Andrew D. Smith

will speak on

Spirals in Spaces of Holomorphic Functions

Time: 3:00PM
Date: Tue 17th September 2024
Location: E0.32 (beside Pi restaurant) [map]

Further information

Abstract: Functions $W(t,z)$ of real time $t \ge 0$ and $z \in \mathbb C$ satisfy the spiral relation:

$$
W(2t, z) = (1+e^z) W(t, z)
$$

For fixed $t$, these are holomorphic functions of $z$ in the region:

$$
\vert \Im z \vert < \cos^{-1} \left[ - \tfrac 1 2 e^{ - \vert \Re z \vert} \right]
$$

Viewed as functions of $t$, for fixed $z$, the functions $W(t,z)$ are H\'{o}lder continuous and nowhere differentiable. They have a time-homogeneity property if $\Re z = 0$, while for $\Im z = \pm \frac 1 2 \pi $ the paths have finite quadratic variation; a property also associated with semi-martingale paths in the theory of stochastic processes.


The $W$ functions can produce beautiful images. Familiar fractal sets: L\'{e}vy's C-curve, Heighway's dragon curve and van Roy's unicorn curve arise as the loci of $W(t,z)$ when $0 \le t \le 1$ and $z = \pm \frac 1 2 i\pi$, that is, functions of $t$ that satisfy both the time-homogeneity and quadratic variation criteria.

https://ucd-ie.zoom.us/j/61064699009

(This talk is part of the Analysis series.)

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