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Dimension and structure of typical compact sets, continua and curves. (English) Zbl 0666.28005
In the sense of Baire categories most compact subsets of a complete metric space \(<X,\rho >\) have Hausdorff and lower entropy dimension 0. If the compact sets having lower entropy dimension \(\geq \delta (>0)\) are dense then most compact sets have upper entropy dimension \(\geq \delta\). Most compact sets C in X have the following property: For any \(x\in C\) and \(0<\epsilon \leq 1\) there are arbitrarily small \(\sigma >0\) such that the “annulus” \(\{y\in X:\quad \epsilon \sigma \leq \rho (x,y)\leq \sigma \}\) is disjoint from C. Thus most compact sets have porosity 1 at any of their points. Similar results on “thinness” in various senses hold for continua, curves and graphs of real continuous functions on [0,1]. [See also A. J. Ostaszewski, Mathematika 21, 116-127 (1974; Zbl 0305.54040), J. A. Wieacker, Math. Ann. 282, 637-644 (1988; Zbl 0636.52004), and the survey of T. Zamfirescu, Rend. Semin. Mat., Torino 43, 67-88 (1985; Zbl 0601.52004).]
Reviewer: P.M.Gruber

28A78 Hausdorff and packing measures
26A21 Classification of real functions; Baire classification of sets and functions
26B15 Integration of real functions of several variables: length, area, volume
28D20 Entropy and other invariants
54C50 Topology of special sets defined by functions
54F45 Dimension theory in general topology
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[1] Besicovitch, A. S., Ursell, H. D.: Sets of fractional dimensions (IV); On dimensional numbers of some continuous curves. J. London Math. Soc.12, 18-25 (1937). · Zbl 0016.01703
[2] Dolzhenko, E. P.: Boundary properties of arbitrary functions. Math. USSR Izv.1, 1-12 (1967). · Zbl 0181.08102
[3] Falconer, K. J.: The Geometry of Fractal Sets. Cambridge: Univ. Press. 1985. · Zbl 0587.28004
[4] Golab, S.: Sur quelques points de la théorie de la longueur. Ann. Soc. Polon. Math.7, 227-241 (1929). · JFM 55.0153.01
[5] Gruber, P. M.: In most cases approximation is irregular. Rend. Sem. Mat. Univ. Politecn. Torino41, 19-33 (1983). · Zbl 0562.41030
[6] Gruber, P. M.: Results of Baire category type in convexity. Ann. New York Acad. Sci.440, 163-169 (1985). · Zbl 0571.52007
[7] Gruber, P. M., Zamfirescu, T.: Generic properties of compact starshaped sets. Proc. Amer. Math. Soc. In print. · Zbl 0683.52008
[8] Hawkes, J.: Hausdorff measure, entropy and the independence of small sets. Proc. London Math. Soc.8, 700-724 (1974). · Zbl 0315.28001
[9] Kahane, J.-P.: Sur la dimension des intersections. In:J. A. Barroso (ed.): Aspect of Mathematics and its Applications, pp. 419-430. Amsterdam: North-Holland. 1986.
[10] Kahane, J.-P., Salem, R.: Ensembles Parfaits et Séries Trigonométriques. Paris: Hermann. 1963.
[11] Kolmogorov, A. N., Tihomirov, V. M.: ?-entropy and ?-capacity of sets in functional spaces. Uspekhi Mat. Nauk14, 3-86 (1959); Amer. Math. Soc. Transl.17, 277-364 (1961). · Zbl 0090.33503
[12] Michael, E.: Topologies on spaces of subsets. Trans. Amer. Math. Soc.71, 152-182 (1951). · Zbl 0043.37902
[13] Ostaszewski, A. J.: Families of compact sets and their universals. Matematika21, 116-127 (1974). · Zbl 0305.54040
[14] Oxtoby, J. C.: Measure and Category. New York-Heidelberg-Berlin: Springer. 1971. · Zbl 0217.09201
[15] Rogers, C. A.: Hausdorff Measures. Cambridge: Univ. Press. 1970.
[16] Taylor, S. J.: The Hausdorff ?-dimensional measure of Brownian paths inn-space. Proc. Cambridge Philos. Soc.48, 31-39 (1953). · Zbl 0050.05803
[17] Wieacker, J. A.: The convex hull of a typical compact set. Math. Ann.282, 637-644 (1988). · Zbl 0636.52004
[18] Zamfirescu, T.: Using Baire categories in geometry. Rend. Sem. Math. Univ. Politecn. Torino43, 67-88 (1985). · Zbl 0601.52004
[19] Zamfirescu, T.: How many sets are porous? Proc. Amer. Math. Soc.100, 383-387 (1987). · Zbl 0625.54036
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