Dynamical Systems
Seminars in 2020-21
In 2020-21, the organiser of the Dynamical Systems seminar was Dr James Waterman.
Seminars
Speaker: Eric Chang (Northwestern University)
Title: A Sierpinski Mandelbrot Spiral
Time: Thursday November 5, 15:00-16:00
Place: online (via MS Teams)
Abstract: The Sierpinski carpet fractal and Mandelbrot set are fascinating images with diverse applications. For iterated functions, fixing a parameter determines the dynamics. However, we will use the dynamical plane to talk about the parameter plane. In particular, for the family of singularly perturbed rational maps $z^n + \lambda / z^d$ for z and $\lambda$ complex, n at least 4 is even, and d at least 3 is odd, we identify patterns of Sierpinski holes and Mandelbrot sets via dynamical constructions, based on the work of Bob Devaney.
Speaker: Mary Rees (University of Liverpool)
Title: The Collatz conjecture
Time: Thursday October 29, 2020, 13:00-14:00
Place: online (via MS Teams)
Abstract: The Collatz map on the integers sends an even integer to n/2 and an odd integer to 3n+1.
Thus 1 is sent to 4, 2 to 1, 3 to 10, 4 to 2, and so on. There is therefore a periodic cycle of 1 to 2 to 4 to 1. The most basic form of the conjecture is:
Does the Collatz orbit of every strictly positive integer end in the periodic orbit of 1?
This problem emerged, initially by word of mouth, in the 1950's. Lothar Collatz had considered maps of this type since the 1930's. Erdos considered it "Hopeless. Absolutely hopeless''. John Conway showed that, in general, such problems could not be solved algorithmically. There are, however, positive results of a probabilistic nature. Terence Tao showed that almost every positive integer attains almost bounded values. He comments in his introduction that removing the second "almost'' would probably be as hard as proving the full conjecture.
Speaker: Gustavo Rodrigues Ferreira (The Open University)
Title: The Permutable Mystery Tour for transcendental meromorphic functions
Time: Thursday October 22, 2020, 13:00-14:00
Place: online (via MS Teams)
Abstract: Which pairs of analytic functions commute? This question has been studied for a century now, and we have several reasonable answers for polynomials and rational functions. For transcendental meromorphic functions (entire and non-entire), however, little is known. In this talk, I will focus on a dynamical approach to this question: do permutable transcendental meromorphic functions have the same Julia set? As we will see, in many cases the answer is "yes", but the most general case remains open. If time allows, we will also see how symmetries of Julia sets relate to this problem.
Speaker: Rafael Alcaraz Barrera (IF-UASLP)
Title: A generalised Blanchard's program
Time: Thursday October 15, 2020, 13:00-14:00
Place: online (via MS Teams)
Abstract: In 1989, F. Blanchard sumarised results regarding a classification theorem for the symbolic dynamics of the class of beta-shifts. In particular, beta shifts of finite type, sofic, beta-shifts with the specification property and synchronised beta-shifts were characterised using the combinatorial properties of the quasi-greedy expansion of 1 in a non-integer base q. During the talk, we describe some recent progress in extending such classification to the class of intermediate beta-shifts on two symbols. If time permits, we will also mention some results and problems regarding such classification result for an arbitrary set of symbols.