A generalized Kolmogorov–Sinai-like entropy under Markov shifts in symbolic dynamics

In this paper, a generalized Kolmogorov–Sinai-like entropy (KSq entropy) in the sense of Tsallis is proposed with a nonextensive parameter q under Markov shifts, which contains the classical Kolmogorov–Sinai (KS) entropy and the Rényi entropy as well as Bernoulli shifts as special cases. To verify the formula of this KSq entropy, a one-dimensional iterative system is chosen as an example of Markov shifts, and its KSq entropy is evaluated by a new refinement method of symbolic dynamics called symbolic refinement which differs from the conventional numerical method. The numerical results show that this KSq entropy is monotonically decreasing as q increases.