# Dynamical Systems

## Seminars in 2021-22

In 2021-22, the organiser of the Dynamical Systems seminar was Dr David Martí-Pete.

### Seminars

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

Title: **Perturbations of non-recurrent exponential maps**

Time: Thursday June 16, 2022, 13:00-14:00

Place: online talk (via Zoom)

Abstract: Consider the exponential family

{λe^{z }: λ∈ℂ\{0}}.

A parameter λ is non-recurrent if 0∉ω(0). We show that the set of non-recurrent parameters has Lebesgue measure zero. Moreover, non-recurrent parameters can be approximated by hyperbolic ones. This is based on joint work with Magnus Aspenberg.

Speaker: **Gustavo Ferreira** (The Open University)

Title: **Uniformity in internal dynamics of wandering domains and some inner functions**

Time: Thursday June 9, 2022, 13:00-14:00

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

Abstract: In recent years, a new strategy for investigating the internal and boundary dynamics of simply connected wandering domains emerged, pioneered by Benini, Evdoridou, Fagella, Rippon, and Stallard. This approach consists of conjugating the orbit of a simply connected wandering domain to the forward composition of a sequence of inner functions of the unit disc and investigating the properties of the latter. They showed that the internal dynamics of a simply connected wandering domain is uniform: the long-term behaviour is the same for all distinct pairs of orbits. After recently showing that their strategy fails for multiply connected wandering domains, we ask ourselves: under what conditions is the internal dynamics of a multiply connected wandering domain uniform? We answer this question by showing that if an open subset of the wandering domain displays uniform internal dynamics, so does the whole wandering domain, and we show how this knowledge can be used to construct new examples. Then, we turn to the related problem of forward composition of inner functions, and obtain new, more detailed information on the nature of possible limit functions.

Speaker:** Insung Park** (ICERM)

Title: **Julia sets having minimal conformal dimension**

Time: Thursday May 19, 2022, 16:00-17:00

Place: online talk (via Zoom)

Abstract: As a fractal embedded in the Riemann sphere, the Julia set of a post-critically finite rational map has conformal dimension between 1 and 2. The Julia set has conformal dimension 2 if and only if it is the entire Riemann sphere. However, the other extreme case, when conformal dimension=1, contains diverse Julia sets, including the Julia sets of post-critically finite polynomials and Newton maps. In this talk, we show that for a post-critically finite hyperbolic rational map $f$, the Julia set $J_f$ has conformal dimension one if and only if there exists an f-invariant graph with topological entropy zero. In the spirit of Sullivan’s dictionary, we can also compare this result with the classification of Gromov-hyperbolic groups whose boundaries have conformal dimension one, which Carrasco-Mackay proved.

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

Title: **Thurston theory: unifying topological and dynamical**

Time: Thursday April 21, 2022, 16:00-17:00

Place: online talk (via Zoom)

Abstract: Thurston proved that a non-Lattés branched cover of the sphere to itself is either equivalent to a rational map (that is: conjugate via a mapping class), or has a topological obstruction. The Nielsen–Thurston classification of mapping classes is an analogous theorem in low-dimensional topology. We unify these two theorems with a single proof, further connecting techniques from surface topology and complex dynamics. This is joint work with Jim Belk and Dan Margalit.

Speaker: **Athanasios Tsantaris** (University of Helsinki)

Title: **Dynamics of Zorich maps**

Time: Thursday March 24, 2022, 16:00-17:00

Place: online talk (via Zoom)

Abstract: In the theory of one dimensional holomorphic dynamics, one of the most well studied families of maps is the exponential family $E_\lambda(z):=\lambda e^z$, $\lambda\in \mathbb{C}\setminus\{0\}$. Zorich maps are the quasiregular higher dimensional analogues of the exponential map on the plane. In this talk we are going to discuss how many well known results about the dynamics of the exponential family generalize the higher dimensional setting of Zorich maps.

Speaker: **Clifford Gilmore** (University of Manchester)

Title: **The dynamics of weighted composition operators on Fock spaces**

Time: Friday December 10, 2021, 14:00-15:00

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

Abstract: The study of weighted composition operators acting on spaces of analytic functions has recently developed into an active area of research. In particular, characterisations of the bounded and compact weighted composition operators acting on Fock spaces were identified by, amongst others, Ueki (2007), Le (2014), and Tien and Khoi (2019). In this talk I will examine some recent results that give explicit descriptions of bounded and compact weighted composition operators acting on Fock spaces. This allows us to prove that Fock spaces do not support supercyclic weighted composition operators. This is joint work with Tom Carroll (University College Cork).

Speaker: **Ana Rodrigues** (University of Exeter)

Title: **Dynamics of piecewise isometries**

Time: Thursday November 25, 2021, 13:00-14:00

Place: online talk (via Zoom)

Abstract: In this talk I will discuss some features of the dynamics of Piecewise isometries (PWIs) which are higher dimensional generalizations of one dimensional IETs, defined on higher dimensional spaces and Riemannian manifolds. In particular, I will introduce the concept of embedding of an IET into a PWI, some particular renormalization scheme and if time allows, the proof of existence of invariant curves for PWIs.

Speaker: **Krzysztof Lech** (University of Warsaw)

Title: **Random quadratic Julia sets**

Time: Thursday November 11, 2021, 13:00-14:00

Place: online talk (via Zoom)

Abstract: For a sequence (c_n) let us consider compositions of functions z^2 + c_n. The definitions of the Julia and Fatou sets are naturally extended to these families of compositions. We shall discuss some results on the connectedness of random Julia sets. In particular the following setting will be of interest: let c_n be chosen with uniform distribution from a disk of radius R centered at 0. Depending on R, what can we say of a typical Julia set, i.e. is it connected/disconnected/totally disconnected?

Speaker: **Lawrence Lee** (University of Manchester)

Title: **L^q-spectra of measures on non-conformal attractors**

Time: Thursday November 4, 2021, 13:00-14:00

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

Abstract: A key aim in fractal geometry is to understand the dimension theory of "irregular" sets and measures. Whilst self-similar and self-affine sets (i.e. sets which are invariant under collections of contracting similarities or contracting affine maps) have been extensively studied, less attention has been given to sets invariant under nonlinear, non-conformal contractions. This is undoubtedly due to the additional challenges posed by working in the nonlinear setting.

In this talk we'll consider a class of measures in the plane, which are supported on attractors of iterated function systems consisting of nonlinear, non-conformal maps with triangular Jacobian matrices. We'll consider a notion of dimension for measures known as the L^q-spectrum, and using ideas from thermodynamic formalism we'll see how this can be calculated in our setting. As a corollary we'll also obtain the box dimension of the sets our measures are supported on. This is joint work with Kenneth Falconer and Jonathan Fraser.

This is a joint talk between the Dynamical Systems and Stochastics seminars.

Speaker: **Mary Rees** (University of Liverpool)

Title: **Collatz orbits of sparser numbers**

Time: Thursday October 14, 2021, 13:00-14:00

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

Abstract: The Collatz map sends an integer n to n/2 if n is even and 3n+1 if n is odd. So the orbit of 1 is 1,4,2,1,4,2,1..

The orbit of 3 is 3,10,5,16,8,4,2,1.. The famously intractable Collatz conjecture, which is thought to have been first circulated by word of mouth in 1950, states that the orbit of every strictly positive integer ends in the cycle 1,4,2..

It is not even known if this problem is decidable. John Conway proved in the 1950's that a more general problem is not decidable. What positive information we have about the Collatz map is largely, although not entirely, of a probabilistic nature. It was shown in the 1970's that, for most positive integers n, in the sense of positive density, the Collatz orbit of n passes below n. Terence Tao's 2019 paper ``Almost all Collatz orbits attain almost bounded values'' is a powerful strengthening of this basic result.

Most work on the Collatz conjecture is focussed on ``typical'' integers, which essentially means those n such that the orbit of n starts by passing below n in a reasonable amount of time. Tao's method shows that the orbits such numbers typically continue to decrease for some time.

This talk will concentrate on numbers which are not necessarily typical -- by looking not at the standard measures of uniform density on intervals of integers, but at other measures on integers: the so called geometric measures, and integers which are typical for such a measure. These are numbers whose Collatz orbits start by increasing, depending on the mean of the geometric measure. A chain of conjectures will be stated which, if true, show that the Collatz orbits of these sparser numbers do later decrease for some time. A first weaker version of the final conjecture in the chain has been proved: not enough to be the basis of an induction, but a step in the right direction.