# Differential Geometry, Part 1 by Chern S., Osserman R. (eds.)

By Chern S., Osserman R. (eds.)

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Note that for all 10' < 10, 8(10) c {x,limsuPQn(x) -liminfQn(x) ~ f} C 8(10') and 8 = {x,Qn(x) converges}. Now define the dimensional indices: do = d(8) = d(lim sup 8(10, N)). 0-+0 dg m N = lim d(S(c:)) = lim d(sup S(c:, N)) 0-+0 0-+0 N 28 Jacques Levy Vehel and Claude Tricot dg msup = limsupd(S(c,N)). c-tO N Proposition 2. For every 0:, fd(O:) :::; do, f~im(o:) :::; dg m , f~imsuP(o:) :::; dg msup . Proof. Use the inclusions Ea C S, Ea(c) C S(2c), Ea(c, N) C S(2c, N). (10) 0 Proposition 3. If the set D is bounded, and d is a stable dimension, then f~im sup and f~im reach the above upper bounds.

Proof. Use the inclusions Ea C S, Ea(c) C S(2c), Ea(c, N) C S(2c, N). (10) 0 Proposition 3. If the set D is bounded, and d is a stable dimension, then f~im sup and f~im reach the above upper bounds. In other words, dg msup E f~imsuP(D) and dg m E f~im(D). Proof. 1. We first show that there exists 0:1 ED such that dg msup :::; f~imsuP(O:l)' Let N E N, c > O. For any x E S(c, N) and any n :::: N, 100n(x) - o:(x)1 :::; c, where o:(x) = ~(liminf O:n(x) + limsuPn O:n(x)). Therefore x E Ea(x)(c, N). This implies that S(c, N) C UaEIR Ea(c, N).

Studia Math. 154 (2003), 207-222. [18] T. Szarek, Invariant measures for nonexpansive Markov operators on Polish spaces. Dissertationes Math. 415 (2003), 1-62. pl J6zef Myjak Dipartimento di Matematica Pura ed Applicata Universita di L'Aquila Via Vetoio, 67-100 L'Aquila Italy and WMS AGH al. pl Progress in Probability, Vol. 57, 23-42 © 2004 Birkhiiuser Verlag Basel/Switzerland On Various Multifractal Spectra Jacques Levy Vehel and Claude Tricot Abstract. We introduce two classes of multifractal spectra, called respectively dimension and continuous spectra.