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Why computers like Lebesgue measure. (English) Zbl 0668.28008

This paper deals with computer chaos versus true trajectory chaos. It has been observed in practice that the histograms of computer simulations seem to display the invariant measure that is absolutely continuous with respect to the Lebesgue measure; and an explanation of this phenomenon is herein proposed. After emphasis on the deficiencies of the random perturbation models, the relation between computer orbits and absolutely continuous invariant measures is exhibited, and then it is shown that a large class of piecewise linear transformations have long periodic trajectory.
Reviewer: G.Jumarie

MSC:

28D20 Entropy and other invariants
37D99 Dynamical systems with hyperbolic behavior
58K35 Catastrophe theory
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[1] Campbell, D.; Crutchfield, J.; Farmer, D.; Jen, E., Experimental mathematics: the role of computation in nonlinear science, Communs Ass. comput. Mach., 28, 374-384 (1985)
[2] Palmore, J. I.; McCauley, J. L., Shadowing by computable chaotic orbits, Phys. Lett., A122, 399-402 (1987)
[3] Benettin, G.; Casartelli, M.; Galgani, L.; Giorgilli, A.; Strelycyn, J. M., II: identification of time averages, Nuovo Cim., 50B, 211-232 (1979)
[4] Coven, E. M.; Kan, I.; Yorke, J. A., Pseudo-orbit shadowing in the family of tent maps (1987), Preprint
[5] Proppe, H.; Byers, W.; Boyarsky, A., Singularity of topological conjugacies between certain unimodal maps of the interval, Israel J. Math., 44, 277-288 (1983)
[6] Góra, P.; Byers, W.; Boyarsky, A., Periodic orbit measures for piecewise expanding transformations (1988), Preprint
[7] Ruelle, D., Differentiable dynamical systems and the problem of turbulence, (Proc. Am. math. Soc., 5 (1981)), 29-42 · Zbl 0474.76052
[8] Boyarsky, A., On the significance of absolutely continuous invariant measures, Physica, 11D, 130-146 (1984) · Zbl 0578.60042
[9] Boyarsky, A., Randomness implies order, J. math. Analysis Applic., 76, 483-497 (1980) · Zbl 0442.60004
[10] Katok, A.; Kifer, Y., Random perturbations of transformations of an interval, J. Analyse Math., 47, 193-237 (1986) · Zbl 0616.60064
[11] Scarowsky, M.; Boyarsky, A., Long periodic orbits of the triangle map, (Proc. Am. math Soc., 97 (1986)), 247-254 · Zbl 0614.28011
[12] Boyarsky, A., Computer orbits, Comput. Math. Applic., 12A, 1057-1064 (1986) · Zbl 0637.58011
[13] Friedman, N.; Boyarsky, A.; Scarowsky, M., Ergodic properties of computer orbits for simple monotonic transformations (1987), Preprint · Zbl 0652.65033
[14] Blank, M. L., Ergodic properties of discretizations of dynamical systems, Soviet Math. Dokl., 30, 449-452 (1984) · Zbl 0589.58012
[15] Nitecki, Z., Differentiable Dynamics (1971), MIT Press: MIT Press Cambridge, Mass
[16] Beck, C.; Roepstroff, G., Effects of phase space discretization on the long-time behavior of dynamical systems, Physica, 25D, 173-180 (1987) · Zbl 0617.65066
[17] Doob, J., Stochastic processes (1953), Wiley: Wiley New York
[18] Lasota, A.; Yorke, J. A., On the existence of invariant measures for piecewise monotonic transformations, Trans. Am. math Soc., 186, 481-488 (1973) · Zbl 0298.28015
[19] Misiurewicz, M., Absolutely continuous measures for certain maps of an interval, Publ. Math. IHES, 53, 17-51 (1981) · Zbl 0477.58020
[20] Knuth, D., (The Art of Computer Programming, Vol. II (1969), Addison-Wesley: Addison-Wesley Reading, Mass)
[21] P. Góra, A. Boyarsky and H. Proppe, Constructive approximations to the absolutely continuous measures invariant under non-expanding transformations. J. statist. Phys.; P. Góra, A. Boyarsky and H. Proppe, Constructive approximations to the absolutely continuous measures invariant under non-expanding transformations. J. statist. Phys.
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