Abstract
We describe the global dynamics of some pointwise periodic piecewise linear maps in the plane that exhibit interesting dynamic features. For each of these maps we find a first integral. For these integrals the set of values are discrete, thus quantized. Furthermore, the level sets are bounded sets whose interior is formed by a finite number of open tiles of certain regular or uniform tessellations. The action of the maps on each invariant set of tiles is described geometrically.
Highlights
A pointwise periodic map is a bijective self-map in a topological space such that each point is periodic
The level sets are bounded and their interior is formed by a finite number of some prescribed tiles of certain regular tessellations, see Figures 1, 2 and 3
If we identify each square with a point, the discrete dynamical system (DDS) restricted to this set is conjugated with the DDS generated by the map h : Z4c2 Ñ Z4c2, hpiq “ ic
Summary
A pointwise periodic map is a bijective self-map in a topological space such that each point is periodic. In this work we revisit the examples and results of Chang et al in the above cited references under the light of their properties as integrable systems For each of these three maps, we obtain a first integral which is defined in certain open and dense set of R2. The level sets are bounded and their interior is formed by a finite number of some prescribed tiles of certain regular tessellations, see Figures 1, 2 and 3 The existence of these quantized integrals with positive measure level sets is quite novel in the context of discrete dynamic systems theory. (c) As a consequence of the above results and the study of the dynamics on the boundary of the level sets, for each map, we characterize the period of every point in terms of the value of its associate first integral and obtain the global dynamics of the map.
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