Invariant sets for discontinuous parabolic area-preserving torus maps

Details Invariant sets for discontinuous parabolic area-preserving torus maps Invariant sets for discontinuous parabolic area-preserving torus maps

1999-08-17

arxiv.org/abs/chao-dyn/9908017v2

Nonlinearity 13 (2000) 819-835

Nonlinear Sciences

Chaotic Dynamics

19 pages in Latex (with epsfig,amssymb,graphics) with 5 figures in eps; revised version: section 2 rewritten, new example and

Scientific paper

10.1088/0951-7715/13/3/317

We analyze a class of piecewise linear parabolic maps on the torus, namely those obtained by considering a linear map with double eigenvalue one and taking modulo one in each component. We show that within this two parameter family of maps, the set of noninvertible maps is open and dense. For cases where the entries in the matrix are rational we show that the maximal invariant set has positive Lebesgue measure and we give bounds on the measure. For several examples we find expressions for the measure of the invariant set but we leave open the question as to whether there are parameters for which this measure is zero.

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