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Summary: PERIODICITY OF CERTAIN PIECEWISE AFFINE PLANAR MAPS
SHIGEKI AKIYAMA, HORST BRUNOTTE, ATTILA PETHO, AND WOLFGANG STEINER
Abstract. We determine periodic and aperiodic points of certain piecewise affine maps in the
Euclidean plane. Using these maps, we prove for { ±1±
5
2
, ±
2, ±
3} that all integer
sequences (ak)kZ satisfying 0 ak-1 + ak + ak+1 < 1 are periodic.
1. introduction
In the past few decades, discontinuous piecewise affine maps have found considerable interest
in the theory of dynamical systems. For an overview, we refer the reader to [1, 7, 12, 13, 17,
18], for particular instances to [29, 16, 25] (polygonal dual billiards), [15] (polygonal exchange
transformations), [10, 31, 11, 8] (digital filters) and [19, 21, 22] (propagation of round-off errors in
linear systems). The present paper deals with a conjecture on the periodicity of a certain kind of
these maps:
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