Entropy of interval maps via permutations. (English) Zbl 1026.37027

Bandt, Christoph; Keller, Gerhard; Pompe, Bernd

Nonlinearity 15, No. 5, 1595-1602 (2002).

Let \(f:I\rightarrow I\), \(I=[0,1]\), be a piecewise monotone map and let \(P\) be the finite partition of \(I\) into finitely many intervals on which the map \(f\) is strictly monotone and define for any \(n\geq 0\) the partition \(P_n=\vee _{i=0}^{n-1}f^{-i}(p_j)\), \(p_j\in P\). Given the numbers \(h_0(n)=\log(\#P_n)/n\) and \(h_{\mu }(n)=(-\sum _{p\in P_n}\mu (p)\log \mu (p))/n\), where \(\mu \) is a probability invariant measure of \(f\), the topological and metric entropies of \(f\) are given by the limits \(h_0=\lim_{n\rightarrow \infty }h_0(n)\) and \(h_{\mu }=\lim_{n\rightarrow \infty }h_{\mu }(n)\), respectively.
In this nice paper, the authors give a definition of topological and metric permutation entropies as follows. Let \(\pi =(k_1,\dots,k_n)\) be a permutation of \(0,1,\dots,n-1\) and writing \(x_0=x\) and \(x_k=f^k(x)\) for \(k>0\), let \(P_{\pi }=\{ x\in I:x_{k_1}<x_{k_2}<\dots<x_{k_n} \}\) and let \(P_n^*\) be the family of all the nonempty sets \(P_{\pi }\) where \(\pi \) ranges the set of permutations of \(0,1,\dots,n-1\). Then \(h_0^*(n)=\log(\#P_n^*)/(n-1)\) and \(h_{\mu }^*(n)=(-\sum _{p\in P_n^*}\mu (p)\log \mu (p))/(n-1)\).
The main result of this paper states that \(h_0=\lim_{n\rightarrow \infty }h_0^*(n)\) and \(h_{\mu }=\lim_{n\rightarrow \infty }h_{\mu }^*(n)\), respectively. Numerical examples with the family of tent maps are made for comparing permutation entropies with their limits.

Reviewer: Jose S.Cánovas (Cartagena)

MSC:

37E05	Dynamical systems involving maps of the interval
37A35	Entropy and other invariants, isomorphism, classification in ergodic theory
54C70	Entropy in general topology

Keywords:

entropy; piecewise monotone maps; permutation entropy