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Measures of chaos and a spectral decomposition of dynamical systems on the interval. (English) Zbl 0812.58062

Various notions of chaos of dynamical systems on the interval have been proposed by different authors. Among them there is the definition of chaotic functions given by Li and Yorke and another notion based on the topological entropy. In this paper the authors use the upper and lower distance distribution functions associated to a continuous function on the interval to measure the degree of chaos of the function.

Let $f$ be a continuous function on the interval. For $x,y\in \left[0,1\right]$, the upper and lower distribution functions, ${F}_{xy}^{*}$ and ${F}_{xy}$, are defined for any $t\ge 0$ as the lim sup and lim inf as $n\to \infty$ of the average number of times that the distance $|{f}^{i}\left(x\right)-{f}^{i}\left(y\right)|$ between the trajectories of $x$ and $y$ is less than $t$ during the first $n$ iterations. Both functions are nondecreasing and may be viewed as cumulative probability distribution functions.

Among the set of all lower distribution functions associated to a given $f$, the authors pick out a subset, ${\Sigma }\left(f\right)$, called the spectrum of $f$. The definition of the spectrum is based on initial results on dynamical systems concerning the notion of scrambled set and periodic decompositions. Thanks to a very simple example the authors are able to illustrate the ideas behind the use of this definition and to measure chaos.

The main results relate previous notions of chaos with the one proposed here: 1) If $f$ has positive topological entropy, then the spectrum is nonempty and finite, and any element of the spectrum is zero on an interval $\left[0,\epsilon \right]$ (thus relating the spectrum with the notion of chaos of Li and Yorke). 2) If $f$ has zero topological entropy, then the spectrum is just the constant unit function.

MSC:
 37D45 Strange attractors, chaotic dynamics 37A30 Ergodic theorems, spectral theory, Markov operators 54C70 Topological entropy 26A18 Iteration of functions of one real variable 37B99 Topological dynamics