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Iteration of piecewise linear maps on an interval

  • James B. McGuire (a1) and Colin J. Thompson (a2)

Abstract

A complete analysis is given of the iterative properties of two piece-piecewise linear maps on an interval, from the point of view of a doubling transformation obtained by functional composition and rescaling. We show how invariant measures may be constructed for such maps and that parameter values where this may be done form a dense set in a one-dimensional subset of parameter space.

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[1]Boyarsky, Abraham and Scarowsky, Manny, “On a class of transformations which have unique absolutely continuous invariant measures”, Trans. Amer. Math. Soc. 255 (1979), 243262.
[2]Collet, Pierre and Eckmann, Jean-Pierre, Iterated maps on the interval as dynamical systems (Progress in Physics, 1. Birkhauser, Boston, Basel, Stuttgart, 1980).
[3]Collet, P., Eckmann, J.-P. and Lanford, O.E. III, “Universal properties of maps on an interval”, Commun. Math. Phys. 76 (1980), 211254.
[4]Feigenbaum, Mitchell J., “Quantitative universality for a class of nonlinear transformations”, J. Statist. Phys. 19 (1978), 2552.
[5]Feigenbaum, Mitchell J., “The universal metric properties of nonlinear transformations”, J. Statist. Phys. 21 (1979), 669706.
[6]McGuire, James B. and Thompson, Colin J., “Distribution of iterates of first order difference equations”, Bull. Austral. Math. Soc. 22 (1980), 133143.
[7]McGuire, James B. and Thompson, Colin J., “Asymptotic properties of sequences of iterates of nonlinear transformations”, J. Statist. Phys. (to appear).
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