GAUSS: The Shadowing Property onn the Unit Interval

Our GAUSS speaker this week is first year math graduate Alex Berrian,
and the title of his talk is "The Shadowing Property on the Unit
Interval."  Check the bottom of this e-mail for his abstract.  The
main point is this: Alex is going to be talking about analysis
research he did as an undergrad in an REU (Research Experience for
Undergraduates) program, but he would also like to spend some time
talking about REU programs.  If you are an undergrad math major then
you should come to GAUSS today to learn about REU programs.  We would
like to get you excited about these excellent opportunities to
Experience real mathematics Research.  If you are one of the many math
grads who at one point was in an REU or involved in an REU, you are
welcome to come join in the conversation.  The final category of
people who should attend are people who like analysis.  Snacks will be
provided.

Check http://math.uiowa.edu/~rkilgore/GAUSS.html for any and all
GAUSS-related news.  If you have any questions, comments, suggestions
concerning GAUSS feel free to e-mail me or Trent.

Your GAUSS co-organizers,
Ross and Trent


Speaker: Alex Berrian
Title: The Shadowing Property on the Unit Interval
Date time and location: Tuesday, November 16, 4:30 PM, 118 MLH

Abstract:

Try this with your TI-83 or similar calculator: Punch in .9 and hit
ENTER. Hit the Cosine button, and then hit ENTER. You get cos(.9),
which is about .62. Hit ENTER again, and you get cos(cos(.9)), which
is about .81. Now hit ENTER a whole lot of times. Eventually you'll
get a number close to .74. We call this process an iteration of the
function cos(x).  Try doing this again, but start with .1 instead.
Guess what? After a while, you get the same number! Do this with any
number between 0 and 1 and you will arrive at the same result. Cool,
right? But wait - the calculator must truncate its answer each time in
order to do its calculations. How do you know that the calculator's
result for the nth iteration of cos(x) is close to the actual value?
One way of characterizing the accuracy of such an iteration is called
the shadowing property, which cos(x) happens to satisfy on the closed
unit interval [0,1], but which some functions do not. I'll explain
research I did during a summer NSF REU program with two other students
that gives a condition for certain functions to satisfy the shadowing
property on [0,1]. Undergraduates who want to know what REU programs
are like, as well as people who enjoy Analysis like me, are very much
encouraged to attend!