From Skew Products to Geometrically Integrable Maps in the Plane

The structure of the space of $C^1$-smooth skew products of interval maps is described [E1].
The concept of geometrically integrable self-maps of the compact plane sets is introduced. Criteria of the geometric integrability are proved and examples of these maps are considered [E2].
Comparison of geometrically integrable maps and skew products properties is given.
The problem of the coexistence of periodic points periods of geometrically integrable maps is solved [E2], [EM].
The concept of a generalized Lorenz self-map of a compact plane set is introduced. The set of generalized Lorenz maps contains the proper subset of "two-dimensional" Lorenz maps that arise in the standard geometric Lorenz model [ABSh].
Solution of the problem of the coexistence of periodic points periods of generalized Lorenz maps is presented [E2].
[E1] L.S. Efremova, Dynamics of skew products of interval maps, Russian Math. Surv., 72:1 (2017), 101–178.
[E2] L.S. Efremova, Geometrically integrable maps in the plane and their periodic orbits, Lobachevskii Journ. Math, 42:10 (2021), 2315–2324.
[EM] L.S. Efremova, E.N. Makhrova One-dimensional dynamical systems, Russ. Math. Surv., 76:5 (2021), 81–146.
[ABSh] V.S. Afraimovich, V.V. Bykov, L.P. Shilnikov, Attractive nonrough limit sets of Lorenz-attractor type, Trudy Moskovskogo Matematich. Obshchestva, 44 (1982), 150–212.