Consider the polynomial $f(z) = z^2 - 2$. We think of it as a function, taking a point $z$ of the complex plane into a different point $f(z)$. Given $z$, we also consider the iterations $f^n(z)$ of $z$ under the map $f$. For example, letting $z = 0$ we get
- $f(0) = -2$,
- $f^2(0) = f(-2) = 2$,
- $f^3(0) = f(2) = 2$, etc…
The next point in the orbit is always obtained by applying $f$ to the previous point.
The Julia Set of $f$ consists of all those points $z$ such that the orbit of $z$ remains bounded. The Julia set of $f(z) = z^2 - 2$ is relatively easy to calculate: it is the interval $[-2,2]$. For any other point $z$, the iterates eventually grow larger and larger, tending to infinity. For example, for the imaginary unit $z = i$ we get
- $f(i) = i^2 - 2 = -1 - 2 = -3$,
- $f^2(i) = f(-3) = 7$,
- $f^3(i) = f(7) = 47$, etc…
Now, what if we replace $2$ by some other number $c$ and consider the dynamics of the innocuous looking function $f_c(z) = z^2 - c$? Unlike the example above, the Julia set of $f_c$ will typically be a complicated set that is fractal in nature. In this talk we will discuss and explore this amazing phenomenon, and the interactions between polynomials and fractals.
Karl-Mikael Perfekt is an associate professor in mathematics at NTNU. His research area is mathematical analysis, a branch of mathematics that deals with limits, approximations, quantities and inequalities.
