The main focus of this dissertation was the Generalized Baker Map, a uniformly hyperbolic chaotic map for which the quantum analog is known. From the Gutzwiller Trace Formula, we know that the energy spectrum of a quantum system whose classical counterpart is chaotic is given, in the limit h → 0, by a sum over pairs of periodic orbits with small action differences. We sought out such pairs of orbits in the Generalized Baker Map. Using a symbolic dynamics exhibited by the Generalized Baker Map, we derived an expression for the classical action. We were then able to use the symbolic dynamics to write the mechanics of the Generalized Baker Map in terms of a collection of mixed generating functions. These generating functions led us to an expression for the classical action of periodic orbits which we could relate, relatively simply, to the ''geometry'' of the symbol sequences. Using the classical action we derived, we found the pairs of periodic orbits (i.e., pairs of symbol sequences) of the Generalized Baker Map which have small action differences between them. By graphically representing these symbol sequence pairs, we were able to match them with the current results for continuous systems. It is speculated that all maps allowing a symbolic dynamics will give the same results.
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