Dynamical determinants and their applications

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Felton, Philip;
  • Subject: QA

This thesis is concerned with situations where we can define trace-class transfer oper-\ud ators, and extract useful information from their determinants.\ud The first topic is on Lyapunov exponents of random products of matrices. We obtain\ud a new expression for the Ly... View more
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    2 Background Material 10 2.1 Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Nuclear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Transfer Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

    3 Lyapunov Exponents of Random Matrix Products 28 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 A Family of Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Transfer Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.4 Dynamical Determinants . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.5 The Discrete Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

    4 Eigenfunctions of Laplacians on Surfaces of Constant Negative Curvature 60 4.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1.1 Hyperbolic Geometry . . . . . . . . . . . . . . . . . . . . . . . 61 4.1.2 The Bowen-Series Transformation . . . . . . . . . . . . . . . . 64 4.1.3 The Hyperbolic Laplacian Operator . . . . . . . . . . . . . . . 67 4.1.4 Transfer Operators . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.5 Computing Eigenvalues . . . . . . . . . . . . . . . . . . . . . . 72 4.2 Computing Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 78

    5 Approximations of Mahler Measures 83 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 De nition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2.1 Polynomials in One Variable . . . . . . . . . . . . . . . . . . . 85 5.2.2 Polynomials in Two Variables . . . . . . . . . . . . . . . . . . 87 5.3 Modifying the Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 88 [15] G. Han and B. Marcus. Derivatives of entropy rate in special families of hidden Markov chains. IEEE Trans. Inform. Theory, 53, no. 7:2642{2652, 2007.

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