1. Julia Sets • Consider the map: xn+1=xn2+c with c=-3/4 • Choose some values of x and iterate many times - what happens ? • See that if initial x lies between the values -1.5 and 1.5 it will always remain so. • But if the initial x lies outside that range the value of x will blow up.

2. Complex Maps • Now can consider a more general set of numbers than just the decimals - complex numbers • x --> z • Maps of complex numbers: zn+1=zn2+c • Can ask what is the set of complex numbers that do not blow up under iteration.

3. Drawing complex numbers • Can use the real and imaginary parts as a pair of (x,y) coordinates in plane • i.e 2+3i is plotted at the point (2,3) • -4i is plotted at point (0,-4) • Usual real numbers lie along the x-axis of this plot.

4. Complex numbers • Arise when we want to generalize the notion of square root. • What is the number which multiplied by itself gives -1 ? • There is no ordinary, decimal (real) number which has this property. • Define a new number called i i2=-1

5. More complex numbers • Now I can square root any negative number • eg. -4=4*-1 sqrt(-4)=sqrt(4)i =2i • The number 2i is an example of an imaginary number • In general a complex number is the sum of a real number and an imaginary number z=x+iy

6. Adding complex numbers • Suppose I have 2 complex numbers z1=c+id and z2=a+ib - how do I add them ? z=z1+z2 • Just add real and imaginary parts separately z=(a+c)+i(b+d)

7. Multiplying ... • z1=a+ib, z2=c+id z= z1*z2 ? z=(a+ib)*(c+id) = (a*c+ib*c+id*a+i2b*d) = (a*c-b*d)+i(b*c+d*a) • Absolute value of z=x+iy |z|2=(x+iy)*(x-iy) • |z|=sqrt(x*x+y*y)

8. Back to Maps ... • OK, so now we have a new sort of number to play with • Consider Maps znew=z2+c (z and c are now complex) • Choose some c eg (-3/4,0) • Can now ask what is the set of z which do not blow up under iteration ? (absolute values that do not get infinite )

9. Drawing them ... • This set of z’s which remain bounded make up the Julia Set associated with the map. • Can draw these points on the plane - it is a fractal! • Black points - values of z which do not get infinite. • Red get big fastest • Yellow get big next fastest • Green next • Blue are slowest to get big.

10. Different Julia Sets • Choosing different c yields different fractals! • All have structure at all length scales and possess a fractional dimension.

11. Mandelbrot Set • For the map znew=z2+c • Choose initial z=(0,0). • Choose a (complex) c. • Does the iteration diverge ? • Compute the set of c’s where the iterates never get big this set is again a fractal -- the famous Mandelbrot set

12. Summary • Generalize the concept of ordinary (real) number to complex numbers z • Consider nonlinear maps of z • Set of z’s which do not diverge under iteration - Julia set and when plotted is seen to have a fractal structure (once again!) • Motion of z on boundary of this set is again chaotic.