January 14, 2014
Math 840, Spring, 2014, HW-1
Instructions The Desire2Learn site for this course is at
These exercises using Maxima are to be done, saved in a le called
Your_Last_Name_HW_1.mac and uploaded to the D2L site for the course
February 23, 2014
Math 840, Spring, 2014, HW-3
These exercises are to be done, typed up, and handed in.
The programming part should be saved in a le called
Your_Last_Name_HW_3.mac and sent to me by email.
For example, if your name is Steven Jones, you wou
February 17, 2014
Math 840, Spring, 2014, HW-2
1. These exercises are to be done, typed up, and uploaded to the HW
Dropbox at d2l.msu.edu.
2. The programming part should be saved in a le called
Your_Last_Name_HW_2.mac and uploaded to the HW D
Some concepts from topological dynamics
Let X be a metric space with metric d and let f : X X be a homeomorphism. The orbit o(x) of x is the set cfw_f n (x) : n Z. The forward orbit
o+ (x) is the set cfw_f n (x) : n Z+ , and the backward orbit o
Let f : S1 S1 be an orientation preserving homeomorphism of S1 . Let
: R S1 be the map (t) = exp(2it). There is a continuous map
F : R R such that
1. F = f
2. F is monotone increasing
3. F id is periodic, with period 1
Moreover, any two such m
April 26, 2012
Further Properties of Topological Entropy
Let f : X X be a continuous self-map of the compact metric space X .
Let x X and n be a positive integer.
An n-orbit O(x, n) is a nite sequence x, f (x), f 2 (x), . . . , f n1 x.
Let > 0. Two n
Consider an orientation preserving homeomorphism f : S1 S1 with irrational rotation number (f ). Let E be the unique minimal set for f . Note
that E equals the non-wandering set of f . Let R = R (f ) be the geometric
rotation through angle (f ).