# Dynamical Systems

## Seminars in 2022-23

In 2022-23, the organisers of the Dynamical Systems seminar were Dr Vasiliki Evdoridou and Dr David Martí-Pete. From January 2023, the seminar was only organised by David, as Vasiliki moved to The Open University.

### Seminars

Speaker: **Lukas Geyer** (Montana State University)

Title: **Hausdorff dimension of Julia sets in class S**

Time: Thursday July 13, 2023, 13:00-14:00

Place: Room MATH-211, hybrid format (via Zoom)

Abstract: The Speiser class S of transcendental entire functions with finitely many singular values naturally decomposes into a countable union of finite-dimensional complex parameter spaces as defined by Eremenko and Lyubich. We show that every hyperbolic component in any such parameter space contains maps whose Julia sets have Hausdorff dimension arbitrarily close to 2. Combining this with previously known results, it follows that Hausdorff dimension of Julia sets in class S attains all values in (1,2]. This is joint work with Jack Burkart.

Speaker: **Weiwei Cui **(Centre for Mathematical Sciences, Lund University)

Title: **Collet-Eckmann rational maps with slowly recurrent critical points**

Time: Wednesday June 14, 2023, 15:00-16:00

Place: Room MATH-211, hybrid format (via Zoom)

Abstract: Let f be a rational map satisfying the Collet-Eckmann condition. We show that such maps are Lebesgue density points of hyperbolic maps if critical points in the Julia set are slowly recurrent. This is a joint work with Magnus Aspenberg and Mats Bylund.

Speaker: **Bernhard Reinke** (University of Liverpool)

Title: **Emergence of tree automorphisms**

Time: Thursday May 11, 2023, 13:00-14:00

Place: Room MATH-103, hybrid format (via Zoom)

Abstract: Emergence is a way to describe the richness of possible statistical behaviours of orbits of a dynamical system. One way to quantify emergence is to consider the upper box dimension of the space of ergodic measures. A system has high (topological) emergence if the space of ergodic measures is infinite dimensional while the underlying space of the dynamical system has finite dimension.

I will give an introduction to emergence in the context of automorphisms of spherically homogeneous rooted trees acting on Cantor sets embedded in the real line. I will provide a criterion for the existence of automorphisms with high emergence based on the degree growth of vertices in the given tree.

Moreover, for sufficiently fast growing trees, we construct examples of conjugacy classes of rooted tree automorphisms such that the action of a random element of the conjugacy class on the ends of the tree has high emergence almost surely.

Speaker: **Kevin Wildrick** (University of Bern)

Title: **Quasiconformal mappings of the Heisenberg group via iterated function systems**

Time: Thursday May 4, 2023, 13:00-14:00

Place: Room MATH-103, hybrid format (via Zoom)

Abstract: The sub-Riemannian Heisenberg group is a non-commutative group structure on three-dimensional Euclidean space equipped with a non-smooth metric adapted to the group structure. Quasiconformal mappings on this group were initially studied in the context of Mostow’s rigidity theorems and have played in an important role in the development of geometric mapping theory on metric spaces. Despite this, an existence theory analogous to that of the planar setting has not yet been developed. In this talk, we will see how to use basic ideas from dynamics to produce new quasiconformal mappings. More precisely, given two suitable iterated function systems of contracting similarities of the sub-Riemannian Heisenberg group, we construct a quasiconformal mapping that maps one invariant set onto the other. We use this construction to produce a quasiconformal mapping of the Heisenberg group that simultaneously raises the dimension of many fibers of a foliation by left-translates of a horizontal subgroup. This provides both a qualitative and quantitative improvement upon previous examples of such mappings. This talk is based on joint work with Jeremy Tyson (University of Illinois at Urbana-Champaign).

Speaker: **Malavika Mukundan** (University of Michigan)

Title: **Dynamical approximation of entire functions**

Time: Tuesday March 7, 2023, 16:00-17:00

Place: Room MATH-106, hybrid format (via Zoom)

Abstract: Postsingularly finite holomorphic functions are entire functions for which the forward orbit of the set of critical and asymptotic values is finite. Motivated by previous work on approximating entire functions dynamically by polynomials, we ask the following question:

Given a postsingularly finite entire function f, can f be realised as the locally uniform limit of a sequence of postcritically finite polynomials?

In joint work with Nikolai Prochorov and Bernhard Reinke, we show how we may answer this question in the affirmative.

Speaker: **Dierk Schleicher** (Aix-Marseille Université)

Title: **Polynomial root finding between numerics and dynamics**

Time: Monday February 20, 2023, 15:00-16:00

Place: Room MATH-029, hybrid format (via Zoom)

Abstract: Since Gauss we know that every polynomial in one variable factors into (complex) roots, and since Ruffini and Abel we know that in general one needs iterative methods to locate these roots. Three of the best known root finding methods are due to Newton-Raphson, to Weierstrass-Durand-Kerner, and to Ehrlich-Aberth. These have been known for a long time. Newton (tries to) converge for one root at a time, while the other are supposed to converge to a vector of all the roots simultaneously.

Common knowledge in the numerical analysis community is that the two high-dimensional methods always work except in cases of obvious symmetry, while Newton has problems. However, until recently very little has been known about the global dynamics of either of these methods.

We present an outline on recent results about the global dynamics of (some of) these methods, and try to describe in more detail the argument for an explicit upper bound on the complexity of Newton’s method.

Speaker: **Jordi Canela Sánchez **(Universitat Jaume I)

Title: **Julia sets with a wandering branching point**

Time: Thursday February 2, 2023, 13:00-14:00

Place: Room MATH-103, hybrid format (via Zoom)

Abstract: According to the Thurston No Wandering Triangle Theorem, a branching point in a locally connected quadratic Julia set is either preperiodic or precritical. Blokh and Oversteegen proved that this theorem does not hold for higher degree Julia sets: there exist cubic polynomials whose Julia set is a locally connected dendrite with a branching point which is neither preperiodic nor precritical. In this talk we reprove this result, constructing such cubic polynomials as limits of cubic polynomials for which one critical point eventually maps to the other critical point which eventually maps to a repelling fixed point. This is a joint work with Xavier Buff and Pascale Roesch.

Speaker: **Eero Saksman** (University of Helsinki)

Title: **On multiplicative Gaussian Chaos**

Time: Monday January 30, 2023, 15:00-16:00

Place: Room MATH-105, hybrid format (via Zoom)

Abstract: I will discuss, in an informal manner, what are Gaussian multiplicative chaos measures, and where do they appear.

Speaker: **Natalia Jurga** (University of St Andrews)

Title: **Hausdorff dimension of self-projective sets**

Time: Thursday January 12, 2023, 13:00-14:00

Place: Room CHEM-BRUN, Brunner Lecture Theatre (Chemistry), hybrid format (via Zoom)

Abstract: A finite set of matrices $A \subset SL(2,R)$ acts on one-dimensional real projective space $RP^1$ through its linear action on $R^2$. In this talk we will be interested in the limit set of $A$: the smallest closed subset of $RP^1$ which contains all attracting fixed points of matrices belonging to the semigroup generated by $A$. Recently, Solomyak and Takahashi proved that if $A$ is uniformly hyperbolic and satisfies a Diophantine property, then the invariant set has Hausdorff dimension equal to the minimum of 1 and the critical exponent. In this talk we will discuss an extension of their result beyond the uniformly hyperbolic setting. This is based on joint work with Argyrios Christodoulou.

Speaker: **Alex Kapiamba** (University of Michigan)

Title: **Elephants all the way down: the near-parabolic geometry of the Mandelbrot set**

Time: Thursday December 8, 2022, 13:00-14:00

Place: Online talk (via Zoom)

Abstract: Understanding the geometry of The Mandelbrot set, which records dynamical information about every quadratic polynomial, has been a central task in holomorphic dynamics over the past forty years. Near parabolic parameters, the structure of the Mandelbrot set is asymptotically self-similar and resembles a parade of elephants. Near parabolic parameters on these "elephants'', the Mandelbrot set is again self-similar and resembles another parade of elephants. This phenomenon repeats infinitely, and we see different parades of elephants at each scale. In this talk, we will explore the implications of controlling the geometry of these elephants. In particular, we will partially answer Milnor's conjecture on the optimality of the Yoccoz inequality, and see potential connections to the local connectivity of the Mandelbrot set.

Speaker: **Rebecca Winarski** (College of the Holy Cross)

Title: **Tree lifting and twisted rabbits**

Time: Thursday November 17, 2022, 13:00-14:00

Place: Room MATH-106, hybrid format (via Zoom)

Abstract: Thurston proved that a post-critically finite branched cover of the plane is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction. We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a topological branched cover is equivalent to. This is joint work with Jim Belk, Justin Lanier, and Dan Margalit. I will discuss recent applications of the tree lifting algorithm to solving infinite families of twisted rabbit problems with Lanier and Mukundan.

Speaker: **Adam Epstein** (University of Warwick)

Title: **Integrality and rigidity for postcritically finite polynomials**

Time: Thursday November 10, 2022, 13:00-14:00

Place: Room MATH-029, hybrid format (via Zoom)

Abstract: It is a well-known fact that the centers of hyperbolic components of the Mandelbrot set are simple roots of their defining polynomials. Using similar arithmetic methods, we prove a version of this result for the multi-dimensional parameter space of higher degree polynomials (of prime power degree). If time permits, we will discuss potential generalisations.

Speaker: **Thomas Richards** (University of Warwick)

Title: **Monodromy and complex Hénon maps**

Time: Thursday November 3, 2022, 13:00-14:00

Place: Room MATH-106, hybrid format (via Zoom)

Abstract: Blanchard, Devaney, and Keen proved that loops in the shift locus of degree *d* polynomials induce automorphisms of the one-sided shift of *d* symbols. Hubbard conjectured that an analogous result holds in Hénon parameter space. In my talk I will discuss this conjecture, and some experimental work aiming to understand it.

Speaker: **Maria Kourou** (University of Würzburg)

Title: **Angular derivatives and petals of semigroups of holomorphic functions**

Time: Thursday October 20, 2022, 13:00-14:00

Place: Online talk (via Zoom).

Abstract: Let $(\phi_t)_{t\geq 0}$ be a one-parameter semigroup of holomorphic self-maps of the unit disk $\mathbb{D}$.

A boundary fixed point $\sigma$ of $(\phi_t)$ is called *repelling* if $\phi_t^{\prime}(\sigma) \in (1, +\infty)$ (in the angular limit sense).

Every repelling fixed point of $(\phi_t)$ corresponds to a *petal* $\Delta$ that is an open simply connected subset of $\mathbb{D}$ where the restriction of $(\phi_{t}) $ is a group of automorphisms.

For every such petal, there exists a conformal mapping $g:\mathbb{D} \to \Delta$ with $ g(\sigma)=\sigma$ that is semi-conformal at $\sigma$; i.e. the angular limit $$\angle \lim_{z \to \sigma} \text{Arg} \frac{\sigma - g(z)}{\sigma - z}=0. $$

If, additionally, the angular derivative $g^{\prime}(\sigma) \in \mathbb{C} \setminus \{0\}$, $g$ is *conformal* at $\sigma$. In this case, the petal $\Delta$ is said to be *conformal* at $\sigma$.

We discuss necessary and sufficient geometric conditions such that the petal $\Delta$ is conformal at its associated repelling fixed point $\sigma$, obtained in a joint work with Pavel Gumenyuk and Oliver Roth.

Speaker: **Myrto Manolaki** (University College Dublin)

Title: **A strong form of Plessner's theorem and applications**

Time: Thursday October 6, 2022, 13:00-14:00

Place: Room MATH-029, hybrid format (via Zoom).

Abstract: Let *f * be a holomorphic function on the unit disc. According to Plessner's theorem, for almost every point *ζ* on the unit circle, either (i) *f * has a finite nontangential limit at *ζ*, or (ii) the image *f *(*S*) of any Stolz angle *S* at *ζ* is dense in the complex plane. In this talk, we will see that condition (ii) can be replaced by a much stronger assertion. This strong form of Plessner's theorem and its harmonic analogue on halfspaces also improve classical results of Spencer, Stein and Carleson. (Joint work with Stephen Gardiner.)

Speaker: **Jonathan Fraser** (University of St Andrews)

Title: **Dimensions of parabolic Julia sets and Kleinian limit sets**

Time: Wednesday September 7, 2022, 13:00-14:00

Place: Room MATH-117, hybrid format (via Zoom)

Abstract: The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. An especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. I will demonstrate that by slightly expanding the family of dimensions considered, a richer and more nuanced correspondence arises. This is joint work with Liam Stuart.