By De-Jun Feng, Ka-Sing Lau

This quantity collects 13 expository or survey articles on subject matters together with Fractal Geometry, research of Fractals, Multifractal research, Ergodic idea and Dynamical structures, chance and Stochastic research, written through the major specialists of their respective fields. The articles are in accordance with papers awarded on the foreign convention on Advances on Fractals and comparable issues, hung on December 10-14, 2012 on the chinese language collage of Hong Kong. the amount bargains insights right into a variety of intriguing, state of the art advancements within the zone of fractals, which has shut ties to and functions in different components resembling research, geometry, quantity idea, likelihood and mathematical physics.

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**Extra resources for Geometry and Analysis of Fractals: Hong Kong, December 2012**

**Example text**

Sur le modele de turbulence de Benoit Mandelbrot. C. R. Acad. Sci. : Sur le chaos multiplicatif. Ann. Sci. Math. Que. : Positive martingales and random measures. Chi. Ann. Math. : Sur certaines martingales de B. Mandelbrot Adv. Math. : Multiplications aléatoires et dimensions de Hausdorff. Ann. Inst. Henri Poincaré Probab. Stat. : Produits de poids aléatoires indépendants et applications. In: Fractal Geometry and Analysis (Montreal, PQ, 1989), pp. 277–324. NATO Adv. Sci. Inst. Ser. C Math. Phys.

From that time on, various properties of singular spectral measures are studied extensively [Dai12, DHJ09, DHS09, DHSW11, LaW02, LaW06]. In particular, many exotic spectra were discovered and they do not appear in their absolutely continuous counterpart. Here, we list some of the interesting ones. -R. Dai et al. (1) There exists a spectrum of a singularly continuous measure μ such that k is also a spectrum of μ for some k = 1; (2) There exists a so that E( ) is a maximal orthogonal collection of exponentials for L 2 (μ), but not a basis; (3) There exists a spectrum of a singularly continuous measure so that its Beurling dimension is zero.

Then define the random measures μ B,n (dx) = 1 E B,n (w)e−HB (w) bn dx, if x ∈ Iw . They form a martingale which converges almost surely to a random measure μ B supported on E B = n≥1 w∈E B,n Iw . Moreover, it is possible to choose (Nk )k≥1 suitably so that all the measures μ B are simultaneously defined and non degenerate conditionally on E B = ∅. The sequence (Nk )k≥1 can also be fixed so that for all γ such that ϕ∗ (γ) ≥ 0, each time B = (qk )k≥1 is such that limk→∞ ϕ Ak (qk ) = γ and limk→∞ ϕ∗Ak (ϕ Ak (qk )) = ϕ∗ (γ), then, conditionally on {E B = ∅}, μ B is exact dimensional with dimension ϕ∗ (γ) and carried both by the set E H (γ) = {x : ν(In (x)) (x|n) = γ} and the set E ν (γ) = x ∈ [0, 1] : limn→∞ log limn→∞ nHlog(b) −n log(b) = ν(B(x,r ) γ ∩ x ∈ [0, 1] : limr →0+ log log(r = γ .