Project Description: |
Given a function f:X -> X, and a point x in X, the orbit of x is the sequence {x, f(x), f(f(x)), . . . }. When a computer is used to generate an orbit, an actual orbit is never obtained due to round-off error. Instead, we get what is called a pseudo-orbit. Given a positive number, d, a d-pseudo-orbit is a sequence of points {x1, x2, x3, . . . } in X with the property that the distance from f(x1) to x2 is less than d, the distance from f(x2) to x3 is less than d, and so on. A d-pseudo-orbit then is the mathematical equivalent of a "sloppy orbit" generated by a computer.
This then raises the question, given a pseudo-orbit, is it close to an actual orbit? That is, given some small positive number e, is there a small enough number d so that the terms of the d-pseudo-orbit all be within e of the terms of an orbit. If so, then we say that the pseudo-orbit is e-shadowed by the actual orbit. If every d-pseudo-orbit can be e-shadowed, then we say that f has the shadowing property.
This is an area of dynamical systems that continues to generate interest and papers. I have two published papers - researched and written with undergraduates - that provide results for dynamical systems based on the unit interval. One of them gives necessary and sufficient conditions for an increasing function to have the shadowing property. Left to consider is finding conditions for which arbitrary continuous functions on the unit interval have the shadowing property. Another possible question for a given dynamical system is to ask what is the probability that a given pseudo-orbit
is e-shadowed.
This summer, I plan to continue researching which functions - on the unit interval or otherwise - have the shadowing property. Students will read various papers on the subject, and then we will decide what particular question we want to pursue.
This research will relay heavily on concepts such as metric spaces, uniform continuity, and equicontinuity. Students should ideally have a background of a year-long course in undergraduate analysis. A background in probability and some computer programming may also be helpful. |
