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Output digitization of simple measure‑preserving linear systems
A.DeStefano,M.Thitsa,C.Martin
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(1 Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, USA;2 Department of Electrical and Computer Engineering, Mercer University, Macon, GA 31207, USA;3 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79430, USA)
摘要:
We examine three simple linear systems from the viewpoint of ergodic theory. We digitize the output and record only the sign of the output at integer times. We show that even with this minimal output we can recover important information about the systems. In particular, for a two-dimensional system viewed as a flow on the circle, we can determine the rate of rotation. We then use these results to determine the slope of the trajectories for constant irrational flow on the two-dimensional torus. To achieve this, we randomize the system by partitioning the state space and only recording which partition the state is in at each integer time. We show directly that these systems have entropy zero. Finally, we examine two four-dimensional systems and reduce them to the study of linear flows on the two-dimensional torus.
关键词:  Ergodic · Linear system · Observability · Cantor set
DOI:https://doi.org/10.1007/s11768-021-00060-0
基金项目:
Output digitization of simple measure‑preserving linear systems
A. DeStefano,M. Thitsa,C. Martin
(1 Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610, USA;2 Department of Electrical and Computer Engineering, Mercer University, Macon, GA 31207, USA;3 Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79430, USA)
Abstract:
We examine three simple linear systems from the viewpoint of ergodic theory. We digitize the output and record only the sign of the output at integer times. We show that even with this minimal output we can recover important information about the systems. In particular, for a two-dimensional system viewed as a flow on the circle, we can determine the rate of rotation. We then use these results to determine the slope of the trajectories for constant irrational flow on the two-dimensional torus. To achieve this, we randomize the system by partitioning the state space and only recording which partition the state is in at each integer time. We show directly that these systems have entropy zero. Finally, we examine two four-dimensional systems and reduce them to the study of linear flows on the two-dimensional torus.
Key words:  Ergodic · Linear system · Observability · Cantor set