Ramified chaotic attractors of smooth geometrically integrable self-maps of a cylinder

We give the definition of a geometrically integrable self-map of a cylinder and formulate criteria for the geometric integrability. Then we prove the geometric integrability of C^1 - smooth self-maps of a cylinder, close to C^1 - smooth skew products and satisfying some additional conditions. Finishing these considerations, we give the example of the family of geometrically integrable cylinder maps so that each map from this family admits the global chaotic attractor, which is a one-dimensional ramified continuum with a complicated topological structure. The global attractor of every map from the family under consideration consists of arcs of two types. On the unique arc of the first type the map is mixing; on arcs of the second type (the family of such arcs is countable) the map is not mixing (for details see [1]). REFERENCES [1] L.S. Efremova, Ramified continua as chaotic attractors of C 1 -smooth self-maps of a cylinder close to skew products, J. Difference Equ. Appl., Special issue "Lozi, He´non and other chaotic attractors. Theory and applications", 28 (2022) (to appear).