Abstract
Consider a discrete time dynamical systemx k+1=f(x k ) on a compact metric spaceM, wheref:M→M is a continuous map. Leth:M→B k be a continuous output function. Suppose that all of the positive orbits off are dense and that the system is observable. We prove that any output trajectory of the system determinesf andh andM up to a homeomorphism. IfM is a compact Abelian topological group andf is an ergodic translation, then any output trajectory determines the system up to a translation and a group isomorphism of the group.
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Dayawansa, W.P., Martin, C.F. & Cheng, D. A state space isomorphism theorem for minimal dynamical systems. Acta Mathematicae Applicatae Sinica 7, 34–37 (1991). https://doi.org/10.1007/BF02080201
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DOI: https://doi.org/10.1007/BF02080201