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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

MSC:

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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References:

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