Dynamical systems from physics are often
continuous time systems, whereas the mathematical models
often use discrete time. When does a discrete-time system embed in a
In the late 1960's, Ornstein constructed the first system with no square root, thus providing an example which not only did not embed in an R-action, it didn't even embed in a rational action.
In the 1950's, Halmos initiated a study of which dynamical properties
generic, in the topological sense. What I plan to do in this talk
is discuss some of the elementary ideas needed to show that the
generic map has a square root; indeed, roots of all orders. The
argument uses a nice topological lemma by Randall Dougherty and a
topological Zero-One Law due to Glasner and yours-truly.
The argument seems tantalizingly close to showing that generically a map extends to a Q-action. However, an example that I will discuss (due to a graduate student, Blair Madore) suggests that it may not be so easy to close the gap.
(30Oct1998: I gave 4 talks on this subject. Professor Vince will continue the topic starting on 04Nov1998, at 12:50PM.)
On Thursday, I
will be giving gave
a gentle introduction to Dissection Theory. No blood will be spilt.
Date: 20 Nov., 1997, at 7PM.
Title: "Dissection theory, Dehn's Theorem and Scissor Congruence"
Place: The Univ. of Florida Math Dept. Lounge, 3rd floor, room 358.
The entire talk will be pictures. I plan to show how to go from any polygon to any other (of equal area) by straight-line cuts. In contrast, Dehn -answering Hilbert's Third Problem- showed that for 3-dimensional polyhedra, the analogous statement is false. In modern language, this is a nice application of linear algebra.
I'll give the modern short proof of this, skipping a detail about the dihedral angle of a tetrahedron. See my Dehn's solution to Hilbert's third problem (pdf) for a complete languorous solution, in only 2+epsilon pages.
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