On effective Birkhoff's ergodic theorem for computable actions of amenable groups

0.2 MB·

by  Nikita Moriakov

ON EFFECTIVE BIRKHOFF’S ERGODIC THEOREM
FOR COMPUTABLE ACTIONS OF AMENABLE GROUPS.
NIKITAMORIAKOV
7
1
0 Abstract. Weintroducecomputableactionsofcomputablegroupsandprove
2 thefollowingversionsofeffectiveBirkhoff’sergodictheorem. LetΓbeacom-
putable amenable group, then there always exists a canonically computable
n
temperedtwo-sidedFølnersequence(Fn)n≥1inΓ. Foracomputable,measure-
a preserving,ergodicactionofΓonaCantorspace{0,1}N endowedwithacom-
J
putableprobabilitymeasureµ,itisshownthatforeveryboundedlowersemi-
3 computablefunctionf on{0,1}NandforeveryMartin-Lo¨frandomω∈{0,1}N
2 theequality
1
lim f(g·ω)= fdµ
S] n→∞|Fn|gX∈Fn Z
D holds,wherethe averages aretaken withrespect toacanonicallycomputable
temperedtwo-sidedFølnersequence(Fn)n≥1. Wealsoprovethesameidentity
.
h for all lower semicomputable f’s in the special case when Γ is a computable
t group of polynomial growth and Fn := B(n) is the Følner sequence of balls
a aroundtheneutralelementofΓ.
m
[

1
v 1. Introduction
5
A classical ergodic theorem of Birkhoff asserts that, if ϕ : X → X is an ergodic
6
3 measure-preserving transformation on a probability space (X,µ), then for every
6 f ∈L1(X) we have
0
n
1. (1.1) lim 1 f(ϕix)= fdµ
0 n→∞nXi=1 Z
7
for µ-a.e. x ∈ X. We refer, e.g., to [EFHN15, Chapter 11] for the proof. A
1
: celebratedresultofLindenstrauss[Lin01]givesageneralizationofBirkhoff’sergodic
v
theorem for measure-preserving actions of amenable groups and ergodic averages,
i
X taken along tempered Følner sequences.
r OnemayalsowonderiftheaveragesinEquation(1.1)convergeforeveryMartin-
a
Lo¨frandomxandeverycomputablef. AnaffirmativeanswerwasgivenbyV’yugin
in [V’y97] for computable f’s. Later,it wasprovedin [BDHMS12]that the ergodic
averagesconverge for all lower semi-computable f’s.
In so far, the effective ergodic theorems have only been proved for actions of Z,
anditisanaturalquestionifonecangeneralizeeffectiveBirkhoff’sergodictheorem
for measure-preservingactions ofmore generalgroups(for instance, the groupsZd,
groupsofpolynomialgrowthandsoon). However,onemustfirstdefinecomputable
actions of groups appropriately. In this article we define computable actions of
computablegroupsinanaturalwayinSection2.5,whichagreeswiththe‘classical’
definition in the case of Z-actions, and obtain the following generalizations of the
resultsfrom[BDHMS12].Firstofall,wederiveageneralizationofKuˇcera’stheorem
in Section 3.1, which is the main technical tool of the article.
Theorem. Let Γ be a computable amenable group and ({0,1}N,µ,Γ) be a com-
putable ergodic Γ-system. Let U ⊂ {0,1}N be an effectively open subset such that
Date:January24,2017.
1
2 N.MORIAKOV
µ(U)<1. Let
U∗ := g−1U
g\∈Γ
be the set of all points ω ∈ {0,1}N whose orbit remains in U. Then U∗ is an
effectively null set.
Using this generalization of Kuˇcera’s theorem and the results of Lindenstrauss,
wederivethefirstmaintheoreminSection3.2. Tosimplifythenotation,wedenote
the averagesby E := 1 .
g∈F |F|
gP∈F
Theorem. Let Γ be a computable amenable group with a canonically computable
tempered two-sided Følner sequence (F ) . Suppose that ({0,1}N,µ,Γ) is a com-
n n≥1
putable ergodic Γ-system. For every bounded lower semicomputable f and for every
Martin-L¨of random ω ∈{0,1}N the equality
lim E f(g·ω)= fdµ
n→∞ g∈Fn Z
holds.
In a special case, when Γ is a computable group of polynomial growth, we are
able to remove the boundedness assumption on f and prove the following version
of effective Birkhoff’s ergodic theorem.
Theorem. Let Γ be a computable group of polynomial growth with the Følner se-
quence of balls around e∈Γ given by
F :={g ∈Γ:kgk≤n} for n≥1.
n
Supposethat({0,1}N,µ,Γ)is acomputableergodic Γ-system. For everylower semi-
computable f and for every Martin-L¨of random ω ∈{0,1}N the equality
lim E f(g·ω)= fdµ
n→∞ g∈Fn Z
holds.
2. Preliminaries
2.1. Computable Amenable Groups. In this section we will remind the reader
of the classicalnotion of amenability and state some results from ergodic theory of
amenablegroupactions. Westressthatallthe groupsthatweconsiderarediscrete
and countably infinite.
Let Γ be a group with the counting measure |·|. A sequence of finite subsets
(F ) of Γ is called
n n≥1
1) a left Følner sequence (resp. right Følner sequence) if for every g ∈Γ
one has
|F △gF | |F △F g|
n n n n
→0 resp. →0 ;
|F | (cid:18) |F | (cid:19)
n n
2) a (C-)tempered sequence if there is a constant C such that for every j
one has
(cid:12)(cid:12) Fi−1Fj(cid:12)(cid:12)<C|Fj|.
(cid:12)i[<j (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Agroupis calledamenable(cid:12)if ithasa(cid:12)left Følnersequence. Asequence offinite
subsets (F ) of Γ is called a two-sided Følner sequence if it is a left and a
n n≥1
right Følner sequence simultaneously.
We refer the reader, e.g., to [SV03] for the standard notions of a computable
function and a computable/enumerable set, which will appear in this article. A
EFFECTIVE BIRKHOFF’S THEOREM 3
sequence of finite subsets (F ) of N is calledcanonically computable if there
n n≥1
is an algorithm that, given n, prints the set F and halts. Formally speaking,
n
k
for a finite set A = {x1,x2,...,xk} ⊂ N, we call the number I(A) := 2xi
iP=1
the canonical index of A. Hence a sequence (F ) of finite subsets of N is
n n≥1
canonicallycomputable ifandonlyifthe (total)function n7→I(F )iscomputable.
n
A group Γ with the composition operation ◦ is called a computable group if,
as a set, Γ is a computable subset of N and the total function ◦ : Γ×Γ → Γ is
computable. ItiseasytoshowthatinacomputablegroupΓtheinversionoperation
g 7→ g−1 is a total computable function. We refer the reader to [Rab60] for more
details.
Any discrete amenable group Γ admits a two-sided Følner sequence. Further-
more, if the group is computable, then there exists a canonically computable two-
sided Følner sequence. To prove that we will need the following result.
Lemma 2.1. Given a discrete amenable group Γ, for any finite symmetric set
K ⊂Γ such that e∈Γ and any ε>0 there exists a finite subset F ⊂Γ such that
(2.1) |KFK|−|F|≤ε|F|.
We refer the reader to [OW87, I.§1, Proposition 2] for the proof.
Lemma 2.2. Let Γ be a computable amenable group. Then there exists a canoni-
cally computable two-sided Følner sequence (F ) .
n n≥1
Proof. First of all, observe that given K ⊂ Γ, ε > 0 as in Lemma 2.1 and a finite
set F ⊂Γ satisfying Equation (2.1), we have
|gF \F|
≤ε
|F|
and
|Fg\F|
≤ε
|F|
for all g ∈ K. Let K be the finite set of the first n elements of the computable
n
groupΓ. Then,foreveryn=1,2,... weapplyLemma2.1tothesetK ∪K−1∪{e}
n n
and ε := 1/n and find the finite set F with the smallest canonical index I(F )
n n n
satisfyingEquation(2.1). Itiseasytoseethat(F ) isindeedatwo-sidedFølner
n n≥1
sequence.
Every Følner sequence has a tempered Følner subsequence. Furthermore, the
constructionofatemperedFølnersubsequencefromagivencanonicallycomputable
Følnersequenceis‘algorithmic’. Theproofisessentiallycontainedin[Lin01,Propo-
sition 1.4], but we provide it for reader’s convenience below.
Proposition 2.3. Let (F ) be a canonically computable Følner sequence in a
n n≥1
computablegroupΓ. Thenthereisacomputablefunctioni7→n s.t. thesubsequence
i
(F ) is a canonically computable tempered Følner subsequence.
ni i≥1
Proof. We define n inductively as follows. Let n := 1. If n ,...,n have been
i 1 1 i
determined, we set F := F . Take for n the first integer greater than i+1
i nj i+1
jS≤i
such that e
1
F △F−1F ≤
ni+1 i ni+1
(cid:12) (cid:12) F
(cid:12)(cid:12) e (cid:12)(cid:12) (cid:12) i(cid:12)
The function i7→ni is total computable. It follows(cid:12)(cid:12)eth(cid:12)(cid:12)at
(cid:12)(cid:12) Fn−j1Fni+1(cid:12)(cid:12)≤2 Fni+1 ,
(cid:12)j[≤i (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4 N.MORIAKOV
hence the sequence (F ) is 2-tempered. Since the Følner sequence (F ) is
ni i≥1 n n≥1
canonicallycomputableandthefunctioni7→n iscomputable,theFølnersequence
i
(F ) is canonically computable and tempered.
ni i≥1
Let us state an immediate corollary.
Corollary 2.4. Let Γ be a computable amenable group. Then there exists a canon-
ically computable, tempered two-sided Følner sequence (F ) in Γ.
n n≥1
Thefollowingresulttells usthatthe limsupofaveragesofboundedfunctionson
an amenable group is translation-invariant.
Lemma 2.5 (Limsup invariance). Let Γ be an amenable group with a right Følner
sequence (F ) and f ∈ℓ∞(Γ,R) be a bounded function on Γ. Then
n n≥1
limsupE f(g)=limsupE f(gh).
g∈Fn g∈Fn
n→∞ n→∞
Proof. A direct computation shows that for all n≥1
1 2|F \F h|·kfk
n n ∞
(cid:12) f(g)− f(g)(cid:12)≤ ,
|F |(cid:12) (cid:12) |F |
n (cid:12)(cid:12)gX∈Fn g∈XFnh (cid:12)(cid:12) n
(cid:12) (cid:12)
andthestatementof(cid:12)thelemmafollowssince(cid:12) (F ) isarightFølnersequence.
n n≥1
Remark 2.6. ThestatementofLemma2.5doesnotholdforgeneralamenablegroups
and unbounded nonnegative functions. As a counterexample, take Γ:=Z with the
tempered two-sided Følner sequence
F :=[−2n,...,2n] for n≥1
n
and define f : Γ → N to be zero everywhere, except for points of the form 2k+1,
where we let
f(2k+1):=2k for all k ≥0.
It is then easy to see that
limsupE f(g)6=limsupE f(g+1).
g∈Fn g∈Fn
n→∞ n→∞
We willresolvethis issue inthe class ofgroupsofpolynomialgrowthin Lemma 2.9
in Section 2.2.
2.2. ComputableGroupsofPolynomialGrowth. LetΓbeafinitelygenerated
discrete group and {γ ,...,γ } be a fixed generating set. Each element γ ∈ Γ can
1 k
bewrittenasaproductγp1γp2...γpl forsomeindexesi ,i ,...,i ∈{1,...,k}and
i1 i2 il 1 2 l
some integers p ,p ,...,p ∈Z. We define the norm of an element γ ∈Γ by
1 2 l
l
kγk:=inf{ |p |:γ =γp1γp2...γpl},
i i1 i2 il
Xi=1
wheretheinfinumistakenoverallrepresentationsofγasaproductofthegenerating
elements. The norm k·k on Γ can, in general,depend on the generating set, but it
is easy to show [CSC10, Corollary6.4.2] that two different generating sets produce
equivalent norms. We will always say what generating set is used in the definition
of a norm, but we will omit an explicit reference to the generating set later on.
WesaythatthegroupΓisof polynomialgrowthifthereareconstantsC,d>0
such that for all n≥1 we have
|B(n)|≤Cnd.
EFFECTIVE BIRKHOFF’S THEOREM 5
Example 2.7. Consider the group Zd for d ∈ N and let γ ,...,γ ∈ Zd be the
1 d
standard basis elements of Zd. That is, γ is defined by
i
γ (j):=δj (j =1,...,d)
i i
for all i = 1,...,d. We consider the generating set given by elements (−1)εkγk
kP∈I
for all subsets I ⊆ [1,d] and all functions ε ∈ {0,1}I. Then it is easy to see by
·
induction on dimension that B(n)=[−n,...,n]d, hence
|B(n)|=(2n+1)d for all n∈N
with respect to this generating set, i.e., Zd is a group of polynomial growth.
Let d∈Z . We say that the groupΓ has polynomial growth of degree d if
≥0
there is a constant C >0 such that
1
nd ≤|B(n)|≤Cnd for all n∈N.
C
Itwasshownin[Bas72]that,ifΓisafinitelygeneratednilpotentgroup,thenΓhas
polynomial growth of some degree d ∈ Z . Furthermore, one can show [CSC10,
≥0
Proposition6.6.6]thatifΓisagroupandΓ′ ≤Γisafiniteindex,finitelygenerated
nilpotent subgroup,havingpolynomialgrowthofdegreed∈Z , then the groupΓ
≥0
has polynomial growth of degree d. The converse is true as well: it was proved in
[Gro81] that, if Γ is a group of polynomial growth, then there exists a finite index,
finitely generated nilpotent subgroup Γ′ ≤ Γ. It follows that if Γ is a group of
polynomial growth with the growthfunction γ, then there is a constant C >0 and
an integer d∈Z , called the degree of polynomial growth, such that
≥0
1
nd ≤|B(n)|≤Cnd for all n∈N.
C
An even stronger result was obtained in [Pan83], where it is shown that, if Γ is a
group of polynomial growth of degree d∈Z , then the limit
≥0
|B(n)|
(2.2) c := lim
Γ n→∞ nd
exists.
Lemma 2.8. Let Γ beagroup ofpolynomial growth. Then (B(n)) is a tempered
n≥1
two-sided Følner sequence in Γ.
Proof. We want to show that for every g ∈Γ
|gB(n)△B(n)|
lim =0.
n→∞ |B(n)|
Let m:=kgk∈Z . Then gB(n)⊆B(n+m), hence
≥0
|gB(n)△B(n)| |B(n+m)|−|B(n)|
≤ →0,
|B(n)| |B(n)|
where we use the existence of the limit in Equation (2.2). Similarly, we use the
relationB(n)g ⊆B(n+m) to show that (B(n)) is a rightFølner sequence. The
n≥1
sequence (B(n)) is tempered, since
n≥1
|B(n−1)·B(n)|≤|B(2n)|≤C2|B(n)|
for all n≥1.
AspromisedinRemark2.6,weprovenowthatthelimsupofaveragesofarbitrary
nonnegative functions on a group of polynomial growth Γ is translation invariant.
6 N.MORIAKOV
Lemma2.9 (Limsupinvariance). Let Γbeagroupof polynomial growth anddefine
the Følner sequence of balls around e∈Γ by
F :={g ∈Γ:kgk≤n} for n≥1.
n
Let f :Γ→R be a nonnegative function on Γ. Then
≥0
limsupE f(g)=limsupE f(gh)
g∈Fn g∈Fn
n→∞ n→∞
for all h∈Γ.
Proof. Let S ⊂Γ be the finite generating set, which is used in the definition of the
norm k·k on Γ. Since the statement of the lemma is ‘symmetric’ and since the set
S generates Γ, it suffices to prove that
limsupE f(g)≥limsupE f(gh)
g∈Fn g∈Fn
n→∞ n→∞
for all h ∈ S ∪S−1. We fix an element h ∈ S∪S−1. It is clear that F h ⊆ F ,
n n+1
hence
1
limsupE f(gh)≤limsup f(g).
g∈Fn |F |
n→∞ n→∞ n g∈XFn+1
But
1 |F |
limsup f(g)=limsup n+1 ·E f(g)
|F | |F | g∈Fn+1
n→∞ n g∈XFn+1 n→∞ n
and |F |/|F |→1 as n→∞, which implies that
n+1 n
1
limsup f(g)=limsupE f(g),
|F | g∈Fn
n→∞ n g∈XFn+1 n→∞
and the proof is complete.
AcomputablegroupΓwithadistinguishedsetofgenerators{γ ,...,γ }willbe
1 k
calledacomputable group of polynomial growthifΓisagroupofpolynomial
growth. Itwillbeessentialfurtherthatthegeneratingsetisknown and fixed. More
precisely, we state the following lemma.
Lemma 2.10. Let Γ be a computable group of polynomial growth with a distin-
guished set of generators {γ ,...,γ }. Then the following assertions hold:
1 k
(a) Thesequenceofballs(B(n)) isacanonicallycomputablesequenceoffinite
n≥1
sets;
(b) The growth function n7→|B(n)|,Z →N is a total computable function;
≥0
(c) The norm k·k:Γ→Z is a total computable function.
≥0
The proof of the lemma is straightforward.
2.3. Ergodic Theory. Let X = (X,B,µ) be a probability space. A measurable
transformation ϕ:X →X is called measure-preserving if
µ(ϕ−1A)=µ(A) for all A∈B.
A measure-preserving transformation ϕ : X → X is called an automorphism if
there exists a measure-preservingtransformation ψ :X →X such that
ϕ◦ψ =ψ◦ϕ=id µ−a.e.
X
We denote by Aut(X) the group of all automorphisms of the probability space X.
GivenadiscretegroupΓ,ameasure-preservingΓ-system1isaprobabilityspace
X=(X,B,µ), endowed with an action of Γ on X by automorphisms from Aut(X).
We denote a measure-preserving Γ-system on a probability space (X,B,µ) by a
1Tosimplifythenotation, theshorterterm‘Γ-system’willalsobeused.
EFFECTIVE BIRKHOFF’S THEOREM 7
triple (X,µ,Γ) and we write g·x, where g ∈Γ,x∈X, to denote the corresponding
action of Γ on elements of X.
Let X = (X,µ,Γ) be a measure-preserving Γ-system on a probability space
(X,B,µ). We saythat X is ergodic(orthat the measureµ onX is ergodic)if, for
all A∈B, the condition
µ(γ−1A△A)=0 for all γ ∈Γ
implies that µ(A)=0 or µ(A)=1. That is, X is ergodic if only the trivial sets are
essentially invariant under Γ.
The simplest ergodic theorem for amenable group actions is the mean ergodic
theorem,whichwestatebelow. Fortheproofwereferthereaderto[Gla03,Theorem
3.33].
Theorem 2.11. Let (X,µ,Γ) be a measure-preserving, ergodic Γ-system, where
the group Γ is amenable and (F ) is a left Følner sequence. Then for every
n n≥1
f ∈L2(X) we have
lim E f ◦g = fdµ,
n→∞ g∈Fn Z
where the convergence is understood in L2(X)-sense.
Pointwise convergence of ergodic averages is much more tricky, in particular,
pointwise ergodic averages do not necessarily converge, unless the Følner sequence
satisfiessomeadditionalassumptions. Thefollowingimportanttheoremwasproved
by E. Lindenstrauss in [Lin01]2.
Theorem 2.12. Let X = (X,µ,Γ) be an ergodic measure-preserving Γ-system,
where the group Γ is amenable and (F ) is a tempered left Følner sequence.
n n≥1
Then for every f ∈L1(X)
lim E f(g·x)= fdµ
n→∞ g∈Fn Z
for µ-a.e. x∈X.
2.4. Computability on Cantor Space and Martin-Lo¨f Randomness. Inthis
section we remind the reader some standard notions of computability on Cantor
space. All of these notions have analogs on computable metric spaces as well, and
we refer to [HR09], [GHR10] for the details.
Throughout the article we fix some enumeration of Q = {q ,q ,q ,...}. We
1 2 2
use the standard notions of a computable real number and of a lower/upper semi-
computablerealnumber. Asequenceofrealnumbers(a ) iscalledcomputable
n n≥1
uniformly in n if there exists an algorithm A:N×N→Q such that
|A(n,i)−a |<2−i for all n,i≥1.
n
We fix some enumeration
{0,1}∗={w ,w ,...}
1 2
of the set if all finite binary words, and let
[w]:={ω :ω =wω′ for some ω′ ∈{0,1}N}⊂{0,1}N
be the cylinder set of all words that begin with a finite word w ∈ {0,1}∗. A set
U ⊆ {0,1}N is called effectively open if there is a recursively enumerable subset
E ⊆N such that
U = [w ].
i
i[∈E
2In fact, a more general statement is proved there, but we only need the ergodic case in this
work.
8 N.MORIAKOV
A sequence (U ) of sets is called a uniformly effectively open sequence of
n n≥1
sets if there is a recursively enumerable set E ⊆N×N such that
U = [w ] for all i≥1.
i j
(i,[j)∈E
A mapping ϕ : {0,1}N → {0,1}N is called computable if (ϕ−1([w ])) is
i i≥1
uniformlyeffectivelyopen,thatis,thereisarecursivelyenumerablesetE ⊆N×N
ϕ
such that
ϕ−1([w ])= [w ] for all i≥1.
i j
(i,j[)∈Eϕ
A function f : {0,1}N →R is called lower semicomputable if the sequence of
≥0
sets (f−1((q ,+∞))) is uniformly effectively open.
n n≥1
LetµbeaBorelprobabilitymeasureon{0,1}N. Wesaythatµisacomputable
measure if
µ([w ]∪[w ]∪···∪[w ])
i1 i2 ik
is computable uniformly in i ,...,i ≥1.
1 k
Suppose that µ is a computable probability measure on {0,1}N. A Martin-Lo¨f
µ-test is a uniformly effectively open sequence of sets (U ) such that
n n≥1
µ(U )<2−n for all n≥1.
n
Any subset of U is called an effectively µ-null set. A point ω ∈ {0,1}N is
n
nT≥1
called Martin-Lo¨f random if it is not contained in any effectively µ-null set.
2.5. Computable Dynamical Systems. Now,letΓ⊆Nbeacomputablegroup,
which acts on {0,1}N by homeomorphisms. We say that the action of Γ is com-
putable if there is a recursively enumerable subset E ⊆Γ×N×N such that
γ−1([w ])= [w ] for all i≥1,γ ∈Γ.
i j
(γ,i[,j)∈E
In general, checking the computability of the action of a computable group Γ
on {0,1}N can be trickier than checking computability of a single transformation.
Imagine a Z-action on {0,1}N with the generating element ϕ ∈ Z. Can it happen
thatbothϕandϕ−1 arecomputabletransformationsof{0,1}N,whilsttheactionof
Zon{0,1}Nisnotcomputable? Fortunately,theansweris‘no’: thefollowinglemma
tellsusthatforanactionofacomputablefinitelygeneratedgroupitsufficestocheck
computability of transformations in a finite symmetric generating set to guarantee
the computability of the action. The lemma also shows that the terminology of
computable group actions which we suggest in this article is compatible with the
classical case, when there is only one computable transformation.
Lemma 2.13. Let Γ be a finitely generated computable group with a finitesymmet-
ric generating set S ⊂Γ. Suppose that Γ acts on {0,1}N by homeomorphisms, and,
furthermore, that for each γ ∈S the transformation
N N
γ :{0,1} →{0,1}
is computable. Then the action of Γ on {0,1}N is computable.
Proof. Given a fixed finite symmetric generating set S = {γ ,γ ,...,γ }, we will
1 2 N
denote by B(n) the corresponding balls around the neutral element e ∈ Γ with
respect to the norm determined by S. Since γ ,γ ,...,γ are computable endo-
1 2 N
morphismsof{0,1}N,therearerecursivelyenumerablesubsetsE ,E ,...,E such
1 2 N
that
γ−1([w ])= [w ] for all k=1,...,N,i≥1.
k i j
(i,j[)∈Ek
EFFECTIVE BIRKHOFF’S THEOREM 9
We will describe an algorithm, which enumerates the set E. At stage n, the
algorithm first computes the finite set B(n) ⊂ Γ by computing all products of the
elements of S of length at most n. For each word
γ γ ...γ =γ ∈B(n)
i1 i2 ik
we have for all i≥1
γ−1([w ])=γ−1γ−1 ...γ−1([w ])=γ−1γ−1 ...γ−1 [w ]=
i ik ik−1 i1 i ik ik−1 i2 j1
(i,j1[)∈Ei1
=γ−1γ−1 ...γ−1 [w ]=
ik ik−1 i3 j2
(i,j1[)∈Ei1(j1,j[2)∈Ei2
= ··· [w ].
jk
(i,j1[)∈Ei1(j1,j[2)∈Ei2(j2,j[3)∈Ei3 (jk−1,[jk)∈Eik
We compute the first n pairs (i,j ) ∈ E , for each of these pairs we compute the
1 i1
first n pairs (j ,j )∈E and so on up to the first n pairs (j ,j )∈E (where
1 2 i2 k−1 k ik
j comes from the one but the last step). The algorithm prints all resulting
k−1
triples (γ,i,j ), and proceeds to the next word (or the next stage, if all words at
k
the current stage have been exhausted).
Since, at each stage n, we look through all products of length at most n, it is
easy to see that
γ−1([w ])= [w ]
i j
(γ,i[,j)∈E
for all i≥1, and, furthermore, the set E is recursively enumerable.
A computable Cantor Γ-system3 is a triple ({0,1}N,µ,Γ), where µ is a com-
putablemeasureon{0,1}NandΓactscomputablyon{0,1}Nbymeasure-preserving
transformations.
Remark 2.14. The notion of a computable action of a computable group which we
suggest directly translates to arbitrary computable metric spaces. Furthermore,
Lemma 2.13 remains valid in the more general setting.
3. Effective Birkhoff’s Theorem
3.1. Kuˇcera’s Theorem. Inthis sectionwe generalizeKuˇcera’stheoremfor com-
putable actions of amenable groups. In the proof we follow roughly the approach
from [BDHMS12], although the technical details do differ.
Theorem 3.1. Let Γ be a computable amenable group and ({0,1}N,µ,Γ) be a
computable ergodic Γ-system. Let U ⊂ X be an effectively open subset such that
µ(U)<1. Let
U∗ := g−1U
g\∈Γ
be the set of all points ω ∈ {0,1}N whose orbit remains in U. Then U∗ is an
effectively null set.
Proof. Let(F ) beacanonicallycomputabletwo-sidedFølnersequenceinΓand
n n≥1
µ(U) < q < 1 be some fixed rational number. Let (I ) be the basis of cylinder
i i≥1
sets in ({0,1}N,µ). Let
(3.1) (i,k)7→n(i,k), N×N→N
3Or a computable Γ-system for short, since we only consider dynamical systems on Cantor
spaceinthisarticle.
10 N.MORIAKOV
be some total computable function, which will be chosen later, and define a com-
putable function m by
m(i,k):= F for i,k≥1.
n(i,k)
(cid:12) (cid:12)
(cid:12) (cid:12)
Since U := U is effectively open, there is a r.e. subset E ⊆ N such that
0 0
U = I is a unionofdisjoint cylinder sets. Since the actionof Γ is computable
0 i
i∈SE0
and since (F ) is canonically computable, the sequence
n n≥1
i7→I ∩ g−1(U )
i 0
g∈F\n(i,1)
is a uniformly effectively open sequence of sets. Let
U1 := Ii∩ g−1(U0),
i∈[E0 g∈F\n(i,1) 
then, clearly, U ⊆ U is an effectively open set and U∗ ⊆ U . Since U is an
1 0 1 1
effectively open set, there is a r.e. subset E ⊆N such that U = I is a union
1 1 i
i∈SE1
of disjoint cylinder sets. Suppose that
(3.2) µIi∩ g−1(U)<qµ(Ii)+q·2−i for all i≥1.
 g∈F\n(i,1) 
The cylinder sets (I ) are pairwise disjoint, hence µ(U )≤qµ(U )+q.
i i∈E0 1 0
We want to apply the same procedure to U and so on to obtain a sequence of
1
uniformly open sets with almost exponentially decaying measure. So, in general,
let k ≥ 1 and suppose that U = I is a disjoint union of cylinder sets for
k−1 i
i∈ESk−1
an r.e. subset E . We let
k−1
Uk := Ii∩ g−1(Uk−1)= Ii∩ g−1(Ij)=
i∈[Ek−1 g∈F\n(i,k)  i∈[Ek−1 g∈F\n(i,k)j∈[Ek−1 
m(i,k)
= Ii∩ gi−,s1(Ijs),
i∈[Ek−1j1,...,jm[(i,k)∈Ek−1 s\=1 
whereg ,g ,...,g isthelistofalldistinctelementsofF . Thesequence
i,1 i,2 i,m(i,k) n(i,k)
of sets
m(i,k)
Ii∩ gi−,s1(Ijs)
s\=1
 i,j1,...,jm(i,k)
isuniformlyeffectivelyopen,soitfollowsthat(U ) isuniformlyeffectivelyopen.
k k≥1
Clearly, U∗ ⊆U ⊆U for every k ≥1. If we show that
k k−1
(3.3) µIi∩ g−1(Uk−1)<qµ(Ii)+qk·2−i for all i≥1
 g∈F\n(i,k) 