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Synchrotron radiation produced by high energy storage rings surpasses all natural Mossbauer sources in spectral brightness within a narrow Mossbauer slice ((GAMMA) (TURN) 10('-8) ev) in the 1(ANGSTROM) wavelength region by 2-4 orders of magnitude. Several methods have been suggested to filter out this intense beam of resonant photons, such as nuclear Bragg reflection and time filtering techniques. This thesis investigates a new interference technique for Mossbauer filtering of synchrotron radiation, the impedance matched grazing incidence films introduced by Hannon et al. We extend their initial study to account for photoabsorption and show new possibilities for the suppression of electronic reflection. In particular the impedance matched condition can still be satisfied even though the reflection amplitudes are now complex and the new damping stabilized solutions previously inaccessible in the zero absorption limit were investigated and compared to impedance-matched solutions. Furthermore we consider coating a resonant Fe('59) film on a nonresonant substrate for the impedance matched and damping stabilized cases as well as alternate types of interference filters such as the half wave films and the ultra thin films. Mossbauer isotopes with strong resonant scattering power in addition to FE('57), namely Sn('119) and Dy('161) are explored as very promising materials in the interference filter design. Potential applications of the intense filtered resonant radiation are also discussed. To test the plausibility of the interference filters in the event that the surfaces are less than ideally smooth, a semiquantitative treatment of the effects of surface irregularities on the reflection properties of real surfaces is presented.

The evolution of phase space densities under the action of nonlinear dynamical systems is studied. The identification of macroscopic properties of these systems is made through their corresponding time dependent phase space density state, in close analogy to statistical mechanics. We focus attention on a property that may arise in either deterministic or stochastically perturbed one dimensional maps, known as asymptotic periodicity (AP). Asymptotically periodic systems exhibit an eventual periodicity in the evolution of their phase space densities. The periodic density cycle that emerges is highly sensitive to the initial density of preparation of the system. Three one dimensional systems possessing AP are studied; the hat map, the quadratic map, and noise perturbed Keener maps. Certain physical properties of these maps are derived highlighting their underlying AP. The idea of using a type of entropy, the conditional entropy, as a measure of localization of a cycle of density states of AP systems is discussed.