Table Of ContentENTROPY OF A SEMIGROUP OF MAPS FROM A
SET-VALUED VIEW
6
1
BINGZHEHOUANDXUWANG
0
2
Abstract. In this paper, we introduce a new entropy-like invariant, named
n
Hausdorffmetricentropy,forfinitelygeneratedsemigroupsactingoncompact
a
metricspacesfromaset-valuedviewandstudyitsproperties. Weestablishthe
J
relation between Hausdorff metric entropy andtopological entropy of asemi-
3
group defined by Biś. Some examples with positive or zero Hausdorff metric
1
entropy are given. Moreover, some notions of chaos are also well generalized
forfinitelygenerated semigroupsfromaset-valuedview.
]
S
D
.
h 1. Introduction
t
a
m Suppose that(X,d) is a compact metric space. Let F = {f1,f2,...,fp} be a
p-tuple of continuous maps from X to itself, and G be the semigroupgeneratedby
[
F. In this paper, we are interested in the dynamical system (X,F)(or (X,G,F)).
1 In classical discrete topological dynamics, the concept of topological entropy for
v
a continuous transformation plays an important role. This notion was introduced
4
by Adler, KonheimandMcAnderew in [1] as aninvariantoftopologicalconjugacy.
9
2 Later, Bowen [7] and Dinaburg [8] gave an equivalent description to this notion
3 on metric space, namely (Bowen’s) metric entropy. For a continuous map f on a
0 compactmetric space, denote h(f) the topologicalentropy (or equivalently, metric
.
1 entropy) of f. For finitely generated semigroups acting on compact metric spaces,
0 there had been some entropy-like invariants introduced. For instance, Friedland
6 introduced the entropy of a graph in [10], which is called Friedland entropy now;
1
Biś in [4] introduced topological entropy of a semigroup, inverse image entropy,
:
v preimage relation entropy and point entropy, and established their relations.
i Theaimofthepresentpaperistointroduceanewentropy-likeinvariant,named
X
Hausdorffmetricentropy,forfinitelygeneratedsemigroupsactingoncompactmet-
r
a ric spaces from a set-valued view. We will give the definition in next section and
studythebasicpropertiesofHausdorffmetricentropyinsection3. Insection4,we
will establish the relations between Hausdorff metric entropy, topological entropy
of a semigroup defined by Biś and max{h(f );f ∈ F}. In fact, two examples are
i i
given. One shows that the Hausdorff metric entropy of a tuple F can be positive
when each element in F has zero entropy; the other one shows that the Hausdorff
metric entropy of a tuple F can be zero when there exists an element in F with
positive entropy. At the last section, some remarks and problems are given. In
particular, some notions of chaos are also well generalized for finitely generated
semigroups.
2000 Mathematics Subject Classification. 37A35,37B40, 54H20,37C85.
Key words and phrases. topological entropy, semigroups, entropy-like invariants, Hausdorff
metric.
1
2 BINGZHEHOUANDXUWANG
2. Definition of Hausdorff metric entropy
Suppose that(X,d) is a compact metric space. Let F = {f ,f ,...,f } be a
1 2 p
p-tuple of continuous maps from X to itself, and G be the semigroupgeneratedby
F. Then for every n≥1,
Fn ={g ◦g ◦···◦g ;g ∈{f ,f ,...,f }, for all j =1,2,...,n.}
n n−1 1 j 1 2 p
isafinitesetofcontinuousmapsatmostpn. PutF0 ={id },whereid istheiden-
X X
tity map on X. Notice that for any point x ∈ X, F(x) = {f (x),f (x)...,f (x)}
1 2 n
isanon-emptycompactsubsetofX,wecangiveanewdefinitionofmetricentropy
from a set-valued view. First of all, let us review some notions with respect to
set-valued spaces.
Let
K(X)={K;K is a non−empty compact subset of X}.
Then the metric d on X induces a metric on K(X). For any A,B ∈K(X), define
dist(A,B)= supd(x,B)= sup inf d(x,y).
x∈A x∈Ay∈B
Furthermore, we define the Hausdorff metric d by
H
d (A,B)=max{dist(A,B),dist(B,A)}.
H
Moreover,(K(X),d )isacompactmetricspace. Inaddition,thetopologyinduced
H
by the Hausdorff metric d on K(X) coincides with the Vietoris topology [18].
H
Denote J the topology induced by the metric d on X. The Vietoris topology J is
V
generated by the base B consisting of sets of the form
m
B(U ,U ,...,U )={K ∈K(X);K ⊆ U , and K U 6=∅, 1≤i≤m.},
1 2 m [ i \ i
i=1
where U ,U ,...,U are non-empty open subsets of X.
1 2 m
Given any n≥0. For any x∈X,
Fn(x)={g ◦···◦g (x);g ∈{f ,f ,...,f }, for all j =1,2,...,n.}∈K(X).
n 1 j 1 2 p
Then Fn can be seemed as a continuous map from X to K(X). In fact, for any
subset A in X,
Fn(A)= Fn(a),
[
a∈A
In particular, if A ∈ K(X) we have Fn(A) ∈ K(X). Thus, F induced naturally a
map F :K(X)→K(X) defined by
e Fn(A)=Fn(A), for any A∈K(X).
Moreover, Fn is a conetinuous map from (K(X),d ) to itself. In addition, there
H
exists a nateural isometric embedding ϕ:X →K(X) defined by
ϕ(x)={x}, for every x∈X,
since for any x,y ∈X,
d(x,y)=d ({x},{y}).
H
Then Fn =Fn◦ϕ. The following lemma guaranteed the continuity of Fn (Fn).
e e
Lemma 2.1. Let (X,d) be a compact metric space and F ={f ,f ,...,f } be a p-
1 2 p
tupleof continuousmaps from X toitself. Then F :K(X)→K(X) is acontinuous
map. e
ENTROPY OF A SEMIGROUP OF MAPS FROM A SET-VALUED VIEW 3
Proof. Given arbitrary element B(U ,U ,...,U ) in the base B. It follows from
1 2 m
thecontinuityoff ,f ,...,f thatforeach1≤i≤mandeach1≤j ≤p,f−1(U )
1 2 p j i
is a non-empty open subset of X and consequently
p
F−1(U )= f−1(U )
i [ j i
j=1
is a non-empty open subset of X. Thus,
F−1B(U ,U ,...,U )=B(F−1(U ),F−1(U ),...,F−1(U ))∈B,
1 2 m 1 2 m
e
which implies F is a continuous map. (cid:3)
e
Now let us introduce a metric entropy for a finite set of continuous maps from
a set-valued view likes the concept of Bowen’s metric entropy. For any x,y ∈ X,
denote
dn(x,y)= max d (Fi(x),Fi(y))
H H
0≤i≤n
For any n ≥ 0 and any ǫ > 0, a subset M in X is said to be a Hausdorff metric
(n,ǫ)-spanning set of X with respect to F, if for each x ∈ X, there exists a point
y ∈M such that
dn(x,y)<ǫ.
H
It follows from the compactness of X and the continuity of F that,
r (n,ǫ,X,F)=min{Card(M);M is a Hausdorff metric(n,ǫ)−spanning set ofX}
H
is a finite positive integer.
Definition 2.2. Let
1
h (G,F)= lim limsup logr (n,ǫ,X,F).
H H
ǫ→0+ n→+∞ n
The quantity h (G,F) is called the Hausdorff metric entropy of a semigroup G
H
generated by F.
We can also describe the Hausdorff metric entropy of a semigroup G generated
by F interms ofHausdorffmetric (n,ǫ)-separatedsets. Namely, for anyn≥0 and
any ǫ >0, a subset E in X is said to be a Hausdorff metric (n,ǫ)-separated set of
X with respect to F, if for any distinct x,y ∈E,
dn(x,y)≥ǫ.
H
It follows from the compactness of X and the continuity of F that,
s (n,ǫ,X,F)=max{Card(E);E is a Hausdorff metric(n,ǫ)−separatedset ofX}
H
is a finite positive integer.
Proposition 2.3. For any semigroup G generated by a finite set F, the following
equality
1 1
lim limsup logr (n,ǫ,X,F)= lim limsup logs (n,ǫ,X,F)
H H
ǫ→0+ n→+∞ n ǫ→0+ n→+∞ n
holds.
4 BINGZHEHOUANDXUWANG
Proof. Let E be a Hausdorff metric (n,ǫ)-separated set of maximal cardinality.
Then E is also a Hausdorff metric (n,ǫ)-spanning set of X, and consequently
r (n,ǫ,X,F)≤s (n,ǫ,X,F).
H H
Now let M be a Hausdorff metric (n,ǫ/2)-spanning set of minimal cardinality.
Define a map ϕ : E →M by choosing for each x ∈ E, some point ϕ(x) ∈ M such
that
ǫ
dn(x,ϕ(x))< .
H 2
Then ϕ is injective and hence the cardinality of E is not greater than that of M,
i.e.,
ǫ
s (n,ǫ,X,F)≤r (n, ,X,F).
H H
2
Therefor, by
ǫ
r (n,ǫ,X,F)≤s (n,ǫ,X,F)≤r (n, ,X,F),
H H H
2
we have
1 1
lim limsup logr (n,ǫ,X,F)= lim limsup logs (n,ǫ,X,F).
H H
ǫ→0+ n→+∞ n ǫ→0+ n→+∞ n
(cid:3)
3. Fundamental properties of Hausdorff metric entropy
First of all, one can see when a finite set of continuous transformation contains
only one element, the Hausdorffmetric entropy is actually the classicaltopological
entropy.
Proposition 3.1. Let (X,d) be a compact metric space and G be a semigroup
generated by F ={f}. Then h (G,F)=h(f).
H
Proof. For any x ∈ X and any n > 0, we have Fn(x) = {fn(x)}. Then for any
x,y ∈X,
d(Fn(x),Fn(y))=d ({fn(x)},{fn(y)})=d(fn(x),fn(y)).
H
Thus, h (G,F)=h(f). (cid:3)
H
Next, similar to the research of the topological entropy of a single map, we will
study the basic properties of the Hausdorff metric entropy for finitely generated
semigroups.
Proposition 3.2. Let (X,d) be a compact metric space and G be a semigroup
generated by a finite set F ={f ,f ,...,f }. Then h (G,F)≥0.
1 2 p H
Proof. It follows from r (n,ǫ,X,F)>0 that
H
1
h (G,F)= lim limsup logr (n,ǫ,X,F)≥0.
H H
ǫ→0+ n→+∞ n
(cid:3)
Proposition 3.3. For every positive integer m, h (G,Fm)=m·h (G,F).
H H
ENTROPY OF A SEMIGROUP OF MAPS FROM A SET-VALUED VIEW 5
Proof. Firstly, one can see that G is also the semigroup generated by Fm. Given
any m > 0, n > 0 and ǫ > 0. A Hausdorff metric (mn,ǫ)-spanning set of X with
respect to F must be a Hausdorff metric (n,ǫ)-spanning set of X with respect to
Fm, then
r (n,ǫ,X,Fm)≤r (mn,ǫ,X,F).
H H
Consequently,
1 m
logr (n,ǫ,X,Fm)≤ logr (mn,ǫ,X,F).
H H
n mn
Thus,
h (G,Fm)≤m·h (G,F).
H H
Onthe otherhand,itfollowsfromthe uniformcontinuityofF(F )thatforany
ǫ > 0, there exists δ > 0 such that if d(x,y) < δ, dm(x,y) < ǫ. Theen a Hausdorff
H
metric (n,δ)-spanning set of X with respect to Fm must be a Hausdorff metric
(mn,ǫ)-spanning set of X with respect to F, and consequently
r (n,δ,X,Fm)≥r (mn,ǫ,X,F).
H H
Thus,
h (G,Fm)≥m·h (G,F).
H H
(cid:3)
Definition 3.4. Let (X,d) be a compact metric space and G be a semigroup
generated by a finite set F ={f ,f ,...,f }. Y ⊆X is called an invariant subset
1 2 p
of X, if for any y ∈ Y, F(y) ⊆ Y, i.e., F(Y) ⊆ Y. Moreover, if Y ⊆ X is an
invariant closed subset of X, we call (Y,F| ) is a subsystem of (X,F).
Y
Proposition 3.5. Let (X,d) be a compact metric space and G be a semigroup
generated by a finite set F = {f ,f ,...,f }. If Y ⊆ X is an invariant closed
1 2 p
subset of X, then h (G| ,F| )≤h (G,F).
H Y Y H
Proof. Given any n > 0 and ǫ > 0. A Hausdorff metric (n,ǫ)-separated set of
Y with respect to F| must be a Hausdorff metric (n,ǫ)-separated set of X with
Y
respect to F, then
s (n,ǫ,Y,F| )≤s (n,ǫ,X,F).
H Y H
Consequently,
1 1
logs (n,ǫ,Y,F| )≤ logs (n,ǫ,X,F).
H Y H
n n
Thus,
h (G| ,F| )≤h (G,F).
H Y Y H
(cid:3)
Definition 3.6. Let (X,d) and (Y,ρ) be two compact metric spaces. Let G and
G′ be two semigroups generated by finite set F = {f ,f ,...,f } and finite set
1 2 p
F′ ={f′,f′,...,f′}respectively. IfthereexistsacontinuoussurjectiveT :X →Y
1 2 q
such that for any x∈X,
{Tf (x),Tf (x)...,Tf (x)}={f′T(x),f′T(x)...,f′T(x)}, i.e.TF(x)=F′(Tx),
1 2 p 1 2 p
then T is called a topological semiconjugacy from (X,F) to (Y,F′). Moreover, if
T is a homeomorphism, we call T a topological conjugacy from (X,F) to (Y,F′).
6 BINGZHEHOUANDXUWANG
Theorem 3.7. Let (X,d) and (Y,ρ) be two compact metric spaces. Let G and
G′ be two semigroups generated by finite set F = {f ,f ,...,f } and finite set
1 2 p
F′ = {f′,f′,...,f′} respectively. If there exists a topological semiconjugacy T
1 2 q
from (X,F) to (Y,F′), then h (G,F) ≥ h (G′,F′). Moreover, T is a topological
H H
conjugacy, then h (G,F)=h (G′,F′).
H H
Proof. For any n > 0, it follows from the uniform continuity of T that for any
ǫ > 0, there exsits δ > 0 such that if dn(x,y) < δ, dn(Tx,Ty) < ǫ. Then, if M
H H
is a Hausdorff metric (n,δ)-spanning set of X with respect to F, T(M) must be a
Hausdorff metric (n,ǫ)-spanning set of Y with respect to F′, and consequently
r (n,δ,X,F)≥r (n,ǫ,Y,F′).
H H
Thus,
h (G,F)≥h (G′,F′).
H H
Notice that if T is a topological conjugacy from (X,F) to (Y,F′), T−1 is a
topological conjugacy from (Y,F′) to (X,F). Therefor,
h (G,F)=h (G′,F′)
H H
(cid:3)
Let (X,d) and (Y,ρ) be two compact metric spaces. Let G and G′ be two
semigroups generated by finite sets F = {f ,f ,...,f } and F′ = {f′,f′,...,f′}
1 2 p 1 2 q
respectively. NowconsidertheCartesianproductspaceX×Y withmetricγ defined
by, for any (x,y),(u,v)∈X ×Y,
γ((x,y),(u,v))=max{d(x,u),ρ(y,v)}.
Denote
F ×F′ ={g×g′;g ∈F,g ∈F′}.
ThenF×F′ isafinitesetofcontinuousmapsonX×Y,anditiseasytoseeforany
n>0, (F ×G)n =Fn×Gn. Moreover,denote G×G′ the semigroupgeneratedby
F ×F′ and h (G×G′,F ×F′) the Hausdorffmetric entropy ofCartesianproduct
H
system (X ×Y,F ×F′).
Lemma 3.8. For any A×B,C×D ∈K(X)×K(Y)⊆K(X ×Y), we have
γ (A×B,C×D)=max{d (A,C),ρ (B,D)}.
H H H
Proof. Firstly,wewillproveforany(x,y)∈X×Y andanyE×K ∈K(X)×K(Y),
γ((x,y),E×K)=max{d(x,E),ρ(y,K)}.
One hand, since E×K is a compact subset of X×Y, there exists a point (e,k)∈
E×K such that
γ((x,y),E×K)=γ((x,y),(e,k))=max{d(x,e),ρ(y,k)}≥max{d(x,E),ρ(y,K)}.
On the other hand , since E and K are compact subsets in X and Y respectively,
there exist e′ ∈E and k′ ∈K such that
d(x,E)=d(x,e′), ρ(y,K)=ρ(y,k′).
Then
max{d(x,E),ρ(y,K)}=max{d(x,e′),ρ(y,k′)}
=γ((x,y),(e′,k′))
≥γ((x,y),E×K).
ENTROPY OF A SEMIGROUP OF MAPS FROM A SET-VALUED VIEW 7
Therefor,
γ((x,y),(E,K))=max{d(x,E),ρ(y,K)}
and consequently,
γ (A×B,C×D)
H
=max{ sup γ((a,b),C×D), sup γ((c,d),A×B)}
(a,b)∈A×B (c,d)∈C×D
=max{ sup max{d(a,C),ρ(b,D)}, sup max{d(c,A),ρ(d,B)}}
(a,b)∈A×B (c,d)∈C×D
=max{dist (A,C),dist (B,D),dist (C,A),dist (D,B)}
d ρ d ρ
=max{d (A,C),ρ (B,D)}.
H H
(cid:3)
Theorem 3.9. The equality
h (G×G′,F ×F′)=h (G,F)+h (G′,F′)
H H H
holds.
Proof. By above lemma, for any (a,b), (c,d)∈X ×Y and any n>0,
γ ((F ×G)n(a,b),(F ×G)n(c,d))=max{d (Fn(a),Fn(c)),ρ (Gn(b),Gn(d))}.
H H H
Then
γn((a,b), (c,d))=max{dn(a,c),ρn(b,d)}.
H H H
One hand, if M is a Hausdorff metric (n,ǫ)-spanning set of X with respect to
F and M′ is a Hausdorff metric (n,ǫ)-spanning set of Y with respect to F′, then
M×M′ is a Hausdorffmetric (n,ǫ)-spanningset of X×Y with respectto F ×F′,
and consequently
r (n,ǫ,X ×Y,F ×F′)≤r (n,ǫ,X,F)·r (n,ǫ,Y,F′),
H H H
Therefor,
h (G×G′,F ×F′)≤h (G,F)+h (G′,F′).
H H H
On the other hand, if E is a Hausdorff metric (n,ǫ)-separated set of X with
respect to F and E′ is a Hausdorff metric (n,ǫ)-separatedset of Y with respect to
F′, then E×E′ is a Hausdorffmetric (n,ǫ)-separatedsetof X×Y with respect to
F ×F′, and consequently
s (n,ǫ,X×Y,F ×F′)≥s (n,ǫ,X,F)·s (n,ǫ,Y,F′),
H H H
Therefor,
h (G×G′,F ×F′)≥h (G,F)+h (G′,F′).
H H H
Thus,
h (G×G′,F ×F′)=h (G,F)+h (G′,F′).
H H H
(cid:3)
8 BINGZHEHOUANDXUWANG
4. Further discussions
We will give an example to show that the Hausdorff metric entropy of a tuple
F can be positive when each element in F has zero entropy. In fact, the following
example is contained in a result of Biś and Urbański (Theorem 4.1 in [5], where
they discussed Biś’s topological entropy of a semigroup).
Example 4.1. Denote X by the unit interval [0,1]. Let F = {f ,f }, where
1 2
f ,f :X →X defined by
1 2
x, if 0≤x≤1/3
f (x)= 3x−2/3, if 1/3≤x≤4/9
1
3x/5+2/5, if 4/9≤x≤1
and
3x/5, if 0≤x≤5/9
f (x)= 3x−4/3, if 5/9≤x≤2/3
2
x, if 2/3≤x≤1
Then
h(f )=h(f )=0 and h (G,F)≥log2>0.
1 2 H
Denote Y the subinterval [1/3,2/3]. Then f−1(Y) = [1/3,4/9]⊆ Y, f−1(Y) =
1 2
[5/9,2/3] ⊆ Y and d(f−1(Y),f−1(Y)) = 1/9. Fix ǫ ∈ (0,1/15). Since every map
1 2
g : X → X, g ∈ G, is a homeomorphism, one can select for every g ∈ G exactly
one point z ∈ g−1(Y). For every n ≥ 0, consider the set A = {z ;g ∈ Fn}. We
g n g
willshowthatA isaHausdorffmetric(n,ǫ)-separatedsetconsistingofexactly2n
n
elements.
Nowitsufficestoprovethat,fortwoarbitraryelementsg 6=hinFn,dn(z ,z )≥
H g h
1/15. Writeg =g ◦g ◦···◦g andh=h ◦h ◦···◦h ,whereg ,h ∈{f ,f }
n n−1 1 n n−1 1 j j 1 2
for all j =1,2,...,n. Since g 6=h, there exists k ∈{1,2,...,n} such that g =h ,
1 1
g =h , ..., g =h and g 6=h . Then
2 2 k−1 k−1 k k
g ◦···◦g (z )∈g−1(Y)
k−1 1 g k
and
g ◦···◦g (z )∈h−1(Y).
k−1 1 h k
Hence,
1 1
d(g ◦···◦g (z ),g ◦···◦g (z )≥ > >ǫ.
k−1 1 g k−1 1 h
9 15
Givenanyw =w ◦w ◦···◦w ∈Fn,wherew ∈{f ,f }forallj =1,2,...,n.
n n−1 1 j 1 2
There are two cases as follows.
Case 1. g =w , g =w , ..., g =w . Then
1 1 2 2 k−1 k−1
w ◦···◦w (z )=g ◦···◦g (z )
k−1 1 g k−1 1 g
and
w ◦···◦w (z )=h ◦···◦h (z ).
k−1 1 h k−1 1 h
Case 2. there exists t ∈ {1,2,...,k − 1} such that g = w , g = w , ...,
1 1 2 2
g =w and g 6=w . Then
t−1 t−1 t t
g ◦···◦g (z )∈g−1(Y) and g ◦···◦g (z )∈g−1(Y),
t−1 1 g t t−1 1 h t
and consequently
11 4 1 4
w ◦g ◦···◦g (z ), w ◦g ◦···◦g (z )∈[ , ] [ , ].
t t−1 1 g t t−1 1 h 15 5 [ 5 15
ENTROPY OF A SEMIGROUP OF MAPS FROM A SET-VALUED VIEW 9
Furthermore,
11 4
w ◦···◦w ◦w ◦···◦w (z ),w ◦···◦w ◦w ◦···◦w (z )∈[ ,1] [0, ].
k−1 t t−1 1 g k−1 t t−1 1 h 15 [ 15
This implies
11 4 1
d(g ◦···◦g (z ),w ◦···◦w (z )≥d(Y,[ ,1] [0, ])= >ǫ.
k−1 1 g k−1 1 h 15 [ 15 15
and
11 4 1
d(g ◦···◦g (z ),w ◦···◦w (z )≥d(Y,[ ,1] [0, ])= >ǫ.
k−1 1 h k−1 1 g 15 [ 15 15
Therefor,
d (Fk−1(z ),Fk−1(z ))>ǫ,
H g h
whichimpliesthatz andz areHausdorffmetric(n,ǫ)-separated. Noticethatthe
g h
map g →z is bijective, then
g
s (n,ǫ,X,F)≥Card(Fn)=2n.
H
In consequence,
h (G,F)≥log2>0.
H
Now we will discuss the relation between Hausdorff metric entropy and Biś’s
topological entropy of a semigroup. Let us review the concept of the topological
entropyh(G,F) defined by Biś in [4]. Let (X,d) be a compact metric space and G
be a semigroup generated by a finite set F = {f ,f ,...,f }. In [4], Biś assumed
1 2 p
the identity map is in F. We will show an equivalent version of this definition
without the assumption. Following [11] we will say that two points x,y ∈ X are
(n,ǫ)-separatedby G (with respect to the metric dn ) if there exists g ∈Fn such
max
that d(g(x),g(y))≥ǫ, e.g.
dn (x,y)=max{dk(x,y);0≤k ≤n}≥ǫ,
max
where
dk(x,y)=max{d(g(x),g(y));g ∈Fk}.
A subset A of X is (n,ǫ)-separated if any two distinct points of A have this
property. Write
s(n,ǫ,X)=max{Card(A);A is an (n,ǫ)−separated subest of X},
and consequently, define
1
h(G,F)= lim limsup logs(n,ǫ,X).
ǫ→0+ n→+∞ n
Thetopologicalentropyh(G,F)alsocanbedescribedby(n,ǫ)-spanningsubset.
A subset B of X is called (n,ǫ)-spanning if for any x ∈ X, there exists a point
a∈A such that
dn (x,a)<ǫ.
max
Write
r(n,ǫ,X)=min{Card(A);A is an (n,ǫ)−spanning subest of X}.
One can see
1
h(G,F)= lim limsup logr(n,ǫ,X).
ǫ→0+ n→+∞ n
10 BINGZHEHOUANDXUWANG
Notice that for any x,y ∈X and k ∈N,
d (Fk(x),Fk(y))≤dk(x,y)
H
and consequently
dk (x,y)≤dk (x,y).
H max
Then an (n,ǫ)-spanning subset must be a Hausdorff metric (n,ǫ)-spanning subset.
Thus, we have the following result.
Proposition 4.2. Let (X,d) be a compact metric space and G be a semigroup
generated by a finite set F ={f ,f ,...,f }. Then
1 2 p
h (G,F)≤h(G,F).
H
The following example will tell us that h (G,F) could be strictly less than
H
h(G,F). In fact, it shows that the Hausdorff metric entropy of a tuple F can be
zero when there exists an element in F with positive entropy.
Example 4.3. Let f be a minimal homeomorphism with positive topological en-
tropy on a compact metric space X. Let F = {f,id } and G be the semigroup
X
generated by F. Then h (G,F)=0.
H
The theoremofJewett-Krieger[14,15]assuresthe existence ofminimalsystems
with positive topological entropy, more concrete examples can be found in [13, 12,
19, 3]. Now it suffices to prove the following result.
Proposition 4.4. Let f be a minimal homeomorphism on a compact metric space
X. Let F ={f,id } and G be the semigroup generated by F. Then h (G,F)=0.
X H
Proof. Notice that for any x∈X and any k ∈N,
Fk(x)={x,f(x),...,fk(x)}
and for any n≥k,
Fn(x)⊇Fk(x).
Givenany ǫ>0. For eachx∈X,by the minimality off, there exists a positive
integer N such that FNx(x) is an ǫ/2-net, i.e.,
x
ǫ
d (FNx(x),X)< .
H
2
Accordingto the compactness of X and the continuity of F, there exists a positive
integer N such that e
ǫ
d (FN(x),X)< , for any x∈X.
H
2
Furthermore, for any x,y ∈X and n≥N,
d (Fn(x),Fn(y))<ǫ.
H
Now one can see that a Hausdorff metric (N,ǫ)-spanning set is also a Hausdorff
metric (n,ǫ)-spanning set for every n≥N. Thus
1
h (G,F)= lim limsup r (n,ǫ,X,F)=0.
H H
ǫ→0+ n→+∞ n
(cid:3)
