# Dynamical Systems Seminars

### On the equality of Hausdorff measure and Hausdorff content - Jonathan Fraser

Abstract: Hausdorff measure and dimension are among the most studied notions used for quantifying the size of a geometric object.  The precise Hausdorff measure (in the appropriate dimension) is particularly difficult to compute in practise and often people are just interested in whether it is zero, positive and finite, or infinite for a given set. The Hausdorff content is a less familiar concept, closely related to the Hausdorff measure, but with a slightly simpler definition.  The Hausdorff content is always a lower bound for the Hausdorff measure and still gives the Hausdorff dimension as the 'critical value'.  In this talk I will discuss both notions, some of their basic properties, and address the general question of 'when are the two quantities equal?' The main example will be self-similar sets but other classes of set will be considered as examples. This is joint work with Abel Farkas (The University of St Andrews).

Abstract: Our goal is to study unique and simultaneous codings generated by special classes of iterated function system consisting of contracting lines. We first consider the case where the attractor K is an interval and study those elements of K$with a unique coding. We prove under mild conditions that the set of points with a unique coding can be identified with a subshift of finite type. As a consequence of this, we can show that the set of unique codings is a graph-directed self-similar set in the sense of Mauldin and Williams. The theory of Mauldin and Williams then provides a method by which we can explicitly calculate the Hausdorff dimension of this set. Secondly, we consider IFS generating codings representing simultaneously two different points in two different bases. We show that for bases sufficiently close to 1 the attractor contains a neighbourhood of the origin. ### Wednesday 1st April 2015 ### Line, spiral, dense - Neil Dobbs Abstract: Generic analytic curves are dense in the plane. For particular paramatrised families of analytic curves, this need not be true (e.g. graphs of complex polynomials), or something stronger could be true (e.g. under the zeta-function, the image of every vertical line in the critical strip is dense). Not many classes of explicit dense curves were known. We show that exponential of exponential of almost every line in the complex plane is dense in the plane, along with some related results. ### Friday 27th March 2015 ### Newton’s method as a practical root finder - Dierk Schleicher Abstract: we all know that Newton’s method is efficient as a local root finder, but has a reputation of being globally unpredictable. We present an algorithm that makes it globally efficient and predictable, and (news of a few days ago) make it possible to find, in practice, all roots of polynomials of degree one million in just a few seconds on a standard personal computer. (Partly joint work with Robin Stoll). ### Wednesday 25th March 2015 ### Stochastic stability and limit theorem for coupled iterated functions systems that contract on average - Charles Walkden Abstract: We define and explore the notion of finitely coupled iterated function system that satisfy a notion of contraction on average. We study the existence and uniqueness of an invariant probability measure for such systems by defining an appropriate transfer operator. Using arguments of Keller and Liverani on properties of perturbations of linear operators, the continuity of the invariant measure as the coupling tends to zero is proved. Other limit theorems, including a central limit theorem, can also be proved. This is joint work with Anthony Chiu. ### Wednesday 4th March 2015 ### Invariance of order for entire functions: Linearisers and the area property - Lasse Rempe-Gillen Abstract: (This is joint work with Adam Epstein.) Let f be an entire function that has only finitely many critical and asymptotic values. Up to topological equivalence, the function f is determined by combinatorial information, more precisely by an infinite graph known as a "line-complex". I will discuss the natural question whether the order of growth of an entire function is determined by this combinatorial information. It turns out that the answer is positive in many cases, but negative in general (by recent work of Bishop). The search for conditions that imply a positive answer to this question leads to the "area property", which turns out to be related to many interesting and important questions in conformal dynamics and function theory. It turns out that Poincare functions (linearising maps) for polynomials with connected Julia sets lead to a wealth of interesting examples. In this connection, I will (time permitting) also touch upon work in progress with Alexandre Dezotti. ### Wednesday 11th February 2015 ### The tongues of the double standard map family - Alexandre DeZotti Abstract: The double standard map family has been introduced by Michał Misiurewicz and Ana Rodrigues in order to study perturbations of doubling maps up to the apparition of a branching. As the name indicates, this family is inspired by Arnold's standard map family and share interesting features with it. ### Wednesday 4th February 2015 ### The transfer operator for the binary Euclidean algorithm - Ian Morris Abstract: The binary Euclidean algorithm is a modification of the classical Euclidean algorithm which replaces division by an arbitrary integer with division by powers of two only. Statistical properties of the classical Euclidean algorithm -- such as the average number of steps required to process a pair of integers both of which are less than N -- can be studied via the thermodynamic formalism of the Gauss map acting on the unit interval. To investigate similar properties for the binary Euclidean algorithm one must instead study the thermodynamic formalism of an IID random dynamical system on the interval. I will describe a recent result on the transfer operator of the binary Euclidean algorithm which can be applied to resolve conjectures of R.P. Brent, B. Valle'{e} and D.E. Knuth. ### Friday 19th December 2014 ### Class B or not Class B, that is the question - Lasse Rempe-Gillen ### Wednesday 9th December 2014 ### Strong Uniform Distribution - Nair Radhakrishnan ### Friday 28th November 2014 ### Arclike Julia Continua of Exponential Maps - Stephen Worsley Abstract: It is understood that the escaping set of exponential maps consists of rays and that the closure of this set is the Julia set. However, when we take the closure of these rays, we find their topological nature becomes more complicated. Under certain restrictions, I have found that we may classify classify some of these continua as arclike. ### Dimensional rigidity for the limiting dynamics of transcendental entire functions - Alexandre DeZotti Abstract: We recall some results on the rigidity of dynamics of transcendental entire functions in the class B near infinity, obtained by Lasse Rempe-Gillen and Gwyneth Stallard. Those results are based on quasiconformal rigidity of the dynamics of class B functions near infinity. Different dimension quantities are known to be invariant in affine equivalence classes of maps. Poincaré maps shows that this invariance may not hold for quasiconformal classes. This is a joint work with Lasse Rempe-Gillen. ### Friday 21st November 2014 ### Extremal length estimates in perturbations of rational dynamics - Tan Lei Abstract: Univalent maps arise as (pseudo-)conjugacies in dynamical perturbations of a rational map. We will give a few controls of their conformal and spherical distortions using extremal length arguments, and show applications in the study of landing properties of parameter rays. This is a joint work with Cui Guizhen. ### Wednesday 12th November 2014 ### A simplified example of the Benoist-Quint theorem on stationary measures - Mark Pollicott Abstract: In 2011, Y. Benoist and J.-F. Quint proved results on the uniqueness of non-atomic stationary measures. In this talk we want to illustrate this approach in the very special case of matrices and tori. Moreover, by making an extra assumption we can further simplify the proof to make it essentially elementary. ### Wednesday 5th November 2014 ### Quasi-symmetric Rigidity of Real Maps - Trevor Clark Abstract: Quasi-symmetric rigidity is the property that whenever two maps are topologically conjugate, they are quasi-symmetrically conjugate. It has many consequences for dynamics in dimension one. I will talk about quasi-symmetric rigidity for a broad class of smooth maps of the interval. This is joint work with Sebastian van Strien. ### Wednesday 29th October 2014 ### Immersion of the dynamical Teichmüller space into the moduli space of rational maps - Matthieu Astorg Abstract: Teichmüller theory's goal is to study deformations of the complex structure of a Riemann surface. In the 80's, McMullen and Sullivan introduced an analogue of this theory in the context of iterations of a rational map f. In particular, they constructed a "dynamical Teichmüller space" which is a simply connected complex manifold, with a holomorphic map F defined on Teich(f) and taking values in the space of rational maps of the same degree as f, and whose image is exactly the quasiconformal conjugacy class of f. A natural question, raised in their article, is to know if this map F is an immersion: it turns out the answer is affirmative. A. Epstein has an unpublished proof of this; we will expose a different approach. ### Wednesday 22nd October 2014 ### On the construction of entire functions in the Speiser class - Simon Albrecht Abstract: We construct entire functions with finite set of singular values and prescribed tracts by using quasiconformal folding, a method introduced by C. Bishop in 2011 ### Wednesday 8th October 2014 ### The size of the Julia set of functions outside the Eremenko-Lyubich class - David Sixsmith (Open University) Abstract: We briefly review some well-known results regarding the size of the Julia sets of some transcendental entire functions. Most of these results concern functions in the Eremenko-Lyubich class. We then discuss two recent results which concern functions outside this class. The first generalises a result of McMullen, and gives a class of transcendental entire functions for which the Julia set has positive area and is a spider's web. The second concerns a class of functions with zeros in a certain sector for which the Julia set has Hausdorff dimension equal to 2. ### Wednesday 1st October 2014 ### Convergence properties of Gronwall area formula for quadratic Julia sets - Alexandre DeZotti Abstract: Using parabolic enrichment, it is shown that Gronwall area formula for the filled Julia set along the boundary of the main cardioid of the Mandelbrot set cannot be well approximated by replacing it by a finite sum. ### Friday 16th May 2014 ### Dynamic rays of the exponential map - Stephen Worsley Abstract: We know that for the family of exponential maps E_a(z) := ae^z , the connected components of the escaping set, which we call dynamic rays, are curves which tend to infinity in one direction along which the dynamics are well understood. However it is not guaranteed how the ray will behave in the other direction, whether it will land or not. In my talk I will discuss the behavior of these rays and show an example of when we can construct a non-landing ray for exponential maps with certain properties. ### Wednesday 14th May 2014 ### Explosion points for exponential maps - Nada Alhabib Abstract: Consider the family of exponential maps, f_a(z) := e^z + a, where a is a complex parameter. For certain parameters, including real values a < -1, the Julia set of this family is well understood. It is an uncountable union of curves, each consisting of a finite endpoint and a ray that connects this endpoint to infinity. John Mayer showed that (for these a) the set of endpoints E has the surprising property that E is totally disconnected but E together with the point at infinity is connected. In this talk, we shall prove that, for any exponential map, the set of “escaping endpoints” (as introduced by Schleicher and Zimmer) together with infinity is connected. ### Wednesday 7th May 2014 ### Diophantine approximation and dynamics - Prof Alexander Gorodnik (Bristol) Abstract: We explore some instances when questions about Diophantine approximation can studied using dynamical systems techniques. In particular, we discuss the Littlewood conjecture and its p-adic analogue proposed de Mathan and Teulie. Our main result, which is a joint work with Vishe, concerns an inhomogeneous version of the de-Mathan-Teulie conjecture. ### Wednesday 30th April 2014 ### Continuity results in complex dynamics and use of the Yoccoz puzzle - Mr Ma Liangang Abstract: My thesis is a study of continuity results of matings of quadratic polynomials. Mating is a construction which involves combining two quadratic polynomials to produce a rational map. Much of my work this year has been about matings with basilica (the star-like ones) polynomials. Rational maps obtained from matings with the basilica are all in the family R_a(z)=\frac{a}{z^2+2z}, a\in\mathbb{C}. An important tool is an adaptation of the Yoccoz puzzle for quadratic polynomials developed by Aspenberg and Yampolsky. The adaptation uses bubble rays. I shall introduce these tools and discuss their use. ### Wednesday 2nd April 2014 ### Satellite renormalization of complex quadratic polynomials - Dr Davoud Cheraghi (Imperial College) Abstract: The renormalization has been one of the main focus of the theory of one-dimensional complex dynamics. It is connected to the conjectures on the density of hyperbolicity and the local connectivity of the Mandelbrot set. For quadratic polynomials, there are two different types of renormalizations—primitive and satellite. The primitive renormalization has been successfully studied to some extent by Kahn and Lyubich, and now there are powerful "a priori" bounds. The satellite type has a very different nature and our knowledge is limited. In this talk, we discuss the difference between the two types of renormalizations and explain recent results on the satellite renormalization. This is a joint work with Prof. Mitsuhiro Shishikura. ### Friday 28th March 2014 ### Rational moment generating functions and polyhedra in R^d. - Prof Vladimir Markovic (Cambridge) Abstract: The problem of reconstructing a measure in R^d from a (truncated) multi-sequence of its moments has important applications, and is in general very hard to solve. We concentrate on a natural case of a measure mu with piecewise-polynomial density supported on a compact polyhedron P, and show that such problems can be solved exactly, due to existence of a natural integral transform of the measure, which is a rational function F_mu(u). The denominator of F_mu(u) is the product of powers of linear functions of the form 1-<u,v>, with v belonging to certain finite set V(P). There are interesting applications of F_mu(u) to compact (not necessarily convex) polyhedra. Let I(P) be the indicator function of P. Then I(P) can be decomposed (up to a measure 0 subset) as a sum, with real coefficients, of I(D), where D runs through simplices with vertices in V(P). This can be viewed as a non-convex generalisation of triangulations of convex polytopes. On the other hand, Laplace transforms of such decompositions arise in the theory of hyperplane arrangements. Further refinements and applications will be discussed. ### Wednesday 26th March 2014 - Joint session with the Access Grid Dynamics Seminar ### Density of Axiom A in Arnol'd's standard family - Prof Lasse Rempe-Gillen Abstract: Consider the self-maps of the circle defined by F_{a,b}(t) := t + a + b*sin(2\pi t) (mod 1), where a and b are real parameters, b>0. This family, known as the standard family, was introduced by Arnol'd in 1961 to model periodically forced nonlinear oscillators, and has since served as one of the simplest models of one-dimensional dynamical systems. I will discuss a recent result (joint with van Strien) establishing the density of Axiom A (or hyperbolic) maps in the region where the maps are non-invertible. (Axiom A maps are those that exhibit the simplest type of dynamical behaviour.) This solves a long-standing open problem. I will also mention connections with recent advances in the dynamics of transcendental entire functions. The talk will begin with a gentle recap of one-dimensional real dynamics, including computer simulations, and should be accessible to postgraduate and undergraduate students with an interest in dynamical systems. ### Wednesday 19th March 2014 ### An overview of group automorphisms as dynamical systems - Prof Thomas Ward (University of Durham) Abstract: Automorphisms of compact metric abelian groups are arguably the second most simple class of dynamical systems. I will try to describe some of the issues that arise in attempting to classify them up to various natural dynamical equivalences. Despite their simple nature, this is largely an account of failing to reach a destination - but the scenery on the journey throws up some beautiful questions and ideas. ### Wednesday 12th March 2014 ### Persistent Markov partitions for rational maps - Prof Mary Rees Abstract: The most famous example of a Markov partition which persists on a region of a parameter space is the Yoccoz puzzle. Each limb of the Mandelbrot set for quadratic polynomials admits a Markov partition which persists throughout the limb. The boundaries of the partition elements are contained in the so-called dynamical rays which land at the$\alpha $-fixed point, together with sections of equipotential. This Markov partition is the zero level of the Yoccoz puzzle for any polynomial in the limb. Successively Markov partitions are given by pulling back under iterates of the polynomial, to give level$n$Markov partitions for each positive integer$n$. Each level$n$partition persists on a region of parameter space with nonempty interior. The different dynamical level$n\$ partitions give successively finer partitions of the limb of the Mandlebrot set. The set of these partitions is often called the Yoccoz parapuzzle.

Since its invention some thirty years ago, the Yoccoz puzzle has been a very important tool in results concerning local connectivity properties of Julia sets and the Mandelbrot set, for example. There have been a number of extensions to other cases. Perhaps the most significant of these are due to Pascale Roesch, starting from her Ph D thesis, when she developed puzzles for the Newton maps for cubic polynomials. Another natural extension was used to study the family of quadratic  rational maps with period two critical point by Aspenberg and Yampolsky, who were able to reproduce many of the results obtained for the Mandelbrot set which used the Yoccoz puzzle. I am going to talk about another extension. The principal  example is of a Markov partition which persists in a neighbourhood of a geometrically finite quadratic rational map with one parabolic cycle. The level zero Markov partition is rather freely defined. It is worth stressing that the properties obtained for puzzle and parapuzzle do not depend on any particular level zero configurations.

### Exploring an arithmetic condition for the topology of infinitely satellite renormalisable quadratic polynomials - Dr Alexandre DeZotti

Abstract: Local connectivity of the Mandelbrot set is known to imply the density of hyperbolic maps in the quadratic family. Infinitely renormalisable maps are the last examples of quadratic polynomials for which the hyperbolic density conjecture has not yet be proved. Some of them are known to have non locally connected Julia sets. We explore the possibility of determining the local connectivity of the associated Julia sets for a subclass of infinitely renormalisable quadratic polynomials according to an arithmetic criterion only based on their combinatorics.

### No unexpected wandering domains for Bishop's example - Sebastien Godillon

Abstract: The construction of transcendental entire functions by quasi-conformal foldings recently provided by Bishop has allowed him to produce the first know example of wandering domain in Eremenko-Lyubich's class. After explaining Bishop's strategy, the main purpose of the talk is to show how a recent result of Lasse Rempe-Gillen and Helena Mihaljevic-Brandt of hyperbolic geometry in transcendental dynamics may be applied to prove that Bishop's example has no other wandering domain than those expected.

### A Newhouse phenomenon in 1D complex dynamics - Lasse Rempe-Gillen

Abstract: Density of hyperbolicity (also known as density of Axiom A) is a central problem in one-dimensional dynamics. In the real one-dimensional case, some celebrated results by Lyubich, Grazcyk-Swiatek, Kozlovski-Shen-van Strien and others have resolved this problem in a variety of settings, but the problem remains open in the complex case even for the Mandelbrot set.

On the other hand, it is well-known that, in real dynamics in several variables, we cannot expect hyperbolicity, or even one-dimensional structural stability (i.e., stability under perturbation of a parameter) to be dense. This is known as the Newhouse phenomenon: there will usually exist open regions in parameter space where a generic map has infinitely many attracting orbits, and the maps are not structurally stable.

We shall show that this situation can also occur in one-dimensional dynamics, when we consider transcendental entire functions with a large (in particular, infinite) set of critical values. This idea is joint work with Adam Epstein.

## Dynamics Learning Seminars 2014

### Dynamic rays of the exponential map - Stephen Worsley

Abstract: We know that for the family of exponential maps E_a(z) := ae^z , the connected components of the escaping set, which we call dynamic rays, are curves which tend to infinity in one direction along which the dynamics are well understood. However it is not guaranteed how the ray will behave in the other direction, whether it will land or not. In my talk I will discuss the behavior of these rays and show an example of when we can construct a non-landing ray for exponential maps with certain properties.

### Introduction to the dynamics of surface homeomorphisms - Dr Toby Hall

Abstract: This is a continuation of the talk on March 14

### Introduction to the dynamics of surface homeomorphisms - Dr Toby Hall

Abstract: This will be the first of 2 or 3 introductory talks on the dynamics of surface homeomorphisms. In this talk I will give an overview of Thurston's famous classification theorem for isotopy classes of surface homeomorphisms, and its dynamical applications. In subsequent talks I will go into some aspects in more detail.

### Friday 28th February 2014

Understanding the dynamics of complex polynomials: Dynamic rays, local connectivity and pinched discs - Prof Lasse Rempe-Gillen

Abstract: Consider a complex polynomial f in one variable. A key technique in the dynamical study of f, pioneered by Douady and Hubbard and finding a key application in celebrated work of Yoccoz, is to use 'external rays' to give a combinatorial tool for understanding the dynamics. I intend to give an overview of this theory and some of the main results. In particular, I will discuss the key role of local connectivity of Julia sets (and the Mandelbrot set) played in this context.

### Friday 14th February 2014

Holomorphic Motions - Prof Lasse Rempe-Gillen

Abstract: As mentioned in Wednesday's talk, *Holomorphic motions* play a key role in one-dimensional complex dynamics. I will discuss and prove the basic results of the theory, and mention some further theorems. If there is time, I will give some examples of applications in transcendental dynamics.