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The Construction of Ergodic Maps with Prescribed Invariant Density and Power Spectrum Poles
  • Andre McDonald ,
  • Anton van Wyk
Andre McDonald
University of the Witwatersrand

Corresponding Author:[email protected]

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Anton van Wyk
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Abstract

We propose an analytical method for constructing ergodic and piecewise linear hat maps with a piecewise constant invariant density selected to approximate a prescribed invariant density, and a rational power spectral density with an arbitrary number of prescribed nonrepeating poles. These semi-Markov maps permit a matrix representation of the Frobenius-Perron operator, referred to as the Frobenius-Perron matrix. We prove that the poles of a piecewise linear hat map’s power spectral density coincide with the unique nonzero and nonunity (in absolute value) eigenvalues of the map’s Frobenius-Perron matrix. A  hat map is constructed such that it possesses a Frobenius-Perron matrix with a selectable eigenspectrum, thereby allowing for the prescription of its poles. A topological conjugate of this map is subsequently constructed. The conjugate map’s poles are equal to those of the original map, but its piecewise constant invariant density is selected to approximate the prescribed invariant density. Similar to previous methods, the novel method does not afford the opportunity to prescribe the zeros of the map’s power spectral density. However, unlike these methods, all of the poles are prescribable and no numerical search is performed during construction. We characterize several piecewise linear hat maps constructed using the novel method.