Invariant of dynamical systems: A generalized entropy

Abstract

In this work the concept of entropy of a dynamical system, as given by Kolmogorov, is generalized in the sense of Tsallis. It is shown that this entropy is an isomorphism invariant, being complete for Bernoulli schemes.

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

“…We will not discuss it since it is not a complete invariant for Kolmogorov automorphism. However, isomorphism of theKSq entropy for Bernoulli shifts will be able to discuss as same as Mesó n and Vericat [18].…”

“…it shows that the both are generalizationes of the KS entropy along the line of abstract dynamics although Renyi and Tsallis entropy are different in the sense of Kolmogorov-Nagumo average [26]. For the Bernoulli scheme, as it is a special case of the Markov scheme, h KSq (T ) does also not necessarily agree with the entropy h KSq (A, T ) by (22), even h KSq (T ) = H q (p) [18] for the Bernoulli scheme of arbitrary q. According to the revised definition (11) we still have h KSq (A, T ) = (1 − q) −1 log[ k−1 j=0 p q j ] to be Renyi entropy in Bernoulli scheme, so the recover (23) must also be available.…”

“…On the one hand, the nonextensive forms of the Tsallis entropy in some important branches of mathematics have been developed. A generalized entropy in the Tsallis sense for the dynamic systems has been presented by Mesón and Vericat first time [18]. The generalization does what Kolmogorov did for Shannon entropy under the Bernoulli scheme, and obtains a mathematical parallel which is proved as an isomorphism invariant [18,19].…”

“…A generalized entropy in the Tsallis sense for the dynamic systems has been presented by Mesón and Vericat first time [18]. The generalization does what Kolmogorov did for Shannon entropy under the Bernoulli scheme, and obtains a mathematical parallel which is proved as an isomorphism invariant [18,19]. On the another hand, the vague relation between the BG entropy of physics and the KS entropy of dynamics has been clarified gradually by works of many authors [20] as research of Tsallis entropy.…”

“…The Tsallis entropy recovers the Shannon entropy when q = 1. We now define a generalized entropy parallel to Ref [18]. for the finite partition partition A = {A 0 , A 1 , .…”