Abstract
Feng and Huang in 2016 defined a new notion called weighted topological entropy (pressure) and obtained the corresponding variational principle for compact dynamical systems. In this paper, it was our hope to carry out a further study from the following three aspects: (1) Inspired from the well-known classical entropy theory, we define various weighted topological (measure-theoretic) entropies and investigate their relationships. (2) The classical entropy formula of subsets and their transformations by factor maps is generalized to the weighted version. (3) A formula which comes from the Brin-Katok theorem of weighted conditional entropy is established.
Highlights
In this paper, we say that (X, d, T ) (or pair (X, T ) for short) is a topological dynamical system (TDS) if (X, d) is a compact metric space and T is a continuous map from X to X
Measure-theoretical entropy of a given measure preserving dynamical system was introduced in 1958 by Kolmogorov [15] and topological entropy of a given topological dynamical system was introduced in 1965 by Adler, Konheim and McAndrew [1]. The connection between these two notions is established through a variational principle which allows us to obtain the topological notion by taking supremum over all T -invariant Borel probability measures of the measure-theoretical entropy
This variational principle was proved by Goodwyn, Dinaburg and Goodman [12, 5, 11], and plays a fundamental role in ergodic theory and dynamical systems [17]
Summary
We say that (X, d, T ) (or pair (X, T ) for short) is a topological dynamical system (TDS) if (X, d) is a compact metric space and T is a continuous map from X to X. Measure-theoretical entropy of a given measure preserving dynamical system was introduced in 1958 by Kolmogorov [15] and topological entropy of a given topological dynamical system was introduced in 1965 by Adler, Konheim and McAndrew [1] The connection between these two notions is established through a variational principle which allows us to obtain the topological notion by taking supremum over all T -invariant Borel probability measures of the measure-theoretical entropy. This variational principle was proved by Goodwyn, Dinaburg and Goodman [12, 5, 11], and plays a fundamental role in ergodic theory and dynamical systems [17].
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