The paper aims to generalize, towards the framework of general topological dynamical systems, some ideas about complexity which are known for the dynamical system \((I,f)\) with \(I=\{x\in\mathbb R:0\leq x\leq 1\}\) and \(f:I\to I\) continuous. In \((I,f)\), the growth rate of the total variation of the iterations of \(f\) is an indicator of the complexity of \(f\). If the total variation of \(f^n\) (the \(n\)th iteration of \(f\)) grows exponentially with \(n\), then \(f\) is said to have rapid fluctuation on \(I\). Rapid fluctuation of \(f\) on \(I\) implies that \((I,f)\) has positive entropy, and the converse is also true provided that \(f\) is piecewise monotonic.
For a general topological dynamical system \((X,f)\) (with \(X\) a complete metric space and \(f:X\to X\) continuous), the authors give first a suitable definition of rapid fluctuation of \(f\) in such a case. This is done (imposing the requirement that \(f\) is a Lipschitz function) by considering Hausdorff measures of subsets of \(X\) (as an analogue of interval lengths to compute the variation of real functions). Afterwards, it is shown in the paper that if \((X,f)\) has a subset which factors onto a one-sided full shift or if \((X,f)\) has a topological horseshoe [J. Kennedy and J. A. Yorke, Trans. Am. Math. Soc. 353, No.6, 2513–2530 (2001; Zbl 0972.37011)], then \(f\) has a rapid fluctuation; this reinforces the idea that rapid fluctuation is a good indicator of complexity, since it is well known that, in both cases, the dynamical system has a positive entropy.
Finally, a predator-prey dynamical system \((\mathbb R^+\times\mathbb R^+,f)\) is studied, and it is shown that it has a rapid fluctuation by proving that \(f^2\) has a topological horseshoe.