The Degenerate Node

The relevant text sections for this lecture are Section 2.1.3, 2.1.4, and 2.4.1.  Please read these prior to meeting on Thursday. Last lecture, we saw that understanding discrete dynamical systems will require an understanding of orbits and how they behave.  As discrete orbits are sequences in a space, convergence properties will be important.  One aspect of measuring convergence is via a metric, and metric spaces will play a vital role in this entire course.  In this lecture, we define a metric and discuss some of its properties.  We also introduce a notion of continuity, Lipschitz continuity, that will facilitate our study.  With that, we can move into our first major topic and theorem of the course:  The Contraction Principle.  While a straightforward concept, there are aspects of it that are a bit tricky, and we will take care to expose some of these thoughtful ideas as we discuss what is a contraction and what information does it convey.  Here are the lecture notes. 

Hey all,  So a couple of years back, I set up this blog to maybe create a forum for me and you guys to interact and discuss the current material from the course.  It was basically a means to attempt to go beyond the standard lecture/office hour/homework model.  Perhaps as you learn and explore, you will find things out on the internet that are relevant to the course.  Sharing them here would expose them to the entire class.  Also, questions about material is always best discussed as a group.  In any case, I will create a thread on each lecture, and also point to thing I find that are outside the text but interesting to think about or see.  You, in turn, are encouraged to participate, by posting comments, finding other examples of some of the topics of this course and sharing, or generally just getting involved.      I sincerely hope that you are already finding the material in, or the perspective of, this course interesting.  I can assure you that eventually, it will being power an

One fun application of phase volume preservation in a dynamical system is billiards.  The polygonal table introduced recently when developing toral flows is an example of a general class of dynamical systems called convex billiards.  Today, we will describe the system and talk about its properties.  One interesting aspect of a convex billiard (essentially the table is a convex domain in the plane) is that it always has periodic orbits of all orders.  This may not be obvious, but we will define the tools that expose why.  And in one particular case, that of an elliptic billiard,  there is a curious side effect of the table;  given a particular placement of a pocket and ball placement, one can aim anywhere and cannot miss making the shot.  Makes for good betting odds, eh? 

Now we will leave circle and toral maps for a bit and change our focus a bit.  We will stay on this notion of recurrence, however, but in a different fashion.  Instead of simply looking for a particular orbit to exhibit certain behavior, let's look at families of orbits simultaneously.  In particular, for a flow or a map, let's look at the orbit of a small open set in state (phase space), as in consider all points in a small open set to be the initial points and follow all of their orbits at once.  Today we will introduce the notion of incompressability of a flow, which is also called phase volume preservation.  All of our toral translations and flows to date preserved volume, as do all isonetries, but contractions do not.  For a flow, this is related to divergence of the vector field of the flow, and we show that Newton;s Second Law of Motion provides a rich set of examples, along with exact differential equations.  Then we show that the ramifications of volume preservatio

Today, we continue with our characterization and classification of toral flows, but move into higher dimensions.  The results are pretty much the same, and again, rely on rigid rotations of each circle in the torus to characterize the flow.  But there is a twist (sorry for the pun) when we look at the time-1 map, an example of what we call a toral translation, or multi-dimensional rotation.  We will detail this today. Next, we will go back to simple circle maps, but move into a more general area of circle homeomorphisms rather than just rigid rotations.  It turns out that even here, there is a sort of rigidity (in that not much can happen) that is somewhat surprising.  For example, is a circle map has a n-periodic point, then it can only have possibly other points of period n, and no other period is possible other than n.  We will talk about why today.  The link is here .  See you today.

Today we move on from maps and flows in the plane and into maps and flows on the circle.  Here again, we start with a relatively simply construction to help set the stage for more complicated stuff later on;  Rigid circle rotations.  Really, they come in only two types, depending on whether the rotation is rational or not (but even this takes a bit of definition.)  But classifying circle rotations allows us to define what it means for an orbit to be dense in a space, and to define what it means for a point in the space to be recurrent with respect to a map.  This leads to an important theorem whose statement actually defines a property;  The Weyl Equidistribution Theorem.  There are two applications here that are quite interesting.  However, due to the timing of this tour, we will not cover them.  Instead, we use circle rotations to begin the discussion of circle flows and toral maps and flows, where things get a little more subtle and complicated.