By Andrzej Lasota, Józef Myjak, Tomasz Szarek (auth.), Christoph Bandt, Umberto Mosco, Martina Zähle (eds.)

Fractal geometry is used to version complex ordinary and technical phenomena in quite a few disciplines like physics, biology, finance, and medication. in view that so much convincing types comprise a component of randomness, stochastics enters the world in a normal approach. This booklet records the institution of fractal geometry as a considerable mathematical idea. As within the prior volumes, which seemed in 1998 and 2000, prime specialists identified for transparent exposition have been chosen as authors. They survey their box of craftsmanship, emphasizing fresh advancements and open difficulties. major subject matters comprise multifractal measures, dynamical structures, stochastic approaches and random fractals, harmonic research on fractals.

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Remark. The same technique shows that fl = (1g)** when fg has compact support. We end this section with some easily proved properties of TCand fC. Proposition 9. 1. T C and fC are increasing and concave. 2. TC(O) = fC(O) = -~(Supp(A)), where Supp(A) = IR - U{u, lui> O,A(u) = +oo}. 42 Jacques Levy Vehel and Claude Tricot 3. If A(u) = log Jl(u)/ log lui with Jl a probability measure, then rC(I) = fC(I) = O. 4. For any sequence (1Jn) tending to zero such that limn-+ oo log1Jn/ log1JnH = 1, r C ( q) = lim infn-+oo log H~n / log 1Jn and fC (q) = lim inf n-+oo log J~n / log 1Jn .

There exists a smallest integer N(x) such that n ~ N{x) ::::} p{2-n) :::; Un{X)min, so that O:n{x) = 0. s, N) = {k-')' /N(k-')') :::; N} is finite, and Eo{c-) = F. If x f/. F, and n is large enough, O:n (x) = 1. Let us take two real numbers c < d and an integer K such that 2-')' < c - 2- K :::; d + 2- K < 1. For all x E [c,dj and n ~ K, then un(x) n F = 0, so that O:n{x) = 1. Therefore [c,dj c E I (2- K ,N) for all N ~ K. We deduce that If x E F, un{x) n F =/:. s < 2- K , sUPNLl(E1(c-,N)) = 1. >.

A dimension may have the following properties: Definition 2. The dimension d is stable if d(E U F) = maxi d(E), d(F)} for all sets E, F in [0,1]. It is a-stable if d(UEn ) = sup{d(En)} n for any countable set family (En). Let us now define our new spectra. For any real number x in [0, 1], set an (x) = A(un(x)). For any real a, let Ea(e,N) = {x,n ~ N:::} lan(x) - al :S e}. Note that Ea(e, N) increases with N, so that UN Ea(e, N) may be written as sUPN Ea(e, N). Also, Ea(e, N) = nn~N Ia(e, n). Let Ea(e) = supEa(e,N) = {x,3Nsuchthat n N liminf 1",,(£, N).