Anthony Manning

Hausdorff dimension for horseshoes

We shall measure how thick a basic set of a C 1 axiom A diffeomorphism of a surface is by the Hausdorff dimension of its intersection with an unstable manifold. This depends continuously on the diffeomorphism. Generically a C 2 diffeomorphism has attractors whose Hausdorff dimension is not approximated by the dimension of its ergodic measures.

A relation between Lyapunov exponents, Hausdorff dimension and entropy

For an Axiom A diffeomorphism of a surface with an ergodic invariant measure we prove that the entropy is the product of the positive Lyapunov exponent and the Hausdorff dimension of the set of generic points in an unstable manifold.

A linear chaotic quantum harmonic oscillator

Abstract We show that a linear quantum harmonic oscillator is chaotic in the sense of Li-Yorke. We also prove that the weighted backward shift map (used as an infinite-dimensional linear chaos model) in a separable Hilbert space is chaotic in the sense of Li-Yorke, in addition to being chaotic in the sense of Devaney.

Dimension of slices through the sierpinski carpet

For Lebesgue typical , the intersection of the Sierpinski carpet with a line has (if non-empty) dimension , where . Fix the slope . Then we shall show on the one hand that this dimension is strictly less than for Lebesgue almost every . On the other hand, for almost every according to the angle -projection of the natural measure on , this dimension is at least . For any we find a connection between the box dimension of this intersection and the local dimension of at .

A Markov partition that reflects the geometry of a hyperbolic toral automorphism

We show how to construct a Markov partition that reflects the geometrical action of a hyperbolic automorphism of the n-torus. The transition matrix is the transpose of the matrix induced by the automorphism in u-dimensional homology, provided this is non-negative. (Here u denotes the expanding dimension.) That condition is satisfied, at least for some power of the original automorphism, under a certain non-degeneracy condition on the Galois group of the characteristic polynomial. The (nu) rectangles are constructed by an iterated function system, and they resemble the product of the projection of a u-dimensional face of the unit cube onto the unstable subspace and the projection of minus the orthogonal (n - u)-dimensional face onto the stable subspace.

How to be sure of finding a root of a complex polynomial using Newton's method

The trouble with Newtons method for finding the roots of a complex polynomial is knowing where to start the iteration. In this paper we apply the theory of rational maps and some estimates based on distortion theorems for univalent functions to find lower bounds, depending only on the degreed, for the size of regions from which the iteration will certainly converge to a root. We can also bound the number of iterations required and we give a method that works for every polynomial and takes at most some constant timesd2(logd)2 log(d3/∈) iterations to find one root to within an accuracy of ∈.

Relating exponential growth in a manifold and its fundamental group

We relate the growth rate of volume in the universal cover of a compact Riemannian manifold to the growth in the fundamental group in terms of word length in a given set of generators and the length of geodesics representing these generators.

The volume entropy of a surface decreases along the Ricci flow

The volume entropy, h(g), of a compact Riemannian manifold (M,g) measures the growth rate of the volume of a ball of radius R in its universal cover. Under the Ricci flow, g evolves along a certain path

A SHORT EXISTENCE PROOF FOR CORRELATION DIMENSION

(g_t, t\geq0)

Curves of fixed points of trace maps

that improves its curvature properties. For a compact surface of variable negative curvature we use a Katok–Knieper–Weiss formula to show that h(gt) is strictly decreasing.