| Summary: | This thesis is concerned with situations where we can define trace-class transfer oper- ators, and extract useful information from their determinants. The first topic is on Lyapunov exponents of random products of matrices. We obtain a new expression for the Lyapunov exponent of a continuous family of matrices, and a slightly different version of existing work for the discrete case. The second topic explores possibilities of using similar theory to approximate eigen- functions of the Laplacian for surfaces of constant negative curvature. The third topic gives a variety of approximations of Mahler measures, which occur in many different areas of mathematics, by manipulating the integrals into a form that can be numerically integrated using work of Pollicott and Jenkinson. The final topic of the thesis works out the details of earlier ideas of Pollicott, to give a method for the numerical approximation of entropy rates of hidden Markov processes.
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