# Iterations of the complex Henon function.

This page introduces a Java applet which allows one to explore the orbits of a particular case of the complex Henon function. The applet was written by Camilla Jordan and Jonathan Jordan. The original reason for investigating this particular case, came from research being carried out by David Jordan. It is possible that your browser will not run the applet.

$$H\begin{pmatrix}z\\w\end{pmatrix}=\begin{pmatrix}\alpha-\beta w-z^2\\z\end{pmatrix}.$$

The case in which we are interested has $$\beta=\frac{\alpha}{|\alpha|}.$$ This means that $$H$$ can be written as the composition of two involutions. (Involutions are self inverse functions.)

### First Activity

The image which appears on the left side of the applet shows (in black) the set of $$\alpha$$ which may have bounded orbits. It can be shown that the points which are not black have unbounded orbits.

• Click with the mouse on this image. The value of $$|\alpha|$$,shown as $$r$$, and $$\arg(\alpha)$$, shown as $$\theta$$ and expressed as a multiple of $$\pi$$, will appear in the appropriate boxes.
• Click on the button "Plot Orbit" and the first 20,000 points of the orbit, starting from (0,0), will be plotted on the right of the applet, unless the orbit is found to be unbounded. Plotting stops as soon as the orbit can be shown to be unbounded.

### Choosing r and q.

• Enter the desired value of r into the r box.
• Enter the desired multiple of p for q into the theta box.
• Click on the button "Plot Orbit" as before.

### Changing the starting point of the orbit.

• Plot an orbit starting from (0,0).
• Click in the right hand area of the window. The message "Ready to start plotting at ¼" will appear.
• Click on "Plot Orbit".

### Colouring

Some orbits are readily partitioned. These partitions can be coloured using the "No. of cosets" box. If, for example you enter 4 in this box, the orbit will have 4 colours which will be plotted cyclically. Try theta as .333333333 (approx 1/3) and enter 6 in this box to see the effect.

If you enter a number, other than 0, in the "Coset to plot" box then only points corresponding to multiples of that number will be plotted. This enables you to get a more detailed look at individual partitions of the orbit.

### Scale etc.

The other buttons enable you to zoom in or out from an orbit, or move about the screen. The new orbit is replotted from scratch each time. "Clear" removes all orbits and "Reset" clears the orbits and puts the screen parameters back to their initial values.