Computability and ergodic theory Mathieu Hoyrup Ergodic - PowerPoint PPT Presentation

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Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition • Let P be a shift-invariant measure over Ω = { 0 , 1 } N : P [ w ] = P [ 0 w ] + P [ 1 w ].

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition • Let P be a shift-invariant measure over Ω = { 0 , 1 } N : P [ w ] = P [ 0 w ] + P [ 1 w ]. • [Birkhoff, 1931] For P -almost every x ∈ Ω, and every w ∈ { 0 , 1 } ∗ , µ x [ w ] := limiting frequency of w along x exists.

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition • Let P be a shift-invariant measure over Ω = { 0 , 1 } N : P [ w ] = P [ 0 w ] + P [ 1 w ]. • [Birkhoff, 1931] For P -almost every x ∈ Ω, and every w ∈ { 0 , 1 } ∗ , µ x [ w ] := limiting frequency of w along x exists. • µ x is itself a shift-invariant probability measure.

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition • Let P be a shift-invariant measure over Ω = { 0 , 1 } N : P [ w ] = P [ 0 w ] + P [ 1 w ]. • [Birkhoff, 1931] For P -almost every x ∈ Ω, and every w ∈ { 0 , 1 } ∗ , µ x [ w ] := limiting frequency of w along x exists. • µ x is itself a shift-invariant probability measure. Question Reading more and more bits of x , can one compute µ x from x ?

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition There are two cases: • µ x is the same for almost all x ’s. In that case, µ x = P almost surely. P is said to be ergodic . • µ x depends on x .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition There are two cases: • µ x is the same for almost all x ’s. In that case, µ x = P almost surely. P is said to be ergodic . • µ x depends on x . P is not ergodic ⇐ ⇒ it can be decomposed into P = λ P 1 + ( 1 − λ ) P 2 with P 1 � = P 2 shift-invariant and 0 < λ < 1 .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition Observation In general µ x cannot be uniformly computed from x . Example Let P = 1 2 ( B p + B q ) with 0 < p � = q < 1 . Every finite sequence is compatible with B p and B q so one can never determine whether µ x = B p or µ x = B q .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition Observation In general µ x cannot be uniformly computed from x . Example Let P = 1 2 ( B p + B q ) with 0 < p � = q < 1 . Every finite sequence is compatible with B p and B q so one can never determine whether µ x = B p or µ x = B q . Definition Let P be a computable shift-invariant measure. P is effectively decomposable if there is a machine M such that for every ε > 0 , P { x : M x ( ε ) computes µ x } ≥ 1 − ε.

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn . . . run 0 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn . . . run 0 . . . run 1 . . . run 2 . . . run 3 . . . run 4 . . . run 5 . . . run 6 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn 7 = 4 ! · 2 ! ) = 1 2 · 2 3 · 1 4 · 3 5 · 2 6 · 4 P ( 7 ! 1 convention: 0 ! = 1

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn 7 = 4 ! · 2 ! ) = 1 2 · 2 3 · 1 4 · 3 5 · 2 6 · 4 P ( 7 ! } ∗ , and more generally for w ∈ { , R ! · B ! R ! · B ! P ( w ) = ( R + B + 1 )! = ( | w | + 1 )! where R is the number of ’s and B the number of ’s in w . 1 1 convention: 0 ! = 1

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn 7 = 4 ! · 2 ! ) = 1 2 · 2 3 · 1 4 · 3 5 · 2 6 · 4 P ( 7 ! } ∗ , and more generally for w ∈ { , R ! · B ! R ! · B ! P ( w ) = ( R + B + 1 )! = ( | w | + 1 )! where R is the number of ’s and B the number of ’s in w . 1 P is a computable shift-invariant measure, so P -almost every sequence x induces a measure µ x . 1 convention: 0 ! = 1

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn 7 = 4 ! · 2 ! ) = 1 2 · 2 3 · 1 4 · 3 5 · 2 6 · 4 P ( 7 ! } ∗ , and more generally for w ∈ { , R ! · B ! R ! · B ! P ( w ) = ( R + B + 1 )! = ( | w | + 1 )! where R is the number of ’s and B the number of ’s in w . 1 P is a computable shift-invariant measure, so P -almost every sequence x induces a measure µ x . Question What does µ x look like? Can it be computed from x ? 1 convention: 0 ! = 1

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn . . . run 0 . . . run 1 . . . run 2 . . . run 3 . . . run 4 . . . run 5 . . . run 6 . . .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Ergodic decomposition The Pólya urn . . . run 0 . . . run 1 . . . run 2 . . . run 3 . . . run 4 . . . run 5 . . . run 6 . . . Each run is equivalent to tossing a coin with some particular bias p , chosen uniformly at random in [ 0 , 1 ]. � 1 R ! · B ! p R ( 1 − p ) B d p P ( w ) = ( R + B + 1 )! = 0

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Effective topology on measures • The space of probability measures over Ω with the metric � 2 −| w | | P [ w ] − Q [ w ] | d ( P , Q ) = w ∈{ 0 , 1 } ∗ is a compact metric space, hence a Baire space .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Effective topology on measures • The space of probability measures over Ω with the metric � 2 −| w | | P [ w ] − Q [ w ] | d ( P , Q ) = w ∈{ 0 , 1 } ∗ is a compact metric space, hence a Baire space . • The subset of shift-invariant measures is closed: P ∈ S ⇐ ⇒ ∀ w , P [ 0 w ] + P [ 1 w ] = P [ w ] so it is a compact metric subspace, hence a Baire space .

Ergodic decomposition A topological observation Weaker result: proof Stronger result: proof Effective topology on measures • The space of probability measures over Ω with the metric � 2 −| w | | P [ w ] − Q [ w ] | d ( P , Q ) = w ∈{ 0 , 1 } ∗ is a compact metric space, hence a Baire space . • The subset of shift-invariant measures is closed: P ∈ S ⇐ ⇒ ∀ w , P [ 0 w ] + P [ 1 w ] = P [ w ] so it is a compact metric subspace, hence a Baire space . • In S , the set E of ergodic measures is a dense Π 0 2 -set : ⇒ ∃ P 0 , P 1 ∈ S such that P 0 � = P 1 and P = P 0 + P 1 • P / ∈ E ⇐ 2 • The Markovian ergodic measures are dense in S E is co-meager in S