Dynamics of Mandelbrot Cascades
نویسنده
چکیده
Mandelbrot multiplicative cascades provide a construction of a dynamical system on a set of probability measures defined by inequalities on moments. To be more specific, beyond the first iteration, the trajectories take values in the set of fixed points of smoothing transformations (i.e., some generalized stable laws). Studying this system leads to a central limit theorem and to its functional version. The limit Gaussian process can also be obtained as limit of an ‘additive cascade’ of independent normal variables. 1. A dynamical system Consider the set A = {0, . . . , b − 1}, where b ≥ 2. Set A ∗ = ⋃ n≥0 A , where, by convention, A 0 is the singleton { } whose the only element is the empty word . If w ∈ A ∗, we denote by |w| the integer such that w ∈ A |w|. If n ≥ 1 and w = w1 · · ·wn ∈ A n then for 1 ≤ k ≤ n the word w1 · · ·wk is denoted by w|k. By convention, w|0 = . Given v and w in A , v∧w is defined to be the longest prefix common to both v and w, i.e., v|n0 , where n0 = sup{0 ≤ k ≤ n : v|k = w|k}. Let A ω stand for the set of infinite sequences w = w1w2 · · · of elements of A . Also, for x ∈ A ω and n ≥ 0, let x|n stand for the projection of x on A . If w ∈ A ∗, we consider the cylinder [w] consisting of infinite words in A ω whose w is a prefix. We index the closed b-adic subintervals of [0, 1] by A ∗: for w ∈ A ∗, we set Iw = ∑ 1≤k≤|w| wkb −k, ∑ 1≤k≤|w| wkb −k + b−|w| . If f : [0, 1] 7→ R is bounded, for every sub-interval I = [α, β] of [0, 1], we denote by ∆(f, I) the increment f(β)− f(α) of f over the interval I. Let P the set of Borel probability measures on R+. If μ ∈ P and p > 0, we denote by mp(μ) the moment of order p of μ, i.e., mp(μ) = ∫
منابع مشابه
Extension of the Douady-Hubbard's Theorem on Connectedness of the Mandelbrot Set to Symmetric Polynimials
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