In modern science, since the spacetime of relativity, human intuition has assigned a special significance for the four dimensional space. Now, it is known that one of the most important discoveries in topology implies that a four dimensional space is more complicated than any lower dimensional one, and, surprisingly, even than higher dimensional spaces [I. Stewart, The problems of mathematics, Oxford Univ. Press (1987)]. The paper under review comes with remarkable interesting connections between chaotic dynamics and dimensionality. It is shown that, under a fairly reasonable assumption, one can conclude that four dimensions mark a special point in chaotic dynamics.
The author states the steps of his reasoning in a very clear and convincing way. The starting point of the analysis is the generally accepted observation that fractals [K.-H. Becker and M. Dörfler, Dynamical systems and fractals. Computer graphics experiments in Pascal, 2nd ed. (Vieweg 1988) (English translation by I. Stewart), Cambridge Univ. Press (1989), 4th ed. (Vieweg (1992)] are the carriers of complex strange behaviour. Secondly, Yorke’s conjecture that single Cantor sets are, in a way of speaking, the backbone of strange behaviour [S. Eubank and J. D. Farmer, Lect. Complex Syst., Santa Fe/1989, 75-190 (1990)] is invoked. In addition, the intuitive fact that in one dimension the simplest fractal set is Cantor’s middle third set [I. Stewart, Does God play dice? (1989; Zbl 0717.00001)] is considered, with the intention to find an equivalent of this set for the case of two dimensions. The result is then generalized in order to obtain an expression for the fractal dimension of such a Cantor-like type of set in an \(n\) dimensional space.
The conclusion is that in a four dimensional phase space, a Cantor-like set will typically have a fractal dimension \(d_ c\approx 4\). This should be a structure which comes very near to a space filling set. Therefore, it might be said that a Cantor-like set is almost ergodic only in a four dimensional space.