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Mary Rees

Publications and source records attributed to Mary Rees.

12 recordsLinked to original sources
arXiv2024

Push-forward of geometric distributions under Collatz iteration: Part 1

Two conjectures are presented. The first, Conjecture 1, is that the pushforward of a geometric distribution on the integers under $n$ Collatz iterates, modulo $2^p$, is usefully close to uniform distribution on the integers modulo $2^p$, if $p/n$ is small enough. Conjecture 2 is that the density is bounded from zero for the incidence of both $0$ and $1$ for the coefficients in the dyadic expansions of $-3^{-\ell }$ on all but an exponentially small set of paths of a geometrically distributed random walk on the two-dimensional array of these coefficients. It is shown that Conjecture 2 implies Conjecture 1. At present, Conjecture 2 is unresolved.

math.PR↗
arXiv2018

An awkward graph

Given a rational map $f:\overline{\mathbb C}\to \overline{\mathbb C}$ and a finite graph $G\subset \overline{\mathbb C}$ such that $f(G)\subset G$ and $f$ is expanding on some neighbourhood of $G$, we show that there is another finite graph $G'\subset \bigcup _{n\ge 0}f^{-n}(G)$ in an arbitrarily small neighbourhood of $G$ such that $f^N(G')\subset G'$ for some integer $N$ but $\bigcup _{i=0}^{N-1}f^{i}(G')$ contains accumulating {\em{plaits}} and {\em{nests}}

math.DS↗
arXiv2016

Persistent Markov partitions and hyperbolic components of rational maps

Markov partitions persisting in a neighbourhood of hyperbolic components of rational maps were constructed under the condition that closures of Fatou components are disjoint in \cite{R1}. Given such a partition, we characterize all nearby hyperbolic components in terms of the symbolic dynamics. This means we can count them, and also obtain topological information. We also determine extra conditions under which all nearby type IV hyperbolic components are given by matings. These are probably the first known results of this type.

math.DS↗
arXiv2013

Persistent Markov partitions for rational maps

A construction is given of Markov partitions for some rational maps, which persist over regions of parameter space, not confined to single hyperbolic components. The set on which the Markov partition exists, and its boundary, are analysed.

math.DS↗
arXiv2012

A connected string of long thick and dominants

We prove that every Teichmuller geodesic of a finite type surface contains a string of intersecting long, thick and dominant segments, such that the distance between consecutive segments is bounded. This is key to obtaining some results about Teichmuller geodesics which mimic those for hyperbolic geodesics. These results have important applications to results about the geometry of hyperbolic three-manifolds.

math.DS↗
arXiv2012

Counting Hyperbolic Components

We give formulas for the numbers of type II and type IV hyperbolic components in the space of quadratic rational maps, for all fixed periods of attractive cycles.

math.DS↗
arXiv2012

The parameter capturemap for V_3

This is a study of the Wittner capture construction for critically finite quadratic rational maps for which one critical point is periodic, and the second critical point is in the backward orbit of the first. This construction gives a way of describing rational maps up to topological conjugacy. It is known that representations as Wittner captures are not unique. We show that, in a certain parameter space which we call $V_3$, the set of maps with exactly $2^r$ representations as a Wittner capture is of density bounded from 0 for each $r\geq 0$, and for each fixed preperiod of the second critical point.

math.DS↗
arXiv2010

A Fundamental Domain for V_3

We describe a fundamental domain for the punctured Riemann surface $V_{3,m}$ which parametrises (up to Möbius conjugacy) the set of quadratic rational maps with numbered critical points, such that the first critical point has period three, and such that the second critical point is not mapped in $m$ iterates or less to the periodic orbit of the first. This gives, in turn, a description, up to topological conjugacy, of all dynamics in all type III hyperbolic components in $V_{3}$, and gives indications of a topological model for $V_{3}$, together with the hyperbolic components contained in it.

math.DS↗
arXiv1992

Problems in holomorphic dynamics

Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichm\"uller space; John Milnor, Thurston's algorithm without critical finiteness; Mary Rees, A possible approach to a complex renormalization problem. 2. Geometry of Julia Sets: Lennart Carleson, Geometry of Julia sets; John Milnor, Problems on local connectivity. 3. Measurable Dynamics: Mikhail Lyubich, Measure and Dimension of Julia Sets; Feliks Przytycki, On invariant measures for iterations of holomorphic maps. 4. Iterates of Entire Functions: Robert Devaney, Open questions in non-rational complex dynamics; Alexandre Eremenko and Mikhail Lyubich, Wandering domains for holomorphic maps. 5. Newton's Method: Scott Sutherland, Bad polynomials for Newton's method

math.DS↗
arXiv1991

A partial description of the parameter space of rational maps of degree two: Part 2

This continues the investigation of a combinatorial model for the variation of dynamics in the family of rational maps of degree two, by concentrating on those varieties in which one critical point is periodic. We prove some general results about nonrational critically finite degree two branched coverings, and finally identify the boundary of the rational maps in the combinatorial model, thus completing the proofs of results announced in Part 1.

math.DS↗