Department of Mathematics
Title : Generalised inverse limits of tent maps
| Speaker: Michael Lockyer Affiliation: University of Auckland Time: 10 am Friday, 16 March, 2012 Location: 303S-561 |
Abstract
| A surjective tent map is a continuous function $f:[0,1]\rightarrow [0,1]$ such that: $$f(x)= \begin{cases} 2x& x\leq \frac{1}{2}\\ 2-2x& x>\frac{1}{2}\\ \end{cases} $$ It is so called because the resulting graph resembles a `tent' with a `peak' at the point $\frac{1}{2}$. It is well known that the inverse limit of this tent map is the famous \emph{Buckethandle Continuum} (or Knaster continuum), an example of an indecomposable continuum. If we alter the function slightly such that its graph retains the tent shape, but has its peak at $a\in (0,1)$, the resulting inverse limit is homeomorphic to the Buckethandle, so is (amongst other things) indecomposable. Using the new techniques of generalised inverse limits, where the bonding maps are upper semicontinuous set valued functions, it is possible to obtain inverse limits of set valued tent maps with peaks at the extreme points of 0 and 1. The resulting inverse limits are very different from the Buckethandle. One of these is homeomorphic to a harmonic fan. The other has been informally called the `monster', and little has previously been known about it. In this talk I will show a construction of the monster, and describe some of its interesting properties. |
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Programmes and Centres
- New Zealand Institute of Mathematics and its Applications (NZIMA)
- Community for Understanding and Learning in the Mathematical Sciences (CULMS)
- Centre for Mathematical Social Science (CMSS)
- Department of Computer Science
- Department of Engineering Science
- Department of Physics
- Department of Statistics
- Auckland Bioengineering Institute
