Entropy Formula for Random $\mathbb{Z}^k$-actions

0.3 MB·

by  Yujun Zhu

Zk
Entropy Formula for Random -actions
Yujun Zhu
College of Mathematics and Information Science, and
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1 Hebei Key Laboratory of Computational Mathematics and Applications,
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2 Hebei Normal University,Shijiazhuang, Hebei, 050024, P.R.China

n
a
J
Abstract: In this paper, entropies, including measure-theoretic entropy and topo-
3
logicalentropy,areconsideredforrandomZk-actionswhicharegeneratedbyrandom

] compositions of the generators of Zk-actions. Applying Pesin’s theory for commuta-
S
D tive diffeomorphisms we obtain a measure-theoretic entropy formula of C2 random
h. Zk-actionsviathe Lyapunovspectraofthe generators. Someformulasandboundsof
at topologicalentropyforcertainrandomZk(orZk+ )-actionsgeneratedbymoregeneral
m maps, such as Lipschitz maps, continuous maps on finite graphs and C1 expanding
[
maps,arealsoobtained. Moreover,asanapplication,wegiveaformulaofFriedland’s

1 entropy for certain C2 Zk-actions.
v
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1 Introduction
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5
0
It is well known that entropies, including measure-theoretic entropy and topological entropy, are
0
. importantinvariantswhichplayimportantrolesinthe study ofcomplexityofthe dynamicalsystems.
1
0 Indeterministicsystems,oneusuallyconsiderthebehavioroftheiterationsofasingletransformation.
7
1 In this paper we are concerned with the entropies of random Zk(k 1)-actions which are generated
≥
v: by random compositions of the generators of Zk-actions.
Xi Throughoutthispaper,weassumethat(X,d)isacompactmetricspaceandC0(X,X)isthespace
r of continuous maps on X equipped with the C0-topology. When X = M is a closed (i.e., compact
a
and boundaryless) C Riemannian manifold, we denote Cr(X,X) or Cr(M,M)(r 1) the space of
∞
≥
Cr maps equipped with the Cr-topology.
Let α:Zk Cr(X,X) (r 0) be a Cr Zk-action on X. We denote the collection of generators
−→ ≥
of α by
= f =α(~e ):1 i k , (1.1)
α i i
G { ≤ ≤ }
(i)
where ~ei = (0, , 1, ,0) is the standard i-th generator of Zk. Clearly, all fi,1 i k, are
··· ··· ≤ ≤
2010 Mathematics Subject Classification: 37A35, 37C85,37H99.
Keywords and phrases: entropyformula;randomZk-action;Lyapunov exponent; Friedland’sentropy.
The author is supported by NSFC(No:11371120), NSFHB(No:A2014205154), BRHB(No:BR2-219) and GC-
CHB(No:GCC2014052).
1
homeomorphisms when r =0, and are Cr diffeomorphisms when r 1. Let
≥
Z ∞
Ω= = (1.2)
Gα Gα
−Y∞
be the infinite productof ,endowedwiththe producttopologyandthe productBorelσ-algebra ,
α
G A
and let σ be the left shift operator on Ω which is defined by (σω) =ω for ω =(ω ) Ω. Given
n n+1 n
∈
ω =(ω ) Ω, we write f =ω and
n ω 0
∈
fσn−1ω fσω fω n>0
◦···◦ ◦
fn := id n=0
ω
 fσ−n1ω◦···◦fσ−−12ω◦fσ−−11ω n<0.
Clearly, each ω induces a nonautonomous dynamical system generated by the sequence of maps
fσiω i Z. There is a natural skew product transformation Ψ : Ω X Ω X over (Ω,σ) which
{ }∈ × −→ ×
is defined by
Ψ(ω,x)=(σω,f (x)). (1.3)
ω
Let α:Zk Cr(X,X) (r 0) be a Cr Zk-actionon X and ν a probability measure on . We
α
−→ ≥ G
can define a probability measure P =νZ on Ω which is invariant and ergodic with respect to σ. By
ν
the induced Cr(r 0) random Zk-action f over (Ω, ,P ,σ) we mean the system generated by the
ν
≥ A
randomly composed maps fn, n Z. It is also called a Cr i.i.d. (i.e., independent and identically
ω ∈
distributed) random Zk-action. We are interested in dynamical behaviors of these actions for P -a.e.
ν
ω or on the average on ω.
When replacing “Zk” and “n Z” in the above definition of random Zk-actions by “Zk”(here
∈ +
Z = 0,1,2, ) and “n Z ” respectively, we can define random Zk-actions similarly.
+ { ···} ∈ + +
The main task of this paper is to investigate the entropies, including measure-theoretic entropy
and topological entropy, of random Zk(or Zk )-actions. A measure-theoretic entropy formula of C2
+
randomZk-actionsisgiveninSection2. Someformulasandbounds oftopologicalentropyforcertain
random Zk(or Zk )-actions are obtained in Section 3. Moreover,as an application, we give a formula
+
of Friedland’s entropy for certain C2 Zk-actions in Section 4. In the remaining of this section, we
introduce the basic notions and state the main results.
1.1 Measure-theoretic entropy of random Zk-actions
Inthis subsectionandthe nextsubsection,we considerthe measure-theoreticentropyandtopological
entropy ofrandomZk-actionsrespectively. The basic notions are derivedfromKifer [14] andLiu [20]
inwhichthe ergodictheoryofgeneralrandomdynamicalsystemsaresystematicallyinvestigated. We
can see [5] for some progress in the research of entropy for random Zk-actions.
Let α:Zk Cr(X,X) (r 0) be a Cr Zk-actionon X. A Borelprobability measure µ on X is
−→ ≥
called α-invariant (resp. ergodic) if it is invariant (resp. ergodic) with respect to each f ,1 i k.
i
≤ ≤
Letν beaprobabilitymeasureon andf theinducedrandomZk-actionover(Ω, ,P ,σ). ABorel
α ν
G A
probability measure µ on X is called f-invariant if
µ(fω−1A)dPν(ω)=µ(A)
ZΩ
for all Borel A X. Clearly, an α-invariant measure must be f-invariant.
⊂
2
Let = P be a finite or countable partition of Ω X into measurable sets then = P ,
k ω k,ω
P { } × P { }
here P = x X :(ω,x) P , is a partition of X.
k,ω k
{ ∈ ∈ }
Definition1.1. Letα:Zk Cr(X,X)(r 0)beaCr Zk-actiononX andν aprobability measure
−→ ≥
on . Let f be the induced random Zk-action over (Ω, ,P ,σ) and µ an f-invariant measure. For
α ν
G A
a finite or countable Borel partition of Ω X, the limit
P ×
n 1
1 −
h (f, ):= lim H (fi) 1 dP (ω), (1.4)
µ P n n µ ω − Pω ν
→∞ ZΩ i=0
(cid:0) _ (cid:1)
where H ( ):= µ(A)logµ(A) for a finite or countable partition of X, exists. The number
µ
Q − Q
A
∈Q
P
h (f):=suph (f, ),
µ µ
P
P
where ranges over all finiteor countablepartitions of Ω X,is called the measure-theoreticentropy
P ×
of f.
In fact, since (Ω, ,P ,σ) is ergodic and measurably invertible, taking integration in (1.4) is
ν
A
extraneous and the limit exists and is constant for P -a.e. ω. However, when considering a random
ν
Zk+-action (here Z+ = {0,1,2,···}) over (Ω,A,Pν,σ), where Ω = GαZ+ = ∞0 G for the set G of
generators with respect to a Zk-action α : Zk Cr(X,X), we must keep the integration in the
+ + −→ Q
corresponding definition.
Remark 1.2. Usually, we often use the following equivalent definition of h (f),
µ
h (f)=suph (f, ),
µ µ
P
P
where ranges over all finite partitions of Ω X in form of Ω in which is a finite partition
P × Q× Q
of X.
It is well known that Pesin’s entropy formula plays an important role in smooth ergodic theory,
including the deterministic case and the random case. For the deterministic case we refer to [27],
[26], [22] [18] and [19], and for the random case, we refer to [2], [3], [20] and [21]. It gives an explicit
formularelatingthemeasure-theoreticentropyandLyapunovexponentsincorrespondingsettings. In
particular, for a general C2 i.i.d. random dynamical system f which is defined by replacing “ ” by
α
G
“Diff2(X)” (i.e., the set of C2 diffeomorphisms) in the above notations for random Zk-action, if
log+ f ,log inf detD f L1(Ω,P ),
ω C2 x ω ν
k k x M| |∈
∈
then for any f-invariant measure µ which is absolutely continuous with respect to the Lebesgue
measure of X, the following entropy formula
h (f)= λ (x)d (x)dµ(x) (1.5)
µ i i
ZXλiX(x)>0
holds, where (λ (x),d (x)) is the essentially non-random Lyapunov spectrum of f. Though the
i i
{ }
Lyapunov spectrum of f is essentially non-random, we can not expect to give some explicit relation
betweenitandthespectraoftheelementsinDiff2(X). Now,foraC2 randomZk-actionf, consists
α
G
of finite elements which are pairwise commutative. A natural question is
3
Question 1. Is there an entropy formula via the Lyapunov spectra of the generators ?
Themainaimofthispaperistogiveapositiveansweroftheabovequestionundertheassumption
that the measure µ is α-invariant and is absolutely continuous with respect to the Lebesgue measure
of X. Hence the measure-theoretic entropy of such C2 random Zk-actions is, in a sense, easer to be
calculated.
Section 2 is devoted to the proof of a measure-theoretic entropy formula, i.e., the main result of
this paper.
Theorem 1. Let α : Zk C2(M,M) be a C2 Zk-action on a d-dimensional closed Riemannian
−→
manifold M and ν a probability measure on . Then for the induced random Zk-action f over
α
G
(Ω, ,P ,σ)andanyα-invariantmeasureµwhichisabsolutelycontinuouswithrespecttotheLebesgue
ν
A
measure m, we have the following entropy formula
k
h (f)= max ν d (x)λ (x)dµ(x), (1.6)
µ i j i,j
ZMJ⊂{1,···,s(x)}j J i=1
X∈ X
where ν = ν(f ) and (λ (x),d (x)) : 1 i k,1 j s(x) is the spectrum of α (see Definition
i i i,j j
{ ≤ ≤ ≤ ≤ }
2.2 for the precise definition).
In particular, if µ is α-ergodic, then we have
k
h (f)= max ν d λ . (1.7)
µ i j i,j
J 1, ,s
⊂{ ··· }j J i=1
X∈ X
(Note that when µ is α-ergodic, then λ (x),d (x) and s(x) are all constant and we denote them by
i,j j
λ ,d and s, respectively.)
i,j j
The strategy to prove Theorem 1 is to adapt the proof of Ruelle’s inequality in Theorem S2.13 of
[13] and the proof of the reverse inequality in section 13 of [23] for any C2 diffeomorphisms to oue
case. In the proof, Pesin’s theory for commutative C2 diffeomorphisms developed by Hu ([11]) and
Birkhoff’s Ergodic Theorem play important roles.
1.2 Topological entropy of random Zk and Zk-actions
+
Let α : Zk (resp. Zk) Cr(X,X) (r 0) be a Cr Zk (resp. Zk)-action on X, ν a probability
+ −→ ≥ +
measure on and f the induced randomZk (resp. Zk)-action over(Ω, ,P ,σ). Define a family of
Gα + A ν
metrics dn :n Z ,ω Ω on X by
{ ω ∈ + ∈ }
dn(x,y)= max d(fi(x),fi(y))
ω ω ω
0 i n 1
≤≤ −
forx,y X. Letε>0,ω ΩandK beasubsetofX. AsetF X issaidtobean(ω,n,ε)-spanning
∈ ∈ ⊂
set ofK ifforanyx K thereexistsy F suchthatdn(x,y) ε.Letr(ω,n,ε,K)denotethesmallest
∈ ∈ ω ≤
cardinality of any (ω,n,ε)-spanning set of K. A subset E K is said to be an (ω,n,ε)-separated set
⊂
of K if x,y E,x = y implies dn(x,y) > ε. Let s(ω,n,ε,K) denote the largest cardinality of any
∈ 6 ω
(ω,n,ε)-separated set of K.
Definition 1.3. Let f be a random Zk (or Zk)-action over (Ω, ,P ,σ) as above. Then the limit
+ A ν
1
h(f):= limlimsup logr(ω,n,ε,X)dP (ω) (1.8)
ν
ε→0 n→∞ nZΩ
exists. The number h(f) is called the topological entropy of f.
4
From [15] we have
1
h(f)= limlimsup logr(ω,n,ε,X)dP (ω). (1.9)
ν
ZΩε→0 n→∞ n
Moreover, we can get the same value of h(f) when we replace “limsup” by “liminf”, or replace
“r(ω,n,ε,X)” by “s(ω,n,ε,X)” in (1.8) and (1.9). If we denote the topological entropy of the
nonautonomous dynamical system {fσiω}i∈Z+ by h(f,ω), then by [17],
1
h(f,ω)= limlimsup logr(ω,n,ε,X),
ε 0 n n
→ →∞
and hence
h(f)= h(f,ω)dP (ω). (1.10)
ν
ZΩ
Generally, we have the following inequality relating topological entropy and measure-theoretic
entropy,
h(f) suph (f), (1.11)
µ
≥ µ
whereµrangesoverallf-invariantmeasures. (1.11)iscalledavariational principle. Formoregeneral
random dynamical systems, (1.11) may hold as an equality, see Theorem 3.1 in [21] for example. In
general, to give an explicit formula of topological entropy, even for a deterministic system, is not an
easy work. One often give the lower or upper bounds of topological entropy via various topological
quantities, such as degrees and volume growth rates, see [13] and [8] for example.
In section 3, we give some formulas and bounds of topological entropy for certain random Zk(or
Zk)-actions generated by more general maps, such as Lipschitz maps (Proposition 3.2), continuous
+
maps on finite graphs and C1 expanding maps (Proposition3.4). By (1.10), topologicalentropy h(f)
is the integral of that of nonautonomous dynamical systems h(f,ω),ω Ω. By some known results
∈
abouttopologicalentropyof nonautonomousdynamicalsystems andBirkhoff’s ErgodicTheorem,we
get the following formulas and bounds.
Proposition. Let α:Zk Cr(X,X) (r 0) be a Cr Zk-action on X, ν a probability measure on
+ −→ ≥ +
with ν =ν(f ), and f the induced random Zk-action over (Ω, ,P ,σ).
Gα i i + A ν
(1) If the generators f ,1 i k, are all Lipschitz maps with the Lipschitz constants L(f ),1
i i
≤ ≤ ≤
i k, respectively, then we have
≤
k
h(f) D(X) ν logL+(f ),
i i
≤
i=1
X
where L+(f )=max 1,L(f ) and D(X) is the ball dimension of X.
i i
{ }
(2) If X is a finitegraph and the generators f ,1 i k, are all homeomorphisms, then h(f)=0.
i
≤ ≤
(3) If X = M is a closed oriented Riemannian manifold and the generators f ,1 i k, are all
i
≤ ≤
C1 expanding maps, then we have
k
h(f)= ν log deg(f ),
i i
| |
i=1
X
where deg(f ) is the degree of f .
i i
5
1.3 Friedland’s entropy of Zk-actions
Friedland’s entropy of Zk-actions was introduced by Friedland [9] via the topological entropy of the
shift map on the induced orbit space. More precisely, let α:Zk Cr(X,X)(r 0) be a Zk-action
−→ ≥
on X with the generators f k . Define the orbit space of α by
{ i}i=1
Xα = x¯={xn}n∈Z ∈ X : for any n∈Z,fin(xn)=xn+1 for some fin ∈{fi}ki=1 .
n Z
(cid:8) Y∈ (cid:9)
This is a closed subset of the compact space X and so is again compact. A natural metric d¯on
n Z
Xα is defined by Q∈
d¯ x¯,y¯ = ∞ d(xn,yn) (1.12)
2n
n= | |
(cid:0) (cid:1) X−∞
for x¯= xn n Z,y¯= yn n Z Xα. We can define a shift map
{ } ∈ { } ∈ ∈
σα :Xα Xα, σα( xn n Z)= xn+1 n Z.
→ { } ∈ { } ∈
Thus we have associated a Z-action with the Zk-action α.
Definition 1.4. Friedland’sentropy of a Zk-action α is defined by the topological entropy of the shift
map σ :X X , i.e.,
α α α
→
1
h(σα)=εlim0linmsupnlog sd¯(σα,n,ε,Xα), (1.13)
→ →∞
where sd¯(σα,n,ε,Xα) is the largest cardinality of any (σα,n,ε)-separated sets of Xα.
Replacing“Z”by“Z ”intheabovestatement,wecangetthedefinitionofFriedland’sentropyfor
+
Zk-actions. Unlike the classical entropy for Zk (or Zk)-actions, Friedland’s entropy is positive when
+ +
thegeneratorshavefiniteentropyassingletransformations. FromtheknownresultsaboutFriedland’s
entropy,we canseethatit isnotaneasytaskto compute it, evenforsome“simple”examples. In[9],
Friedlandconjectured thatfor a Z2-actionon the circle S1 whose generatorsare T =px(mod 1) and
+ 1
T =qx(mod 1), where p and q are two co-prime integers, its entropy
2
h(σ )=log(p+q).
α
Soon afterwards Geller and Pollicott [10] answered this conjecture affirmatively under a weaker con-
dition “p,q are different integers greater than 1”. In [7] and [34], the formulas of Friedland’s entropy
forthe “linear”Zk andZk-actionsonthe torusviathe eigenvaluesofthe generatorsaregivenrespec-
+
tively. We can see that all the known formulas of Friedland’s entropy as we mentioned above were
obtained for special (“linear”) actions on simple manifolds (tori). A natural question is
Question 2. Is there a formula of Friedland’s entropy for general smooth Zk and Zk-actions via the
+
Lyapunov spectra of the generators ?
In section 4, applying the entropy formula (1.7) for random Zk-actions and use the techniques,
especially the variational principle for pressures and Birkhoff’s Ergodic Theorem, in [7] and [34], we
give some formulas and bounds of Friedland’s entropy for certain Zk-actions. Hence, the Friedland’s
entropy formulas in [10], [7] and [34] are all special cases of the following result.
6
Theorem 2. Let α : Zk C2(M,M) be a C2 Zk-action on a d-dimensional closed Riemannian
−→
manifoldM. Let andΩbeasin(1.1)and(1.2)respectively, andΨtheskewproducttransformation
α
G
as in (1.3). If there is a measure with maximal entropy of Ψ in the form of P µ, where P =νZ is
ν ν
×
theproduct measureofsomeBorelprobability measureν on withν =ν(f )andµisanα-invariant
α i i
G
measure on M which is absolutely continuous with respect to the Lebesgue measure m. Then
k k
h(σ ) ν logν + max ν d (x)λ (x)dµ(x). (1.14)
α i i i j i,j
≤−i=1 ZMJ⊂{1,···,s(x)}j J i=1
X X∈ X
Furthermore, if µ is α-ergodic and for each pair of generators f and f , 1 i = j k, the set
i j
≤ 6 ≤
of their coincidence points, i.e. Coinc(f ,f ) which is defined by x M : f (x) = f (x) , is of µ
i j i j
{ ∈ }
measure zero, then we get the following formula of Friedland’s entropy
k
h(σ )= max log exp d λ . (1.15)
α j i,j
J 1, ,s
⊂{ ··· } (cid:16)Xi=1 (cid:0)Xj∈J (cid:1)(cid:17)
2 A measure-theoretic entropy formula of C2 random Zk-actions
We first recall some fundamental properties of C2 Zk-actions. Let M be a d-dimensional closed C
∞
Riemannianmanifold. We denoteby , , andd(, )the innerproduct,thenormonthetangent
hh· ·ii k·k · ·
spaces and the metric on M induced by the Riemannian metric, respectively.
Let α : Zk C2(M,M) be a C2 Zk-action on M with the generators f ,1 i k, as in (1.1)
i
−→ ≤ ≤
and µ be an α-invariant measure. By the Multiplicative Ergodic Theorem ([25]), for each f there
i
exists a measurable set Γ with f (Γ ) = Γ and µ(Γ ) = 1, such that for any x Γ , there exist a
i i i i i i
∈
decomposition
r(x,fi)
T M = E (x)
x j
j=1
M
r(x,fi)
intosubspacesE (x)ofdimensiond (x,f )(where d (x,f )=d),andnumbersλ (x,f )< <
j j i j i 1 i
···
j=1
λ (x,f ) which satisfy the following properties:P
r(x,fi) i
(1) for 1 j r(x,f ),v E (x) 0 ,
i j
≤ ≤ ∈ \{ }
1
lim log Dfn(x)v =λ (x,f );
n n k i k j i
−→±∞
(2)E (x),d (x,f )andλ (x,f )allmeasurablydependonxandthefollowinginvarianceproperties
j j i j i
Df (x)E (x)=E (f (x)) and λ (f (x),f )=λ (x,f ),
i j j i j i i j i
hold for each 1 i k,1 j r(x,f ).
i
≤ ≤ ≤ ≤
The above numbers λ (x,f ), ,λ (x,f ) are called the Lyapunov exponents of f at x, and
1 i ··· r(x,fi) i i
the collection (λ (x,f ),d (x,f )) : 1 j r(x,f ),x Γ is called the spectrum of f . For each
j i j i i i i
{ ≤ ≤ ∈ }
1 i k and each x Γ , denote
i
≤ ≤ ∈
Es(x,f )= E (x) and Eu(x,f )= E (x).
i j i j
λj(xM,fi)<0 λj(xM,fi)>0
In [11], Hu discussed some ergodic properties of C2 Z2-actions concerning Lyapunov exponents
andentropies. He gavea versionofPesin’stheoryforthis case. Moreprecisely,by the commutativity
7
of the generators, he gave a family of refined decompositions of the above E (x) into subspaces
j
{ }
related to the Lyapunov exponents of both of the generators f and f , then constructed a family of
1 2
Lyapunov charts and applied them to obtain the subadditivity of the entropies. We first introduce
some fundamental properties for C2 Zk-actions, they all derive from [11].
Proposition 2.1. Let α : Zk C2(M,M) be a Zk-action on M with the generators f ,1 i k
i
−→ ≤ ≤
and µ be an α-invariant measure. Then there exists a measurable set Γ Γ with f (Γ) = Γ for
i i
⊂
each i (in this case we call Γ is α-invariant) and µ(Γ) = 1, such that for any x Γ, there exists a
∈
decomposition of tangent space into subspaces
r(x,f1) r(x,fk)
T M = E (x)
x ··· j1,···,jk
jM1=1 jMk=1
satisfying the following properties:
(1) if E (x)= 0 , then for 0=v E (x) and 1 i k,
j1,···,jk 6 { } 6 ∈ j1,···,jk ≤ ≤
1
lim log Dfn(x)v =λ (x,f );
n n k i k ji i
−→±∞
(2) for each E (x), we have the following invariance properties
j1,···,jk
Dfi(x)Ej1,···,jk(x)=Ej1,···,jk(fi(x)) and λji(fi′(x),fi)=λji(x,fi),
where 1 i,i k.
′
≤ ≤
Notice that the subspaces E (x), 1 j r(x,f ),1 i k, of T M may not be pairwise
j1,···,jk ≤ i ≤ i ≤ ≤ x
different, and the Lyapunov exponents of f with respect to different subspaces may be coincide. For
i
simplicity of the notations, we relabel these subspaces by F (x),1 j s(x), such that
j
≤ ≤
s(x)
T M = F (x)
x j
j=1
M
in which F (x) = 0 for each 1 j s(x), and rename the corresponding Lyapunov exponents
j
6 { } ≤ ≤
of f with respect to F (x) by λ (x). If denote d (x) = dimF (x), then clearly s(x)d (x) = d.
i j i,j j j j=1 j
Therefore, the items (1) and (2) in Proposition 2.1 become
P
(1) for 0=v F (x),1 j s(x),
′ j
6 ∈ ≤ ≤
1
lim log Dfn(x)v =λ (x);
n n k i k i,j
−→±∞
and
(2) for each F (x), we have the following invariance properties
′ j
Dfi(x)Fj(x)=Fj(fi(x)) and λi,j(fi′(x))=λi,j(x),
where 1 i,i k.
′
≤ ≤
Definition 2.2. We call the collection
(λ (x),d (x)):1 i k,1 j s(x),x Γ
i,j j
{ ≤ ≤ ≤ ≤ ∈ }
the spectrum of α.
8
When the measure µ in Proposition 2.1 is ergodic with respect to α, then the above λ (x),d (x)
i,j j
and s(x) are all constant a.s., which are then denoted by λ ,d and s respectively.
i,j j
Before starting to prove Theorem 1, we first give some remarks and examples about it.
Remark 2.3. Let α : Zk C2(M,M) be a C2 Zk-action on M and ν a probability measure on
−→
. Let f be the induced random Zk-action over (Ω, ,P ,σ) and µ a f-invariant measure which is
α ν
G A
absolutely continuous with respect to the Lebesgue measure m.
(1) When k =1, f is no longer a random system but a deterministic system generated by a single
diffeomorphism, and (1.6) is exactly the well-known Pesin’s entropy formula as in [26],
h (f)= λ (x)d (x)dµ(x),
µ j j
ZM{λj(x)X:λj(x)>0}
where (λ (x),d (x)) is the Lyapunov spectrum of f.
j j
{ }
(2) From the formula (1.6), we can give the relation between the entropy of f and those of the
generators as follows
k
h (f) ν h (f ). (2.1)
µ i µ i
≤
i=1
X
Particularly, if
Es(x,f ) Eu(x,f )= 0 ,µ a.e.x for all 1 i,j k,
i j
∩ { } − ≤ ≤
then the equality in (2.1) holds. Moreover, the equality in (2.1) also holds in this particular case
whenever µ is an “SRB” measure, i.e., it has smooth conditional measures on the unstable manifolds
of f.
(3) When µ is α-ergodic, (1.7) can be regarded as a function of the distribution ν on the set of
generators of α. Precisely, let Sk 1 = (ν , ,ν ) : k ν2 = 1 be the k 1-dimensional unit
− { 1 ··· k t=1 1 } −
sphere and denote Sk 1 = (ν , ,ν ) Sk 1 : ν > 0,1 i k . we can define a function h (f)
− 1 k − i P µ
∗ { ··· ∈ ≤ ≤ }
on Sk 1 by
−
∗ k
h (f)((ν , ,ν ))= max ν d λ .
µ 1 k i j i,j
··· J 1, ,s
⊂{ ··· }j J i=1
X∈ X
Hence applying the theory of conditional extremum, we can get the extremum of the entropy of f, and
from the corresponding distributions we can identify the contributions of different generators to the
complexity of the action f.
Example 2.4. Let α be a Zk-action on the torus X = Td with the generators f k which are
{ i}i=1
induced by non-singular integer matrices A k . Since A A = A A for any 1 i,j k, then
{ i}i=1 i j j i ≤ ≤
by Theorem A of [11], we can write Rd as a direct sum of subspaces V V which are all
1 s
···
invariant under each A ,1 i k, and moreover, for each 1 i k and 1 j s, the eigenvalues
i L L
≤ ≤ ≤ ≤ ≤ ≤
of A =A all have the same norm. For each 1 j s denote d =dimV and λ the common
i,j i|Vj ≤ ≤ j j i,j
norm of the eigenvalues of A . In fact, the collection (logλ ,d ) : 1 i k,1 j s is
i,j i,j j
{ ≤ ≤ ≤ ≤ }
exactly the spectrum of α. Therefore, by Theorem 1, for any probability measure ν on and the
α
G
corresponding random Zk-action f over (Ω, ,P ,σ), we have
ν
A
k
h(f)=h (f)= max ν d logλ (2.2)
m i j i,j
J 1, ,s
⊂{ ··· }j J i=1
X∈ X
(recall that m is the Lebesgue measure).
9
In a particular case, for the Z2-action α on the torus T2 with the generators f ,f which are
1 2
{ }
induced by the hyperbolic automorphisms
2 1 1 1
A1 = and A2 =A−11 = − ,
1 1 ! 1 2 !
−
respectively, and the induced random Z2-action f over (Ω, ,P ,σ), we have
ν
A
3+√5
h(f)= ν ν log ,
1 2
| − | 2
where ν =ν(f ).
i i
In the remaining of this section we will prove the main theorem, i.e., Theorem 1, of this paper.
We always assume that α : Zk C2(M,M) is a C2 Zk-action on M, ν is a probability measure
−→
on with ν = ν(f ), f is the induced random Zk-action over (Ω, ,P ,σ) and µ is an α-invariant
α i i ν
G A
measure which is absolutely continuous with respect to the Lebesgue measure m.
It is well known that for any C2 diffeomorphism g on M and any invariant measure µ which is
absolutely continuous with respect to m, we have the so-called Pesin’s entropy formula
h (g)= d (x)λ (x)dµ(x),
µ j j
ZMλjX(x)>0
where (λ (x),d (x)) is the spectrum of g. For the proof of the inequality
j j
{ }
h (g) d (x)λ (x)dµ(x),
µ j j
≤
ZMλjX(x)>0
which is called Ruelle’s inequality, we refer to Theorem S2.13 of [13], and for the proof of the reverse
inequality we refer to section 13 of[23]. We will adapt the methods in [13] and[23] to get the desired
entropy formula (1.6) for random Zk-actions.
Lemma 2.5 (Lemma 1 of [2]). (1) Define fn by (fn)i = fni,i Z,ω Ω, for n N (here
ω ω ∈ ∈ ∈
N= 1,2, ). Then for all n N one has
{ ···} ∈
h (fn)=nh (f). (2.3)
µ µ
(2) If is a sequence of finite partitions of M with lim diam =0, then
p p + p
{Q } → ∞ Q
h (f)= lim h (f, ). (2.4)
µ µ p
p + Q
→ ∞
Suppose ε (0,1) and ρ : Ω M (0,ǫ) is a measurable function. Given ω Ω,x M and
ε
∈ × −→ ∈ ∈
n>0, define two subsets by
B (ω,ε,x)= y M :dn(x,y) ε (2.5)
n { ∈ ω ≤ }
and
B (ω,ρ ,x)= y M :d(fi(x),fi(y)) ρ (σiω,fi(x)),0 i n 1 . (2.6)
n ε { ∈ ω ω ≤ ε ω ≤ ≤ − }
Lemma 2.6. Let = ρ : ε (0,1) be a family of measurable functions on Ω M such that for
ε
H { ∈ } ×
any ω Ω, ρ (ω, ) monotonically decreases as ε 0. Then
ε
∈ · −→
1
h (f) lim limsup logm(B (ω,ρ ,x))dµ(x), P a.e.ω Ω. (2.7)
µ n ε ν
≥ZMε−→0 n→∞ −n − ∈
10