Quantization of Anosov Maps

Abstract

Cat maps, linear automorphisms of the torus, are standard examples of classically chaotic systems, but they are periodic when quantized, leading to many untypical consequences. Anosov maps are topologically equivalent to cat maps despite being nonlinear. Generalizing the original quantization of cat maps, we establish that quantum Anosov maps have typical spectral fluctuations for classically chaotic systems. The periodic orbit theory for the spectra of quantum Anosov maps is not exact, as it is for cat maps. Nonetheless our calculations verify that the semiclassical trace of the propagator is accurate well beyond any "log time" cutoff.