Theory for Multimodal Maps

We study the dynamics of the renormalization operator for multimodal maps. In particular, we prove the exponential convergence of this operator for infinitely renormalizable maps with same bounded combinatorial type.

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

Bounds on the Generalised Acyclic Chromatic Numbers of Bounded Degree Graphs

We give upper bounds for the generalised acyclic chromatic number and generalised acyclic edge chromatic number of graphs with maximum degree d, as a function of d. We also produce examples of graphs where these bounds are of the correct order.