Basic facts concerning actions of amenable groups on compact spaces

19 Pages·2011·0.14 MB·English

BASIC FACTS CONCERNING ACTIONS OF
AMENABLE GROUPS ON COMPACT SPACES
TOMASZ DOWNAROWICZ
based on the seminal paper by
D. Ornstein and B. Weiss
Entropy and isomorphism theorems for actions of amenable groups
J. d’Anal. Math., 48 (1987), 1–141.
and a joined work with
GuoHua Zhang
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PRELIMINARIES ON AMENABLE GRUOPS
A group G is amenable if there exists a finitely additive
left-invariant probability measure on G. Abelian groups,
nilpotent groups, solvable groups, groups with polynomial
or subexponential growth are amenable. A group that con-
tains the free subgroup with two generators is not amenable.
Here we will use this equivalent definition:
DEFINITION 1. A countable, infinite, discrete group
G is called amenable if it has a Følner sequence i.e., a
sequence (F ) of finite sets F ⊂ G (n ≥ 1) satisfying,
n n≥1 n
for every g ∈ G, the condition
|F ∩ gF |
n n
−→ 1.
|F | n→∞
n
• gF = {gf : f ∈ F}
• | · | denotes the cardinality of a set
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A related very important notion is this:
DEFINITION 2: Let E be a finite subset of G
and choose δ ∈ (0,1). We will say that a finite set F is
(E,δ)-invariant if
|F△EF|
≤ δ,
|F|
• EF = {ef : e ∈ E,f ∈ F}
• △ denotes the symmetric difference
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Some trivial observations
• If E contains the unity of G then (E,δ)-invariance is
just the condition
|EF| ≤ (1 + δ)|F|.
• If a set F is (E,δ)-invariant, so is Fg, for every g ∈ G.
• It is clear, that if (F ) is a Følner sequence then for
n
every finite set E ⊂ G and every δ > 0, F is eventually
n
(i.e., for sufficiently large n) (E,δ)-invariant.
• If (F ) is a Følner sequence and E is a finite set then
n
both (EF ) and (E ∪ F ) are Følner sequences as well.
n n
(In this manner we can easily produce a Følner sequence
containing the unity.)
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DEFINITION 3: Fix an arbitrary (usually infinite) set
H ⊂ G. For every finite set F we will denote
|H ∩ Fg|
D (H) = inf
F
g∈G |F|
(notice that the multiplication by g is now on the right)
and we define
D(H) = sup{D (H) : F ⊂ G,|F| < ∞}.
F
D(H) will be called the lower Banach density of H.
LEMMA 1: If (F ) is a Følner sequence then for every
n
set H ∈ G we have
D(H) = lim D (H).
Fn
n→∞
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Proof. Fix some δ > 0 and let F be a finite set such that
D (H) ≥ D(H) − δ. Let n be so large that F is (F,δ)-
F n
invariant. Given g ∈ G, we have
|H ∩ Ffg|
≥ D (H),
F
|F|
for every f ∈ F . This implies that there are at least
n
D (H)|F||F | pairs (f′,f) with f′ ∈ F,f ∈ F such that
F n n
f′fg ∈ H. This in turn implies that there exists at least
one f′ ∈ F for which there are not less than D (H)|F |
F n
corresponding fs in F (see figure),
n
i.e., |H ∩ f′F g| ≥ D (H)|F |.
n F n
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Since f′ ∈ F and F is (F,δ)-invariant (and hence so is
n
F g), we have
n
|H ∩ f′F g| ≤ |H ∩ FF g| ≤ |H ∩ F g| + δ|F |,
n n n n
which yields
|H ∩ F g| ≥ (D (H) − δ)|F |.
n F n
We have proved that D (H) ≥ D (H) − δ ≥ D(H) − 2δ,
Fn F
which ends the proof. (cid:3)
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DEFINITION 4: Let {A } be a (possibly infinite) family
α
of finite sets. We will say that this family is ε-disjoint if
there exist pairwise disjoint sets A′ ⊂ A such that, for
α α
every α,
|A′ | ≥ (1 − ε)|A |.
α α
The following lemma plays the key role in many dynam-
ical constructions (entropy, topological entropy, symbolic
extensions, etc.)
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LEMMA 2: Let (F ) be a Følner sequence. Then for ev-
n
ery ε ∈ (0, 1) and n ∈ N there exist k ≥ 1 and some num-
2 0
bers n ≥ n ≥ ··· ≥ n = n +1, and sets C ,C ,...,C
k k−1 1 0 k k−1 1
contained in G such that the family
{F c : 1 ≤ i ≤ k, c ∈ C }
ni i
is ε-disjoint, and its union
k
H = F C
ni i
[
i=1
has lower Banach density larger than 1 − ε.
Proof. Too long!
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SUBADDITIVITY
Let us consider a a non-negative function f defined on finite
subsets of G.
We say that f is monotone if F ⊂ F′ =⇒ f(F) ≤ f(F′).
We say that f is left-invariant if f(F) = f(Fg) for any
g ∈ G.
We say that f is subadditive if f(F ∪F′) ≤ f(F)+f(F′).
EXAMPLES:
• Given a subset H ⊂ G, the function f(F) = sup |H∩Fg|
g
is non-negative, monotone, left-invariant and subadditive.
This function is used to define upper Banach density.
• In a classical dynamical system (X,T) the functions
f(F) = H(UF)
or
f(F) = H (PF)
µ
are non-negative, monotone, left-invariant and subadditive.