Dynamical Systems - 8/4/10

Today, we were treated to a quick introduction to dynamical systems, an area of math which focuses on repeated iteration of maps to study long-term or asymptotic behavior associated with this iteration.

We started by taking a simple example. Consider a real valued function f from R to R. Then consider the sequence of points (x, f(x), f(f(x)) ....) We call this sequence the orbit of the point x.

Let us take f(x) = kx(1-x), which is the equation for a quadratic function. Also consider the identity map, g(x) = x.

For now, we take k >0.

Depending on our specific value of k, we have a parabola which intersects the g(x) either once or twice.

We are interested in the fixed points of f. As the name implies, these are points where the map f leaves a point in its domain fixed, or in symbols: f(x1) = x1.

We wish to study asymptotic behavior of functions. To do this, we pick a point x on the real line and first apply the function. This gives us a new point f(x). Then we find f(f(x)). Since we have the identity map also drawn in (in this case, g(x)), this process is applying f, moving horizontally to g, moving vertically to f, and so on.

We see that repeating this process, we notice two general patterns: In some situations, we may continue onward towards positive or negative infinity in an unbounded fashion. However, for other sections of the real line, we see that there is a convergence towards certain fixed points.

We call a fixed point x attracting if the magnitude of its derivative at x is less than 1 and we say a fixed point is repelling if the magnitude of its derivative at x is greater than 1.

We see then that the derivative of the function at these fixed points greatly affects this convergence behavior.

Afterwards, we moved on to dynamics of complex functions which map the complex plane to itself.

We first took f: C -> C (complex plane to itself) given by f(z) = z^2 + c for some complex number c.

If we let c = 0, then points inside the unit disk spiral towards the origin. Points outside the unit disk spiral to infinity. We see that points on the unit circle itself get mapped to other points on the unit circle (their length is preserved, but they rotate.) Thus the mapping leaves the unit circle invariant. Furthermore, we see that iterates of the map can map a small section of the unit disk onto the whole unit disk.

We now consider the effect of varying our complex number c. For c very small, we obtain a similar picture: the interior of the curve spirals inward to the center, the curve itself is invariant, and the outer regions escape to infinity.

For very large x, most points escape outward to infinity.

We then construct the Mandelbrot set as the set of points c such that the n-th iterate of the map f does not go to infinity when evaluated at the point c itself for technical reasons.

We take all parameters c and iterate at the point c itself.

Similar to our notion of fixed points, we say a point z in the complex plane is periodic with a period n if the n-th iterate of f applied at z = z.

We concluded by seeing various pictures of the Mandelbrot set and fractals constructed through similar dynamical systems in the complex plane.

Universal Covers and Fundamental Domains

Hi, here are the notes from 7/28 that I've been digesting for a while. As usual, feel free to make edits for clarification/or to correct any errors.

We start by recalling the notion of a universal covering space (envisioned as the typical "stack of pancakes" form). We specifically consider the torus.

If we draw the curves that generate the torus (one that encircles the "hole" and one that passes through the hole), we note that these curves intersect at a single point. This feature is preserved upon lifting to the universal covering space.

As such, we can envision the universal covering space as an infinite grid (reflecting the 2 degrees of freedom - the x and y directions.)

In the covering space, we first fix a base point. We take the set of all closed loops which start and end at the base point. We consider a binary operation which consists of "loop concatenation", that is, we first navigate one loop and then the next to form our new compound loop.

We see that this operation on this set of closed loops from a given base-point forms a group.

Take We consider a deck transformation to be a homeomorphism from the universal cover to itself that preserves the covering map. We can think of the deck transformation as permuting the fibers within the "stack of pancakes."

Now consider the universal covering space of the 2-torus (which has two holes.)

If we draw two figure eights that intersect in the middle of the two holes of the 2-torus, we see that this intersection picture must lift above to the universal cover. If we look at the universal cover as a tiling of the Poincare disk, we see this means the disk is tiled by octagons that meet up 8 to a point.

We can use a simple continuity argument to establish one possible angle measure for these octagons. We know that for a small octagon in the center of the Poincare disk, the interior angle measures will be very nearly that of an octagon in the Euclidean plane, namely 135 degrees for a regular octagon. Near the edges of the Poincare disk, we know that the same regular octagon will have an interior angle measure near zero. Thus by continuity, there must be octagons that tile the Poincare disk which have an interior angle of 45 degrees for instance. We call each of these tiles a fundamental domain.

Now we switch to a brief discussion of measure. Suppose we have a group acting on a surface (which is locally like the plane or like a non-Euclidean plane)

We say a set on the surface has Lebesgue measure 0 if and only if for all positive epsilon, there exists a covering (possibly infinite) by round disks such that the total sum of the area of such disks is less than or equal to epsilon. Thus the total area of the disks is bounded by epsilon.

We also say that a set with nonzero Lebesgue measure has positive measure.

We say a set D of positive measure is called a measurable fundamental domain if the the group actions of elements in the fundamental group map to disjoint sets; that is, our set of measure 0 is the disjoint union of D with all possible images under group actions of the fundamental group. In our earlier example concerning the universal cover of the plain torus, the measurable fundamental domain would consist of the interior of a single square within the grid.

We then defined a geometric fundamental domain as a region with a "nice" boundary (one that consists of a finite number of smooth curves, and thus has measure 0) whose interior is a measurable fundamental domain.

We considered the Dirichlet fundamental domain (related to the concept of Voronoi regions.) Physically, these ideas are realized in soap films. The basic notion is that we scatter various points in a plane, and then divide the plane up into regions associated with the closest point. This partitions the plane into such regions.

We considered next rotations and taking powers of a rotation to form a group. We can then ask, does this group have a measurable fundamental domain. For rational rotations, we see clearly that fundamental domains exists- a rational rotation maps out "sectors" about its point of rotation.

For irrational rotations, the picture is more interesting. We see that the order of the group elements is infinite; that is, any power of an irrational rotation never returns to the identity rotation. As such, these sectors (domains) are not disjoint under rotation- any arbitrarily small sector is revisited by an irrational rotation.

This leads us to the Poincare Recurrence Theorem: For any infinite area-preserving action on a surface of finite area, there is no measurable fundamental domain.

Professor Sullivan then noted that generally, any arbitrary group action on a space can be broken up into a recurrence region, a fundamental domain part, and a set of measure 0.

Finally, we stated the Shottky Lemma: Suppose G1 and G2 are two group actions on a surface S with geometric fundamental domains F1 and F2. Suppose S = F1 u F2 (S is the union of F1 with F2.) Then the group generated by G1 and G2 consists of all of the words and it has fundamental domain equal to the intersection of F1 with F2. We denote this group by G1 * G2 and call it the free product of G1 and G2. (It is "free" because group elements do not interact outside of the individual groups.)

We concluded with some example of the lemma as well as a rough sketch of the proof:

Suppose we start at a point p. This point p must either be in F1 or F2. Suppose it is in F1. Then applying the group action means p cannot end up in F1 (since F1 is a fundamental domain.) So it must land outside F1. But since S is the union of F1 with F2, p must land in F2. Thus we see any point in the intersection of F1 with F2 is always mapped outside of itself by a group action, which means it is the fundamental domain of this free product.