Robust transitivity and ergodicity of transformations on the real line and the real plane.
Robust transitivity. Ergodicity. Expansivity. Alternating Systems.
In this work we describe the dynamics of a large class of transitive transformations on the real line and the real plane. We introduce the notion of Alternating Systems to understand problems like robust transitivity and ergodicity with respect to the Lebesgue measure. Then, we look for sufficient conditions to obtain robust expansivity, existence of ergodic absolutely continuous probabilities in the unidimensional case and physical measure in the bidimensional case (In both cases, with respect to induced transformations on bounded parts of the space). Important examples like the Boole transformation and the Henon-Devaney map can be treated using this approach.