Discrete-Time Transitivity and Accessibility: Analytic Systems

A basic open question for discrete-time nonlinear systems is that of determining when, in analogy with the classical continuous-time “positive form of Chow’s Lemma,” accessibility follows from transitivity of a natural group action. This paper studies the problem and establishes the desired implication for analytic systems in several cases: (i) compact state space, (ii) under a Poisson stability condition, and (iii) in a generic sense. In addition, the paper studies accessibility properties of the “controllability sets” recently introduced in the context of dynamical systems studies. Finally, various examples and counterexamples are provided relating the various Lie algebras introduced in past work.