By De-Jun Feng, Ka-Sing Lau
This quantity collects 13 expository or survey articles on issues together with Fractal Geometry, research of Fractals, Multifractal research, Ergodic conception and Dynamical structures, likelihood and Stochastic research, written by means of the prime specialists of their respective fields. The articles are in accordance with papers provided on the overseas convention on Advances on Fractals and similar issues, hung on December 10-14, 2012 on the chinese language collage of Hong Kong. the amount deals insights right into a variety of fascinating, state of the art advancements within the sector of fractals, which has shut ties to and functions in different parts equivalent to research, geometry, quantity idea, likelihood and mathematical physics.
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Additional resources for Geometry and Analysis of Fractals: Hong Kong, December 2012
Sample text
1]. Remark 6 (1) The measures of type μ B can be used to control the Hausdorff and packing dimensions of the wider family of sets Eν (γ; γ ) = x ∈ [0, 1] : lim inf n→∞ log ν(In (x)) log ν(In (x)) = γ, lim sup =γ , −n log(b) −n log(b) n→∞ 0 ≤ γ ≤ γ , (see [AB14] for the details). (2) Consider the branching random walk H (w), w ∈ A ∗ , for itself, and do not assume any integrability properties for H (or W ). Set f (n, γ, ) = n −1 logb # w ∈ A n : Hn (w) ∈ [γ − , γ + ] . n log(b) The same approach as above yields that with probability 1, Mandelbrot Cascades and Related Topics 27 for all γ ∈ R, dim E H (γ) = lim lim inf f (n, γ, ) →0+ n→∞ = lim lim sup f (n, γ, ) = f (γ).
Probab. Theory Relat. : Random conformal weldings. Acta Math. : Hausdorff and packing spectra, large deviations, and free energy for branching random walks. in Rd . To appear in Comm. Math. Phys (2014). : Log-infinitely divisible multifractal processes. Commun. Math. Phys. 236, 449–475 (2003) 42 [Bar00] J. : Continuity of the multifractal spectrum of a random statistically selfsimilar measure. J. Theor. Probab. : Comparing multifractal formalisms: the neighboring boxes conditions. Asian J. Math.
This approach can be adapted to get the result when α = 1 and ν = μ (see [BKNSW14a]). s. 5]. 8). The asymptotic behavior of (Fn )n≥1 is far from being completely understood in general, and deserves to be further explored. An interesting related and earlier work is [DES93] about the mean field theory of directed polymers with random complex weights whose modulus is independent of the argument; under this kind of assumptions, renormalization results in the space of distributions have been obtained very recently in [LRV00] for complex Gaussian multiplicative chaos on Rd when the modulus and argument which are independent.