| EPSRC Reference: |
GR/R64650/01 |
| Title: |
Approximating spectral data of ruelle transfer operations |
| Principal Investigator: |
Jenkinson, Professor O |
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| Department: |
Sch of Mathematical Sciences |
| Organisation: |
Queen Mary University of London |
| Scheme: |
Fast Stream |
| Starts: |
15 April 2002 |
Ends: |
14 May 2003 |
Value (£): |
61,744
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
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A central object of Ruelle's thermodynamic formalism, which plays an important role in the ergodic theory of expanding real-analytic dynamical systems, is the transfer operator L. Its spectral properties yield important dynamical and geometrical invariants, such as entropy, Lyapunov exponents, invariant measures, and Hausdorff dimension. The actual calculation of spectral data of L, however, poses considerable difficulties. The proposed research will focus on a promising method for estimating eigendata of L, known as the finite section method. One of our aims is to establish rigorous error bounds for the approximation of both eigenvalues and eigenvectors. We intend to achieve this end by exploiting the positivity of the transfer operator, together with the fact that it is trace-class. The finite section method will be used to compute dynamical and geometric invariants such as Lyapunov exponents and Hausdorff dimension. A particular application will be to the rigorous approximation of the absolutely continuous invariant measure for the Jacobi-Perron algorithm (the most natural multi-dimensional continued fraction algorithm). This will be used to prove the almost everywhere exponential convergence conjecture for this algorithm. The Hausdorff dimension calculations will centre on limit sets of infinite branch conformal iterated function systems, such as the one associated to the Apollonian packing. Abstract spectral properties of transfer operators will also be investigated. When acting on appropriate ces of analytic functions, we aim to establish sufficient conditions for the spectrum to be real, positive, or simple.An application of this will be to the linearised Feigenbaum renormalisation operator, whose spectrum is conjectured to be real.
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Potential use in non-academic contexts |
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| Description |
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Sectors submitted by the Researcher |
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