| EPSRC Reference: |
EP/I019030/1 |
| Title: |
Non-Generic Teichmueller Dynamics |
| Principal Investigator: |
Ulcigrai, Dr C |
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| Department: |
Mathematics |
| Organisation: |
University of Bristol |
| Scheme: |
First Grant - Revised 2009 |
| Starts: |
01 May 2011 |
Ends: |
31 October 2013 |
Value (£): |
68,011
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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Imagine playing billiard in a table whose shape is not rectangular as usual, but polygonal, for example in an octagonal table. Imagine that the ball is a point that moves without friction, so that it keeps moving forever if it does not hit any pockets. This is a mathematical billiard. These idealized billiards are studied by mathematicians because they arise naturally as models of various systems in the physical sciences. Moreover, they represent one of simplest models of chaos. Billiard trajectories that never hit a pocket become quickly complicated and chaotic. One of the main goals in dynamical systems and ergodic theory is to describe and quantify this chaotic properties and to be able to predict thei asymptotic behaviour.One of the objectives of this research is to prove chaotic properties of a class of billiards in polygons. The goal of understanding billiards in polygons has led to the development of very deep mathematical theories, in particular to the study of an abstract space (the moduli space) of deformations of surfaces. The study of moduli spaces has independent interest and connections and applications to different areas of fundamental research in mathematics (as algebraic geometry and number theory) and in theoretical physics. The beautiful philosophy that has guided the research in the field is that properties of a single billiard are reflected by the properties of the dynamics in the moduli space. Unfortunately, some of the deep results obtained for these abstract spaces do not apply directly to billiards, which form a small set in this abstract space.In our proposal we plan to develop methods and tools to solve some of the open questions for polygonal billiards and other concrete systems which cannot currently be attacked by the standard tools. We have recently developed a renormalization technique which allows to attack these problems. We plan to get very precise information which is not usually accessible for generic systems. Other related objectives concern the properties of maps of the interval, called interval exchange transformations, which arise naturally in connection with billiards and have been studied as a basic model of dynamical system. We plan to investigate long-standing open problems about their spectral properties and a new generalization of classical Diophantine properties.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.bris.ac.uk |