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

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.

### 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
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