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| EPSRC Reference: |
GR/S75581/01 |
| Title: |
Algebraic invariants & the structure of pseudo-Anosov foliations |
| Principal Investigator: |
Dr T Hall |
| Other Investigators: |
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| Researcher Co-investigator: |
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| Project Partner: |
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| Department: |
Mathematical Sciences |
| Organisation: |
University of Liverpool |
| Scheme: |
Standard Research |
| Starts: |
01 February 2004 |
Ends: |
31 January 2007 |
Value (£): |
158,537
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| EPSRC Research Topic Classifications: |
| Algebra and Geometry |
Mathematical Analysis |
| Non-linear Systems Mathematics |
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| EPSRC Industrial Sector Classifications: |
| No relevance to Underpinning Sectors |
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| Related Grants: |
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| Panel History: |
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Summary |
Pseudo-Anosov maps, introduced by Thurston, play a central role in the study of nonlinear dynamical systems defined on surfaces. A pseudo-Anosov preserves a pair of foliations of the underlying surface, and also has certain algebraic invariants associated to it. The aim of this proposal is to study the interplay between the structure of the foliations, the dynamics, and the algebraic invariants. Several mathematicians have worked on this important and open-ended problem in the last 20 years, but a new impetus has been provided by the powerful techniques recently developed by Band.
The proposed research comes under the "Nonlinearity" theme of the EPSRC's current research priorities relating to the Mathematical Foundation.
The total cost of this research would be 155910, the majority of which would be spent on employing a post-doctoral research associate to work full time on the problem for 3 years.
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| Final Report Summary |
"Chaotic" dynamical systems have been the subject of intensive study, both mathematical and experimental, in recent years. The research carried out in this project centred on the study of such chaotic systems arising on surfaces: two-dimensional objects such as the surface of a sphere, of a doughnut (or torus), or of more complicated bodies (like a torus, but with more than one hole). The main aim is the study of the interplay between various objects associated with such systems.
* The first such objects are "covering spaces". A surface can have a "cover", a new surface which can be thought of as winding round the original one a certain number of times. For instance, one could take a torus, cut through it, make a second copy, and glue the two cut tori together on their cut ends. This would create a new torus, twice as big as the original, which is a "double cover" of it. Performing this with more copies of the original torus yields a triple cover, a quadruple cover, and so on. Continuing this process indefinitely yields an infinite cover, in which infinitely many cut tori are glued together to form an infinite cylinder. This cylinder can then itself be cut along its length to form new covers, culminating in the plane, which is obtained by gluing together infinitely many copies of the cut cylinder. This is called the universal cover of the torus, since no further coverings are possible.
Chaotic systems defined on a surface may "lift" to give new chaotic systems defined on the covers of the surface.
* The second type of object studied is pseudo-Anosov maps, introduced by Thurston. These are chaotic systems which have the important property of "dynamical minimality": that is, perturbing them gradually can only make their behaviour more complicated, never simpler. Because of this property they play a central role in the study of chaotic systems on surfaces.
* Pseudo-Anosov maps have "invariant foliations": that is, collections of (one-dimensional) lines filling the whole surface, with the property that each such line is sent by the system to another such line, having been stretched or contracted by a fixed amount. These foliations are a key tool in understanding the behaviour of pseudo-Anosovs.
* Pseudo-Anosov maps also have "Markov partitions": collections of rectangles drawn in the surface which are sent over each other in a particularly simple way by the system. Markov partitions give a very detailed understanding of the behaviour of the pseudo-Anosov, but they are hard to find in general.
* Finally, there are certain "algebraic invariants" associated with pseudo-Anosov maps. These are much easier to determine, but often give less information about the behaviour of the pseudo-Anosov.
Almost all of the results obtained during this research concern the interplay between these various objects, and what they tell us about the behaviour of the original chaotic system.
One main result which is relatively easy to explain concerns the "topological entropy" of a chaotic system, which is a measure of just how chaotic it is. There is an established technique for getting a "lower bound" on this quantity (that is, a value which the topological entropy is certainly at least as big as): this technique uses algebraic invariants of a lift of an appropriate pseudo-Anosov to a certain cover. The new result gives the precise conditions (in terms of the invariant foliations) under which the lower bound is actually equal to the topological entropy, and also provides an abbreviated way of calculating it.
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| Further Information: |
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| Organisation Website: |
http://www.liv.ac.uk |
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