Exponential Decay of Expansive Constants

0.1 MB·

by  Peng Sun

EXPONENTIAL DECAY OF EXPANSIVE CONSTANTS
1 Peng Sun
1
0 ChinaEconomicsandManagement Academy
2 CentralUniversityofFinanceandEconomics
No. 39CollegeSouthRoad,Beijing,100081, China
n
a
J
4 Abstract. Amapf onacompactmetricspaceisexpansiveifandonlyiffn
isexpansive. Westudytheexponentialrateofdecayoftheexpansiveconstant

] and find some of its relations withother quantities about the dynamics, such
S
asdimensionandentropy.
D
h. 1. Expansive maps
t
a Let X be a compact metric space. A homeomorphism (continuous map) f is
m called expansive if there is γ > 0 such that d(fn(x),fn(y)) < γ for all n ∈ Z
[ (n ≥ 0) implies x = y. We call the largest γ the expansive constant of f, denoted

by γ(f), as it depends on f.
1
v Inthis paperweassumethatf is anexpansivehomeomorphism,ifnotspecified.
1 Resultsandtheir proofsforexpansivecontinuousmapsareverysimilarandwillbe
5 omitted.
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Our discussion on expansive constants builds on the following proposition:
0
1. Proposition 1.1. For every n∈Z (n∈N for continuous maps), f is expansive if
0 and only if fn is expansive.
1
1 Proof. From the definition, it is trivial that f is expansive if and only if f−1 is
: expansive, and for n∈N, if fn is expansive, then f is expansive.
v
i Nowassumethatf isexpansivewithexpansiveconstantγ(f). Asf iscontinuous,
X there is ǫ>0 such that d(x,y)<ǫ implies dn(x,y)<γ(f), where
f
r
a dn(x,y)= max d(fk(x),fk(y)).
f
0≤k≤n−1
So d((fn)k(x),(fn)k(y))<ǫ for all k ≥0 implies d(fkn+j(x),fkn+j(y))<γ(f) for
all k ≥0 and 0≤j ≤n−1. fn is expansive with expansive constant no less than
ǫ. (cid:3)
From this proposition, if f is expansive, we can talk about γ(fn), the expansive
constant of fn. It is not difficult to make the following observations:
Lemma 1.2. If f is expansive, then the following hold.
(1) (For homeomorphisms only) γ(f)=γ(f−1).
(2) For every n∈N, γ(fn)≤γ(f).
(3) If d(x,y)<τ implies dn(x,y)<γ(f), then γ(fn)≥τ.
f
1991 Mathematics Subject Classification. Primary: 37B40,54F45. Secondary: 37D20.
Key words and phrases. Lebesguenumber,dimensiontheory,topological entropy.
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2 PENGSUN
It is natural to think about how γ(fn) varies as n increases. For Lipschitz maps
we have a very rough estimate:
Lemma 1.3. If f is expansive and Lipschitz with Lipschitz constant L(f), then
L(f)>1.
Proposition 1.4. Let f be expansive and Lipschitz with Lipschitz constant L(f),
then for every n∈N, γ(fn)≥γ(f)·(L(f))−(n−1).
Proof. d(x,y)≤γ(f)·(L(f))−(n−1) implies dn(x,y)≤γ(f). (cid:3)
f
Corollary1.5. Iff isabi-Lipschitz homeomorphism withLipschitzconstantsL(f)
and L(f−1), then for every n∈N,
γ(fn)≥γ(f)· min max{(L(f))−j,(L(f−1))−(n−j)}
0≤j≤n
2. Exponential decay of expansive constants
As in the usual way to study a quantity about the dynamics, we consider the
asymptotic exponential decay rate of the expansive constant γ(fn).
Definition 2.1. If f is expansive, then let
1
h+(f)=limsup− logγ(fn)
E n
n→∞
1
h−(f)=liminf− logγ(fn)
E n→∞ n
From now on we use h∗ to denote either h+ or h−, when the argument works
E E E
for both cases. Some simple facts about them are listed below.
Lemma 2.2. If f is expansive, then
(1) h∗(f)≥0.
E
(2) h−(f)≤h+(f).
E E
(3) (For homeomorphisms only) h∗(f)=h∗(f−1).
E E
Proposition 2.3. If f is expansive, then for every n ∈ N, h+(fn) ≤ nh+(f) and
E E
h−(fn)≥nh−(fn).
E E
Proof.
1 1
h+(fn)=limsup− logγ(fkn)≤nlimsup− logγ(fj)=nh+(f).
E k j E
k→∞ j→∞
The other one is analogous. (cid:3)
Proposition 2.4. If f is continuous map that is expansive and Lipschitz, then
h∗(f)≤logL(f). If f is homeomorphism that is expansive and bi-Lipschitz, then
E
logL(f)·logL(f−1)
h∗(f)≤ .
E logL(f)+logL(f−1)
In particular, if L=max{L(f),L(f−1)}, then h∗(f)≤ 1logL.
E 2
Proof. This is a corollary of Proposition 1.4 and Corollary 1.5. (cid:3)
Proposition 2.5. Let f : X → X and g : Y → Y be expansive and there is a
bi-Lipschitz conjugacy h:X →Y between them. Then h∗(f)=h∗(g).
E E
DECAY OF LEBESGUE NUMBERS 3
Proof. For every n and k, d(gkn(x),gkn(y))<(L(h))−1·γ(fn) implies
d(h(fkn(h−1(x))),h(fkn(h−1(x))))<γ(fn),
which provides h−1(x) = h−1(y), hence x = y. So γ (g) ≤ (L(h))−1 · γ(fn).
n
Similarargumentshowsγ(fn)≤(L(h−1)−1·γ (g). Takinglimits,wehaveh∗(f)=
n E
h∗(g). (cid:3)
E
Corollary 2.6. h∗(f) is invariant under strongly equivalent metrics.
E
3. Relations with the exponential decay of Lebesgue numbers
The exponential decay of Lebesgue numbers has been discussed in [2]. We are
somewhatsurprisedbytherelationbetweenitandthedecayofexpansiveconstants
we observe.
Recall that for every open cover U of a compact metric space X, the Lebesgue
number δ(U) is defined as the largest positive number such that every δ(U) ball
is covered by some element of U. Let f be a continuous map on X. Let Un =
f
n−1f−j(U) and δ (f,U)=δ(Un). Define
Wj=0 n f
1
h−(f,U)=liminf− logδ (f,U),
L n→∞ n n
1
h+(f,U)=limsup− logδ (f,U),
L n n
n→∞
h−(f)=suph−(f,U)
L L
U
and
h+(f)=suph+(f,U).
L L
U
Theorem 3.1. If f is expansive, then h∗(f)≤h∗(f).
E L
Proof. Let U be an open cover such that diamU < γ(f). Then for each n > 0,
d(x,y) < δ (f,U) implies d(fj(x),fj(y)) < diamU < γ(f) for 0 ≤ j ≤ n − 1.
n
So if d(fkn(x),fkn(y)) < δ (f,U) for every k, then d(fm(x),fm(y)) < γ(f) for
n
every m = kn+j, which runs over all integers. This forces x = y as γ(f) is the
expansiveconstantoff. Soγ(fn)≤δ (f,U). Takethelimitandweobtainh∗(f)≤
n E
h∗(f,U) ≤ h∗(f). (The last relation is in fact an equality from [2, Corollary
L L
3.9]) (cid:3)
4. Relations with entropy and dimension
The most important fact we observe is that the product of h∗(f) and box di-
E
mension also bounds topological entropy. By Theorem 3.1, this bound is (strictly,
see Theorem 4.3 for example) better than [2, Theorem 4.7]. But this result only
makes sense when f is expansive.
Theorem 4.1. Let f be expansive on a compact metric space X. dim±X are the
B
upper andlower box dimensions of X andh(f)is thetopological entropyof f. Then
h−(f)·dim+X ≥h(f) and h+(f)·dim−X ≥h(f).
E B E B
Proof. We only show the first inequality. Proof of the other is similar.
The resultis trivialif h(f)=0. Assume h(f)>0. Takeanyλ>dim+X. There
B
is ε > 0 such that ε < ε implies that there is an open cover U of X such that
0 0
diamU < ε and |U| ≤ ε−λ. If n large enough such that expnh(f) = exph(fn) >
4 PENGSUN
ε−λ, thenU is nota generatorunder fn (see for example,[1, Section5.6]),as there
are at most ε−kλ elements in k−1f−jn(U), which makes
Wj=0
k−1
1
h(fn,U)= lim H(_ f−jn(U))≤log(ε−λ)<h(fn).
k→∞k
j=0
There is A∈ ∞ f−jn(U) that containsat leasttwo points, say, x and y. Then
Wj=−∞
for every j ∈ Z, d(fjn(x),fjn(y)) < ε. ε is not an expansive constant for fn and
γ(fn)<ε.
NowtakeN >−λlogε0. Foreveryn>N,takeε<ε suchthatexp(n−1)h(f)<
h(f) 0
ε−λ <expnh(f). Then
logγ(fn) logε (n−1)h(f)
− >− >
n n nλ
Takethelowerlimitwegeth−(f)·λ>h(f). Theresultfollowssinceλisarbitrarily
E
chosen. (cid:3)
The idea of the proof is a byproduct of the following problem considered by the
author.
Problem. Letf beanexpansivehomeomorphism onacompactmetricspace. What
is the smallest possible number of elements in a generator? How is this number
related to other properties of f?
It is sure that this number should be at least exph(f). So the best result we
can expect is the integer no less than exph(f), and this is the case for full shifts.
However,even for subshifts of finite types we have no idea at this moment.
As for subshifts of finite types, we observe the following fact.
Proposition 4.2. Let σ be a subshift of finite type. Let q > 1 and the metric
A
on ΩA be defined by d(ω,ω′)=q−min{|j||ωj6=ωj′}. Then h∗E(σA)·dimHΩA =h(σA),
where dim Ω is the Hausdorff dimension of Ω .
H A A
Remark. It is well-known that under the above assumptions,
2 1
dim+Ω =dim−Ω =dim Ω = lim logkAkk.
B A B A H A logq k→∞k
(or 1 lim 1 logkAkk for one-sided shifts.)
logq k→∞ k
Proof. We only show the result for two-sided shifts. Proof for one-sided shifts is
analogous.
We know that h(σ ) = lim 1logkAkk. If h(σ ) = 0, the result is trivial.
A k→∞ k A
Otherwise, it is enough to show that h∗(σ )= logq.
E A 2
If d(ω,ω′) < q−n, then we must have ω = ω′ for all −n ≤ j ≤ n. So
j j
d(σk(2n−1)(ω),σk(2n−1)(ω)) < q−n for all k ∈ N implies ω = ω′, hence γ(σ2n−1) ≥
A A A
q−n. In fact, γ(σl )≥q−n for all l≤2n−1. So h−(σ )≥ logq.
A E A 2
By Theorem 4.1, h+(σ )≤ logq. The result follows. (cid:3)
E A 2
The aboveresultis notso difficult but veryinteresting. Itseems that forcertain
expansive dynamical systems the topological entropy may also be given by the
product h∗(f)·dim (X). Though the box dimensions may be different from the
E H
Hausdorffdimensionwhenthemetricischanged,webelievethattheresultinvolving
Hausdorffdimensionisalwaystrue. ConsideringCorollary2.6,theresultistruefor
DECAY OF LEBESGUE NUMBERS 5
many other commonly used metrics on symbolic spaces. A proof for all equivalent
metrics is in process.
Moreover, as every Anosov diffeomorphism is expansive and has Markov parti-
tion, we may expect the following result:
Conjecture 1. Let f be an Anosov diffeomorphism on a m-dimensional compact
manifold, then h(f)=mh∗(f).
E
It would be a big surprise if the conjecture is true. Nevertheless, the following
fact might boost our confidence, at least a little bit.
Theorem 4.3. The conjecture is true for hyperbolic linear automorphisms on the
2-torus with the standard metric.
Proof. Let λ > 1 and λ−1 be the eigenvalues. For every µ > λ, there is a metric
that is strongly equivalent to the standard metric, such that the diffeomorphism
f is a bi-Lipschitz map with Lipschitz constants L(f) < µ and L(f−1) < µ. By
Proposition2.4andCorollary2.6, h∗(f)< 1logµ. Since µ isarbitrarilytakenand
E 2
by Theorem 4.1, we have h∗(f)= 1logλ= 1h(f). (cid:3)
E 2 2
References
[1] (MR0648108) P.Walters,“AnIntroduction toErgodicTheory,”Springer-Verlag,1982.
[2] P.Sun,Exponential Decay of Lebesgue Numbers,preprint.
E-mail address: [email protected]