Is the dimension of chaotic attractors invariant under coordinate changes? (English) Zbl 0588.58042
Several different dimensionlike quantities, which have been suggested as being relevant to the study of chaotic attractors, are examined. In particular, we discuss whether these quantities are invariant under changes of variables that are differentiable except at a finite number of points. It is found that some are and some are not. It is suggested that the word ”dimension” be reversed only for those quantities have this invariance property.
MSC:
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 82B05 | Classical equilibrium statistical mechanics (general) |
References:
| [1] | B. Mandelbrot,Fractals: Form, Chance, and Dimension (Freeman, San Francisco, 1977) · Zbl 0376.28020 |
| [2] | B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982). · Zbl 0504.28001 |
| [3] | J. D. Farmer, E. Ott, and J. A. Yorke,Physica 7D:153(1983). |
| [4] | J. Balatoni and A. Renyi,Publ. Math. Inst. Hung. Acad. Sci. 1:9 (1956) [English translation inThe Selected Papers of A. Renyi, Vol. 1, (Akademia, Budapest, 1976), p. 588]. |
| [5] | H. G. E. Hentschel and I. Procaccia,Physica 8D:435 (1983). |
| [6] | P. Grassberger, Generalized dimensions of strange attractors, Wuppertal, preprint (1983). · Zbl 0593.58024 |
| [7] | S. M. Ulam,Problems in Modern Mathematics (John Wiley and Sons, New York, 1960), p. 74. · Zbl 0086.24101 |
| [8] | J. C. Alexander and J. A. Yorke, Dynamical Systems and Ergodic Theory, in press. |
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