Pointwise periodic maps with quantized first integrals

We describe the global dynamics of some pointwise periodic piecewise linear maps in plane that exhibit interesting dynamic features. For each these we find a first integral. integrals set values are discrete, thus quantized. Furthermore, level sets bounded whose interior is formed by finite number open tiles certain regular or uniform tessellations. The action on invariant described geometrically. • three integrable plane. explicit quantized integrals. necklaces tilings. prove existence an and dense where defined: set. completely terms complementary simple geometrical terms.