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Mathematics Institute

Talks

Abstract: The homotopy shadowing framework of Ishii-Smillie allows us to construct conjugacies between dynamical systems which are not ‘close’ in the sense of structural stability. We do this by considering more general multivalued dynamical systems. Complex Hénon maps are polynomial diffeomorphisms of $\mathbb{C}^2$ and a result of Friedland and Milnor tells us that these maps are interesting to study in terms of the dynamics. We will present an alternative proof, in the homotopy shadowing framework of Ishii-Smillie, of the well known theorem that for Hénon maps $H_{p,a}$ which are small perturbations of hyperbolic polynomials $p$, the topology of the Julia set of $p$ determines the Julia set of $H_{p,a}$

Abstract: Dynamical systems is an area of mathematics which studies how things evolve over time. One could think of the movement of planets, the growth of some animal population, or the swinging of a pendulum. In practice, it is usually concerned with the iteration of functions. Complex dynamics studies this iteration when our functions are defined over the complex numbers. Usually we are interested in studying where this iteration is "chaotic" or unpredictable, and we can begin studying this with very few prerequisites.

A significant motivation in the complex case is the striking images one can produce of the regions where our function's behaviour is chaotic. It is this aspect which caused a renaissance in the field during the 1980's. The pictures not only captured the imagination of mathematicians, but made their way into popular culture. A central object of this study, the Mandelbrot set, is perhaps one of the most recognisable mathematical images to non-mathematicians.

I will give a gentle introduction to this field and its interesting history, and discuss how the seemingly simple case of quadratic functions still has very difficult open problems. Along the way, I will show many pictures, illustrating the beauty of this field. If time permits, I will also discuss my work in higher dimensions.