A Spectral Representation for the Entropy of Topological Dynamical Systems
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A Spectral Representation for the Entropy of Topological Dynamical Systems M. Rahimi1 Received: 15 November 2019 / Revised: 19 June 2020 / Accepted: 3 October 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we follow an approach which considers the entropy of a dynamical system as a linear operator. We assign a linear operator on Lp spaces using a kernel entropy function. The case p = 2 is of special interest, since we may relate the entropy of the system in terms of the eigenvalues of the operator. The special case p = 1 also results in a local entropy map. Keywords Entropy · Invariant · p-entropy kernel operator Mathematics Subject Classification (2010) 37A35
1 Introduction Traditionally, entropy is known as a non-negative extended real number assigned to a dynamical system and measures the rate of increase in dynamical complexity as the system evolves with time [1, 6, 15, 17, 20]. It may be defined locally on the points of the space which its integral equals the entropy of the system [2, 3, 7–9, 13, 16, 17]. Many generalizations of the concept of entropy are studied extensively [5, 14, 18, 19, 21]. Also, the concept of scaled entropy is discussed in [23, 24] in order to characterize the complexity of systems with zero entropy. In all of these approaches, the entropy is considered as a non-negative extended real number. Unlike the classical viewpoints, in [11, 12], the entropy is considered as a linear operator between Banach spaces which contains the Kolmogorov entropy in its nature. A similar approach may be applied to define an integral type operator between Lp spaces for continuous topological systems on compact metric spaces. To do this, we need to define a kernel function corresponding to such systems.
M. Rahimi
[email protected] 1
Department of Mathematics, Faculty of Science, University of Qom, Qom, Iran
M. Rahimi
In this paper, following this approach, for any compact topological dynamical system T : X → X, we introduce a kernel function p (p ≥ 1) using the metric of the space. Then, we define a linear operator p between Lp -spaces, namely, p-entropy kernel operator. The case p = 2 is of special interest. In this case, we have a Hilbert-Schmidt operator on a separable Hilbert space. Finally, we express the Kolmogorov entropy in terms of the eigenvalues of 2 . So, we will have a spectral representation of the entropy of compact topological systems of finite entropy. In Section 2, we recall some preliminary facts which will be used in the paper. In Section 3, we define the concept of p-entropy kernel operator and prove some of its properties. In Section 4, we consider the special case p = 2 and will express the Kolmogorov entropy in terms of the eigenvalues of the operator.
2 Preliminary Facts Let (X, d) be a compact metric space, BX the Borel σ -algebra of X, and μ a probability measure on BX . Let also T : (X, d) → (X, d) be a continuous map preserving μ. For n ∈ N, the metric dn is defined by dn (x, y) := max d(T j (x), T j (y)). 0≤j ≤n
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