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Effective fractal dimension in the hyperspace and the space of - - PowerPoint PPT Presentation
Effective fractal dimension in the hyperspace and the space of - - PowerPoint PPT Presentation
Effective fractal dimension in the hyperspace and the space of probability distributions (informal talk) Elvira Mayordomo Universidad de Zaragoza, Iowa State University AIM, August 13th 2020 Why algorithmic randomness? Why algorithmic
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Why algorithmic randomness?
detect randomness
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Why algorithmic randomness?
detect randomness produce randomness
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Why algorithmic randomness?
detect randomness produce randomness mimic randomness
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Information content in a separable space
Let (X, ρ) be a separable metric space. Let D be a dense set and f : {0, 1}∗ ։ D
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Information content in a separable space
Let (X, ρ) be a separable metric space. Let D be a dense set and f : {0, 1}∗ ։ D What is the information content of x ∈ X?
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Information content in a separable space
Let (X, ρ) be a separable metric space. Let D be a dense set and f : {0, 1}∗ ։ D What is the information content of x ∈ X?
Definition
Let x ∈ X, n ∈ N. The Kolmogorov complexity of x at precision n is Kf
n(x) = inf
- K(q)
- q ∈ D, ρ(x, q) ≤ 2−n
.
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Effective dimension in a separable space
(X, ρ) is a separable metric space, D is a dense set, and f : {0, 1}∗ ։ D
Definition
Let x ∈ X, cdimf (x) = lim inf
n
Kf
n(x)
r .
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Effective dimension in a separable space
(X, ρ) is a separable metric space, D is a dense set, and f : {0, 1}∗ ։ D
Definition
Let x ∈ X, cdimf (x) = lim inf
n
Kf
n(x)
r .
Definition
Let E ⊆ X, cdimf (E) = sup
x∈E
cdimf (x).
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Effective dimension in a separable space
(X, ρ) is a separable metric space, D is a dense set, and f : {0, 1}∗ ։ D
Definition
Let x ∈ X, cdimf (x) = lim inf
n
Kf
n(x)
r .
Definition
Let E ⊆ X, cdimf (E) = sup
x∈E
cdimf (x). Both definitions relativize to any oracle B by using KB(w)
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Point to set principle for separable X
((X, ρ) is a separable metric space, D is a dense set and f : {0, 1}∗ ։ D)
Theorem (ptsp separable spaces)
Let E ⊆ X. Then dimH(E) = min
B⊆{0,1}∗ cdimf ,B(E).
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PTSP
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PTSP
Let us answer questions on dimH using effective dimension
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PTSP
Let us answer questions on dimH using effective dimension Questions on dimH(E) are questions on the randomness of x ∈ E
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The hyperspace
Let (X, ρ) be a separable metric space Let K(X) be the set of nonempty compact subsets of X together with the Hausdorff metric distH defined as follows distH(U, V ) = max
- sup
x∈U
ρ(x, V ), sup
y∈V
ρ(y, U)
- .
(ρ(a, B) = inf{ρ(a, b)|b ∈ B})
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Known result
Theorem (McClure 1996)
Let E ⊆ X be self-similar. Let ψs(t) = 2−1/ts. Then dimψ
H(K(E)) ≤ dimH(E).
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Constructive exact dimension
Definition
Let x ∈ X. The f -constructive dimensionϕ of x is cdimf ,ϕ(x) = inf{s | ∃∞n Kf
n(x) ≤ log(1/ϕs(2−n))}.
Definition
Let x ∈ X. The f -constructive strong dimensionϕ of x is cDimf ,ϕ(x) = inf{s | ∀∞n Kf
n(x) ≤ log(1/ϕs(2−n))}.
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- J. Lutz, N. Lutz, EM 2020
Theorem (Hyperspace dimension theorem)
Let E ⊆ X be an analytic set. Let ϕ be a gauge family, let ˜ ϕs(t) = 2−1/ϕs(t). Then dim ˜
ϕ P(K(E)) ≥ dimϕ P(E).
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Prokhorov metric space
Let P(X) be the set of Borel probability measures on X dP(µ, ν) = inf {α > 0 |µ(A) ≤ ν(Aα) + α ∧ ν(A) ≤ µ(Aα) + α} Aα = {x |ρ(A, x) < α}, ∅α = ∅ Notice that dP(δx, δy) = min(ρ(x, y), 1).
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Information content in Prokhorov metric space
((X, ρ) is a separable metric space, D is a dense set and f : {0, 1}∗ ։ D) M = α1δa1 + . . . + αkδak
- k ∈ N, ai ∈ D, αi ∈ Q ∩ [0, 1],
k
- j=1
αk = 1
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Why Prokhorov metric space I
Definition
Γ ⊆ P(X) is tight if for any ǫ > 0 there is K compact with µ(K) ≥ 1 − ǫ for any µ ∈ Γ.
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Why Prokhorov metric space I
Definition
Γ ⊆ P(X) is tight if for any ǫ > 0 there is K compact with µ(K) ≥ 1 − ǫ for any µ ∈ Γ.
Theorem (Prokhorov theorem)
For Γ ⊆ P(X), Γ is compact if and only if Γ is tight
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Why Prokhorov metric space II
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Why Prokhorov metric space II
Theorem (Riesz representation theorem)
Let (X, ρ) be compact and Hausdorff. If ϕ : C(X) → R is positive and ϕ = 1 then there is a unique µ ∈ P(X) with ϕ(f ) =
- fdµ
for all f ∈ C(X).
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References
Jack H. Lutz and Neil Lutz, Who asked us? How the theory
- f computing answers questions about analysis, Ding-Zhu Du