An operator theoretical approach to the sequence entropy of dynamical systems

Abstract

In this paper, given a sequence of positive integers, we assign a linear operator on a Hilbert space, to any compact topological dynamical system of finite entropy. Then we represent the sequence entropy of the systems in terms of the eigenvalues of the linear operator. In this way, we present a spectral approach to the sequence entropy of the dynamical systems. This spectral representation to the sequence entropy of a system is given for systems with some additional condition called admissibility condition. We also prove that, there exist a large family of dynamical systems, satisfying the admissibility condition.

Keywords:

• Compact operator
• entropy
• sequence entropy

Acknowledgments

The authors would like to thank the referees for their comprehensive and useful comments which helped in the improvement of this work to the present form. We are also grateful to Professor Jose S Cánovas for of his useful comments and discussions.

Disclosure statement

No potential conflict of interest was reported by the author(s).