Reconstruction of a dynamical system is an important task in applications. This amounts to constructing a dynamical system that approximates the given time series data so that the error is minimal. Usually, time series data is a collection of points in a metric space, and the term "approximation" here refers to a quantitative one. We can naturally ask what happens if observation is in a more general setting, such as a partially ordered set. Can we still consider approximations in such a case and obtain a reconstruction scheme?
Here, we consider problems of this kind for dynamical systems taking values in topological concrete categories, which are categories with underlying set structure and have nice properties. Examples include topological or measure-theoretic dynamical systems.
Using this framework, we formulate the structure of time series data and data generation by dynamical systems. We show that a qualitatively exact reconstruction is possible if one can observe the exact state of the system for infinite time, as expected. This result justifies the use of the word "reconstruction" here. This is joint work with S. Das (Texas Tech).