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 randomness? detect randomness

Why algorithmic randomness? detect randomness produce randomness

Why algorithmic randomness? detect randomness produce randomness mimic randomness

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

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 ?

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 K f � � q ∈ D , ρ ( x , q ) ≤ 2 − n � � n ( x ) = inf K ( q ) .

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 , K f n ( x ) cdim f ( x ) = lim inf . r n

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 , K f n ( x ) cdim f ( x ) = lim inf . r n Definition Let E ⊆ X , cdim f ( E ) = sup cdim f ( x ) . x ∈ E

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 , K f n ( x ) cdim f ( x ) = lim inf . r n Definition Let E ⊆ X , cdim f ( E ) = sup cdim f ( x ) . x ∈ E Both definitions relativize to any oracle B by using K B ( w )

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 B ⊆{ 0 , 1 } ∗ cdim f , B ( E ) . dim H ( E ) = min

PTSP

PTSP Let us answer questions on dim H using effective dimension

PTSP Let us answer questions on dim H using effective dimension Questions on dim H ( E ) are questions on the randomness of x ∈ E

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 dist H defined as follows � � dist H ( U , V ) = max sup ρ ( x , V ) , sup ρ ( y , U ) . x ∈ U y ∈ V ( ρ ( a , B ) = inf { ρ ( a , b ) | b ∈ B } )

Known result Theorem (McClure 1996) Let E ⊆ X be self-similar . Let ψ s ( t ) = 2 − 1 / t s . Then dim ψ H ( K ( E )) ≤ dim H ( E ) .

Constructive exact dimension Definition Let x ∈ X . The f -constructive dimension ϕ of x is cdim f ,ϕ ( x ) = inf { s | ∃ ∞ n K f n ( x ) ≤ log(1 /ϕ s (2 − n )) } . Definition Let x ∈ X . The f -constructive strong dimension ϕ of x is cDim f ,ϕ ( x ) = inf { s | ∀ ∞ n K f n ( x ) ≤ log(1 /ϕ s (2 − n )) } .

J. Lutz, N. Lutz, EM 2020 Theorem (Hyperspace dimension theorem) Let E ⊆ X be an analytic set. ϕ s ( t ) = 2 − 1 /ϕ s ( t ) . Then Let ϕ be a gauge family, let ˜ dim ˜ ϕ P ( K ( E )) ≥ dim ϕ P ( E ) .

Prokhorov metric space Let P ( X ) be the set of Borel probability measures on X d P ( µ, ν ) = inf { α > 0 | µ ( A ) ≤ ν ( A α ) + α ∧ ν ( A ) ≤ µ ( A α ) + α } A α = { x | ρ ( A , x ) < α } , ∅ α = ∅ Notice that d P ( δ x , δ y ) = min( ρ ( x , y ) , 1).

Information content in Prokhorov metric space (( X , ρ ) is a separable metric space, D is a dense set and f : { 0 , 1 } ∗ ։ D )  �  k �   � � M = k ∈ N , a i ∈ D , α i ∈ Q ∩ [0 , 1] ,  α 1 δ a 1 + . . . + α k δ a k α k = 1 � �  j =1 �

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.

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

Why Prokhorov metric space II

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 ) = fd µ for all f ∈ C ( X ) .

References Jack H. Lutz and Neil Lutz, Who asked us? How the theory of computing answers questions about analysis, Ding-Zhu Du and Jie Wang (eds.), Complexity and Approximation: In Memory of Ker-I Ko, pp. 48-56, Springer, 2020. https://arxiv.org/abs/1912.00284 Jack H. Lutz and Elvira Mayordomo, Algorithmic fractal dimensions in geometric measure theory, Springer, to appear. https://arxiv.org/abs/2007.14346 Jack H. Lutz, Neil Lutz, and Elvira Mayordomo The Dimensions of Hyperspaces. https://arxiv.org/abs/2004.07798 Onno van Gaans, Probability measures on metric spaces. https: //www.math.leidenuniv.nl/~vangaans/jancol1.pdf