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Lap number entropy formula for piecewise affine and projective maps in several dimensions. (English) Zbl 1142.37015

Nonlinearity 20, No. 12, 2755-2772 (2007); erratum 21, No. 6, 1411 (2008).
The authors extend the lap number formula for calculating the topological entropy of a piecewise monotone interval or circle map from M. Misiurewicz and W. Szlenk [Stud. Math. 67, 45–63 (1980; Zbl 0445.54007)]. They give analogous formulae for calculating the topological entropy of special classes of maps in higher dimensions. Namely, the topological entropy of so-called piecewise projective homeomorphisms of the \(2\)-sphere is shown to be given by the growth rate of the number of projective pieces under iteration. In higher dimensions and for non-invertible maps the authors obtain only an upper bound for the entropy. An analogous lap number formula is proved for the topological entropy of piecewise affine homeomorphisms of the plane, such as Lozi maps. Again, in higher dimensions and for non-invertible maps the authors obtain only an upper bound for the entropy; they also provide counterexamples showing that the inequality is not an equality in general.

MSC:

37B40 Topological entropy
37B10 Symbolic dynamics

Citations:

Zbl 0445.54007
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