Dynamical Systems
LMS Scheme 3 meeting on Holomorphic Dynamics
Wednesday 22nd June 2022
Important notice: There will be a train strike on Tuesday June 21 and Thursday June 23, so please keep this in mind when planning your trip. As of June 20, train tickets are being sold online for Wednesday June 22, and it is possible to travel from London and return on the same day.
The University of Liverpool will hold a meeting of the LMS Scheme 3 network on Holomorphic Dynamics. The coordinator of the network is Dr Daniel Nicks from the University of Nottingham. This meeting is organised by Dr Vasiliki Evdoridou and Dr David Martí-Pete.
This will be a hybrid meeting, and it will be possible to attend the meeting online (via Zoom). If you would like to attend the meeting (either in person or online), please contact Vasiliki or David. There is a limited amount of funding to support the travel expenses of PhD students and early-career researchers from the universities in the network (Imperial College London, The Open University and University of Nottingham).
Schedule
The schedule of this meeting will be the following:
12:00 - 13:00 lunch at the Victoria Gallery Museum
13:00 - 13:50 Rohini Ramadas (University of Warwick)
14:00 - 14:25 Matteo Tabaro (Imperial College London)
14:30 - 15:00 coffee break
15:00 - 15:40 James Waterman (Stony Brook University)
15:50 - 16:15 Caroline Davis (Indiana University) — online talk
16:20 - 17:10 Phil Rippon (The Open University)
The talks will take place at room MATH-211 of the Mathematical Sciences Building, indicated by number 206 on the campus map. If you plan to come by car, please read the information about the on-campus visitor car parking.
Abstracts and slides
Caroline Davis (Indiana University)
Almost all matings are carpets
Abstract: Mating is a popular way of combining two polynomials' dynamics into a single rational map. Despite that the most familiar examples don't display Sierpinski carpet topology, many experts believe that, in some sense, a generic mating ought to be a carpet. In this talk we will remind the audience what a carpet is, then sketch a proof that given any two quadratic primitive PCF hyperbolic polynomials whose mating is a rational map, the probability that its Julia set is a carpet goes to 1 as the periods of the to-be-mated polynomials go to infinity. As an upshot, we conclude with what the proof techniques can indicate about the structure of Per_n(0). This is joint work in preparation with Insung Park.
Download the slides: 2022 LMS Scheme 3 Holomorphic Dynamics - Caroline Davis
Rohini Ramadas (University of Warwick)
Irreducibility of Gleason polynomials implies irreducibility of Per_n
Abstract: Per_n is a punctured (nodal) Riemann surface parametrizing quadratic rational maps with an n-periodic critical point; its irreducibility over C is an open question. The Gleason polynomial G_n is the polynomial whose roots are {c such that 0 is n-periodic under z^2+c}; its irreducibility over Q is an open question. I will discuss recent work, finding a smooth point with Q-coordinates “at infinity” on Per_n, and using this to conclude that if G_n is irreducible over Q, then Per_n is irreducible over C.
Download the slides: 2022 LMS Scheme 3 Holomorphic Dynamics - Rohini Ramadas
Phil Rippon (The Open University)
The Borel–Cantelli lemmas and applications in complex dynamics
Abstract: In this largely expository talk we discuss the Borel–Cantelli lemmas and various extensions and generalisations of these, and we describe some ways in which these lemmas have been applied in complex dynamics, in particular in joint work with Anna Benini, Vasiliki Evdoridou, Nuria Fagella and Gwyneth Stallard.
Download the slides: 2022 LMS Scheme 3 Holomorphic Dynamics - Phil Rippon
Matteo Tabaro (Imperial College London)
Semi-hyperbolicity implies existence of ACIPs for real multimodal maps
Abstract: Our work focuses on understanding the existence of absolutely continuous invariant probability measures (ACIPs) for real multimodal maps in order to investigate Palis’ conjecture. In the last few decades, a plethora of sufficient analytic conditions for the existence of ACIPs have been developed: Collet-Eckmann condition, Nowicki-van Strien summability condition, Martens-Nowicki condition or the BRSS large derivative condition, just to name a few. Topological conditions are also known: in [1] Rivera-Letelier and Shen showed that Topological Collet-Eckmann is sufficient for the existence of ACIPs for real C3 multimodal maps. We prove a similar result exploiting substantially different techniques. Our main theorem states semi-hyperbolic real C3 multimodal maps with negative Schwarzian derivative with no periodic attractors support ACIPs. We also show that there exist real multimodal maps which admit ACIPs thanks to our condition, but do not satisfy any of the previously listed analytic conditions.
Bibliography
[1] J. Rivera-Letelier and W. Shen, "Statistical properties of one-dimensional maps under weak hyperbolicity assumptions", Annales Scientifiques de l'École Normale Superieure, vol. 47, no. 2, pp. 1027–1083, 2014.
Downolad the slides: 2022 LMS Scheme 3 Holomorphic Dynamics - Matteo Tabaro
James Waterman (Stony Brook University)
Maverick points on the boundary of wandering domains
Abstract: Wandering domains, or wandering Fatou components, are a central object of study in the iteration of transcendental entire functions. We will introduce several basic properties of wandering domains. Moreover, focusing on behavior on the boundary of these domains, we will discuss the existence of points on the boundary of a wandering domain with accumulation behavior distinct from that of the wandering domain itself. We call such points maverick points. This is joint work with David Martí-Pete and Lasse Rempe.
Downolad the sides: 2022 LMS Scheme 3 Holomorphic Dynamics - James Waterman
Recordings
Group picture
