Upper estimates for stable dimensions of fractal sets with variable number of foldings

0.21 MB·English

by  Eugen Mihailescu

UPPER ESTIMATES FOR STABLE DIMENSIONS OF FRACTAL SETS WITH
VARIABLE NUMBERS OF FOLDINGS
EUGENMIHAILESCUANDBERNDSTRATMANN
3 Abstract. For a hyperbolic map f on a saddle type fractal Λ with self-intersections, the number of f-
1 preimagesofapointxinΛmaydependonx. Thismakesestimatesofthestabledimensionsmoredifficult
0 thanfordiffeomorphismsorformapswhichareconstant-to-one. Weemploythethermodynamicformalism
2 in order to derive estimates for the stable Hausdorff dimension function δs on Λ, in the case when f is

n conformal on local stable manifolds. These estimates are in terms of a continuous function on Λ which
a boundsthepreimagecountingfunctionfrombelow. Asacorollaryweobtainthatifδs attainsitsmaximal
J
possible value in Λ, then the stable dimension is constant throughout Λ, whereas the preimage counting
9
function is constant on at least an open and dense subset of Λ. In particular, this shows that if at some

] pointinΛ thestabledimensionisequal totheanalogue of thesimilaritydimensioninthestable direction
S atthatpoint,thenf behavesverymuchlikeahomeomorphismonΛ. Finally,wealsoobtainresultsabout
D
thestableupperboxdimensionforthesetypeoffractals. Weendthepaperwithadiscussionoftwoexplicit
. examples.
h
t
a
m Mathematics Subject Classification 2000: 37D35, 37F15 37D45, 37F10.
[

1 Keywords: Non-invertible hyperbolic maps; thermodynamic formalism; basic sets of saddle type; stable
v
dimensions; conformal maps; fractals with overlap;Hausdorff dimension; box-counting dimension.
7
2
8
1
.
1
0 1. Introduction and statement of results.
3
1 In this paper we investigate fractal sets Λ of saddle type which are invariant under a non-invertible C2-
:
v
endomorphism f of a Riemann manifold M into itself. These fractals are basic sets of f, meaning that Λ is
i
X compactandf-invariantsuchthatf| istopologicallytransitiveandsuchthatthereexistsaneighbourhood
Λ
r U of Λ satisfying Λ = fn(U). The fact that f is non-invertible produces complicated overlaps and
a n∈Z
foldings within Λ, which influence the Hausdorff dimension of the sections through Λ and the number of
T
overlaps does not necessarily has to be constant. We will always assume that f is hyperbolic on Λ in the
sense of Ruelle [17], that is, for each backward orbit xˆ=(x,x ,x ,...) of x in Λ, where f(x )=x and
−1 −2 −1
f(x )=x ∈Λforalli∈N,there existsacontinuoussplitting ofthe tangentbundleoverthe spaceΛˆ
−(i+1) −i
ofallbackwardorbitsofelementsofΛ,calledthenatural extension(orinverselimit)ofthetuple(Λ,f),into
stable spaces Es and unstable spaces Eu. It is well-known that Λˆ is a compact metric space and that the
x xˆ
lift fˆ: Λˆ → Λˆ of f to Λˆ, given by fˆ(xˆ) := (f(x),x,x ,x ,...), is a homeomorphism. Note that natural
−1 −2
extensionsplay animportantrolein the study ofthe dynamicsofendomorphisms(see for instance[17, 11]).
As in the diffeomorphism case, for a hyperbolic endomorphism f on Λ there exist local stable manifolds
Ws(x) and local unstable manifolds Wu(xˆ), for each x ∈ Λ and xˆ ∈ Λˆ. Note that there may be infinitely
r r
many local unstable manifolds through a given point in Λ and, unlike in the diffeomorphism case, these do
not necessarily give rise to a foliation.
1
2 EUGENMIHAILESCUANDBERNDSTRATMANN
In this paper we will consider the stable dimension at x∈Λ, which is given by
δs(x):=dim (Ws(x)∩Λ),
H r
where dim refers to the Hausdorff dimension. To give estimates for the stable dimension is by far more
H
delicatethanfortheunstable dimensionδu(xˆ):=dim (Wu(xˆ)∩Λ). Infact, in[11]itwasshownthatδu(xˆ)
H r
is constant on Λˆ and that its value is given by the unique zero of the pressure function P (−tlog|Df |),
fˆ|Λˆ u
where |Df (x)| denotes the norm of the derivative of f restricted to Eu. However, for the stable dimension
u xˆ
we can not expect that a similar formula holds in general.
Before we state our main result, let us point out that in this paper we consider a special type of hyperbolic
endomorphisms which will be called c-hyperbolic. A map f is c-hyperbolic on Λ if it is hyperbolic as an
endomorphism over Λ, if it is conformal on all local stable manifolds and if Λ does not contain any critical
points of f.
Also, let us introduce the preimage counting function ∆:Λ→N, which is given for each x∈Λ by
∆(x):=Card f−1(x)∩Λ
One immediately verifies that ∆ is upper semi-contin(cid:0)uous and b(cid:1)ounded on Λ (see e.g. [12, Lemma 1]).
Moreover,the stable potential function Φs on Λ is defined by Φs(x):=log|Df (x)|, where |Df (x)| denotes
s s
the norm of the derivative of f restricted to Es. We are now in the position to state the main result of this
x
paper.
Theorem 1. Let f : M → M be a C2-endomorphism which is c-hyperbolic on a basic set Λ of f and for
which there exists a continuous function ω : Λ → R such that ∆(x) ≥ ω(x), for all x ∈ Λ. It then follows
that
δs(x)≤t ,
ω
where t refers to the unique zero of the pressure function t 7→P(tΦs−logω) associated with the potential
ω
function tΦs−logω.
Let us point out that one of the difficulties in proving this theorem is that the map f is not necessarily
expandingandthatits inversebranchesdonotnecessarilycontractsmallballs. Infact,somedirectionsmay
be even expanding in backward time. Another difficulty is that the number of preimages of a point that
remain in Λ is not always constant.
The reader might like to recall that in their pioneering work Bowen [4] and Ruelle [19] employed the
thermodynamic formalism in order to derive dimension formulae for rational maps. In fact, in the diffeo-
morphism case, it turned out that the stable and the unstable dimension can in general be computed both
as the zero of the pressure function of the stable potential, respectively the unstable potential (see [8]);
(for further applications of the thermodynamic formalism in dimension theory, we refer to [1], [14]). It is
importantto note that for an endomorphismf in higher dimension, a hyperbolic basic set is not necessarily
totally invariant. This is of course significantly different from the case of Julia sets of rational maps in the
complex one dimensional case. Examples of perturbations of toral endomorphisms which are Anosov and
whoseunstable manifoldsdepend onthe whole prehistoryweregivenin[15]. Another classofnon-invertible
hyperbolic maps with crossedinvarianthorseshoeswas given by Bothe in [2]. Also, Simon [21] gaveanother
class of non-invertible endomorphisms, for which the Hausdorff dimension of the associated attractors can
be computed with the help of a pressure formula just as in the invertible case. Examples of non-linear
hyperbolic skew products of Cantor sets with overlaps in their fibres were given in [10], where the strongly
non-invertiblecharacterofthesemapshasbeenestablished,andwhereitwasshownthattheseskewproducts
are far awayfrom being constant-to-one. In [10] it was also shownthat there are maps for which there exist
Cantor sets in each of their fibres, such that through each point of these sets there pass uncountably many
UPPER ESTIMATES FOR STABLE DIMENSIONS OF FRACTAL SETS WITH VARIABLE NUMBERS OF FOLDINGS 3
differentlocalunstablemanifolds. Alsoletusmentionthatyetanotherclassofc-hyperbolicendomorphisms
can be found by considering hyperbolic basic sets of saddle type, for holomorphic maps f : P2C → P2C on
the 2-dimensional complex projective space ([11]).
The paper continues by showing that an applicationof Theorem 1 gives rise to the following proposition.
In here, we consider the situation in which δs attains a maximal value and show that in this case, δs has to
be constant throughout Λ and that ∆ has to be equal to its least value d on an open dense subset.
Proposition 1. If in addition to the assumptions in Theorem 1 we have that the minimal value of ∆ on Λ
is equal to d and that there exists a point x ∈ Λ at which δs is equal to the unique zero t of the pressure
d
function t7→P(tΦs−logd), then ∆ is equal to d on an open dense subset of Λ and δs(y) is equal to t , for
d
all y ∈Λ.
Note that the latter proposition can be applied in particular in the case in which d is equal to 1 and
wherethereisnooverlap. Inthissituationthestabledimensionisequaltothesimilaritydimension,andthe
proposition then guarantees that there exists an open dense set of points in Λ at which f has precisely one
preimage in Λ. Therefore,in this case the map behaves almost like a homeomorphismwhen restrictedto Λ.
This particular situation is somewhat parallel to a result of Schief [20], although the setting and proofs are
completely different. We summarise these results in the following corollary.
Corollary 1. Let f : M → M be a C2-endomorphism which is c-hyperbolic on a basic set Λ of f and
for which there exists a point x ∈ Λ such that δs(x) is equal to the unique zero t of the pressure function
1
t7→P(tΦs). Then there exists an open dense set of points in Λ at which f has precisely one preimage in Λ.
Moreover, we have that δs(y)=t , for all y ∈Λ.
1
Also, in Corollary 3 from Section 4 we will show how our Corollary 1 can be applied to a class of
translations of horseshoes with overlaps,previously studied by Simon and Solomyak in [22].
Let us now remark that a combination of Theorem 1 with the main theorem in [12] gives rise to the
following result.
Corollary 2. If in addition to the assumptions in Theorem 1 we have that the preimage counting function
∆ is locally constant on Λ, then it follows that δs(x)=t , for all x∈Λ. Here, t is given as in Theorem 1.
ω ω
Finally, we consider the stable upper box dimension βs(x) which is given by the upper box-counting
dimension dim (Ws(x)∩Λ) of the set Ws(x)∩Λ, for each x ∈ Λ. (For a general discussion of the upper
B r r
box dimension for fractal sets we refer to [9] and [14]). We show that this dimension function is constant
throughout Λ and that in the situation in which ∆ is bounded from below, similarly as in Theorem 1, one
derives an upper bound for its value. These results are summarised in the following proposition.
Proposition 2. Let f : M →M be a C2-endomorphism which is c-hyperbolic on a basic set Λ of f. Then
the following hold.
(a) If there exists a continuous function ω :Λ→R such that ∆(x)≥ω(x), for all x∈Λ, then we have,
with t given as in Theorem 1,
ω
βs(y)≤t , for all y ∈Λ.
ω
(b) The function βs is constant on Λ.
In particular the above results apply for hyperbolic basic sets of saddle type for holomorphic maps f :
P2C → P2C. We will end the paper by giving two further explicit examples in which the above results can
be applied. Our first example will be concerned with certain horseshoes with overlaps in R3 considered in
[22]. The second example will be on basic sets for a family of hyperbolic skew products studied in [10].
4 EUGENMIHAILESCUANDBERNDSTRATMANN
We close this introduction with some comments on how the results in this paper relate to previous work
in this area.
In[6](seealso[23]and[13])Falconerstudiedself-affinefractalswithoverlapsobtainedfromfinitelymany
linear contractions T (x)=λ x,i=1,...,ℓ in R satisfying 0<|λ |<1 and |λ |<1. He showed that
i i i 1≤i≤ℓ i
the Hausdorff dimension of the invariant set of the family of translated contractions {T +a ,:1≤i≤ℓ} is
P i i
equal to s, for Lebesgue almost all (a ,...,a ) ∈ R×...×R; where s represents the similarity dimension,
1 ℓ
defined as the solution of the equation
|λ |s =1
i
1≤i≤ℓ
X
We remark that this result may be extended also to similarities on Rn. However, the result fails if the
condition |λ | < 1 is not satisfied, as observed by Edgar ([5]), who based his argument on a result
1≤i≤ℓ i
1/2 0
by PrzytycPki and Urban´ski ([16]). Indeed, if T = T = and if |λ| > 1, then for Lebesgue
1 2 0 λ ! 2
almost every a = (a ,a ) ∈ R2 the attractor Λ(a) of the system {T +a ,T +a } stays to be the same;
1 2 1 1 2 2
and moreover if 1/λ is a Pisot number (that is, an algebraic integer such that the absolute value of all its
algebraicconjugatesis less than1), then dim (Λ(a))<2−(log(1/λ))/log2 (see e.g. [23]). This shows that
H
fractals originating from overlapping constructions can have Hausdorff dimension less than their similarity
dimension.
In [20] Schief considered self-similar fractal sets K and showed that if for the similarity dimension σ of
K one has that the σ-dimensional Hausdorff measure Hσ(K) is positive, then K satisfies the strong open
set condition, that is, the system behaves similar to a homeomorphism on K. Note that this result is in the
spirit of our results in this paper, although the setting and the ideas of our proofs differ significantly from
the approach in [20]. More precisely, the assumptions in Proposition 1 are much weaker than the ones in
[20]. Namely, in order to obtain the ”almost injectivity” of the system associated with Λ, we only require
that the stable dimension δs(x) is equal to the zero t of the pressure function t→P(tΦs), for some x∈Λ;
1
we do notrequire thatHt1(Wrs(x)∩Λ)>0. In ourcase t1 is the analogueof the similarity dimensionin the
stable direction, in the sense that it represents the dimension which one would obtain if the system would
be invertible. In particular if there exists some x ∈ Λ for which Ht1(Ws(x)∩Λ) > 0 is positive, then we
r
have that the stable dimension is everywhere equal to t and that there exists an open dense set of points
1
in Λ which have precisely one preimage in Λ.
Finally, in [12] Mihailescu and Urbanski studied c-hyperbolic maps on Λ for which ∆ is bounded from
above by acontinuousmapη onΛ. Inthatpaper the authorsobtainthe resultthatδs(x)≥t forallx∈Λ,
η
where t refers to the unique zero of the pressure function t → P(tΦs−logη). Note that the proof for the
η
upper estimateinthis paperis verydifferentfromthe proofforthe lowerestimatesin[12]. However,wecan
combine these two estimates, as done in Corollary 2, to obtain that if the preimage counting function ∆ is
locally constant on Λ, then the stable dimension is equal to t throughout Λ.
∆
2. Proof of Theorem 1.
For ease of exposition, let us first consider the situation in which ω is locally constant and takes on
only two different positive integer values on Λ, namely d on the set V and d on the set V . We then
1 1 2 2
have that V ∪ V = Λ and that V and V are two disjoint compact subsets of Λ. Hence, there exists
1 2 1 2
ε > 0 such that the distance d(V ,V ) between V and V is greater than ε . For x ∈ Λ and n ∈ N, let
0 1 2 1 2 0
B (x,ε) := {y ∈ Λ : d(fi(y),fi(x)) < ε,0 ≤ i ≤ n−1} refer to the n-Bowen ball centred at x of radius
n
ε > 0. Note that for 0 < ε < ε we have that if y ∈ B (x,ε) then fi(y) and fi(x) both belong to either
0 n
UPPER ESTIMATES FOR STABLE DIMENSIONS OF FRACTAL SETS WITH VARIABLE NUMBERS OF FOLDINGS 5
V or V , for each 0 ≤ i ≤ n−1. Recall that Φs(x) := log|Df (x)|,x ∈ Λ. Now, let t > t be fixed. By
1 2 s ω
definition of t , we have that there exists β >0 such that
ω
P(tΦs−logω)<−β
Hence, by choosing ε > 0 sufficiently small, there exists a constant C > 0 such that for each n ∈ N large
enough, there exists a minimal (n,ε)-spanning set E for Λ such that
n
1
(1) (diam U (z))t· <Ce−βn <1,
n ∆(f(z)·...·∆(fn(z))
zX∈En
where we have set U (z) := fn(B (z,ε))∩Ws(x)∩Λ. Note that in here we have used the fact that the
n n r
set U (z) is the intersection of an unstable tubular neighbourhood with the fixed stable manifold Ws(x).
n r
Also, we used that |Dfn(z)| is uniformly comparable to diam U (z), which follows from the fact that f is
s n
conformal on local stable manifolds.
In the sequel let us put W :=Ws(x)∩Λ. Hence, the aim is to show that dim (W)≤t, for each t>t .
r H ω
ThemainideaoftheproofistoextractsuitablecoversofW outofthelargesetofcoverswhicharegivenby
taking n-preimages,suchthat at eachstep a different sum will be minimised. Note that we saythat a point
y is a k-preimage of x if fk(y)=x. Eachsuchn-preimagewill be included in a Bowenball oftype B (z,ε),
n
for some z ∈E . This procedure is delicate, since at eachstep the number of preimages of points belonging
n
to Λ varies. The idea is to consider the k iterates of n-preimages,then to subdivide Λ into various different
parts and finally, to find suitable covers of these parts which minimise certain sums at the k-th level.
First, note that since Λ is covered by the set of Bowen balls {B (z,ε): z ∈E }, it follows that {U (z):
n n n
z ∈E }coversW. However,thiscoverisfartoorichandwewillhavetoextractasuitablesubcover. Indeed,
n
by using a well known theorem by Besicovitch (see for e.g. [9]), there exists a subcover {5U (z):z ∈G(0)}
n
of W such that {U (z) : z ∈ G(0)} consists of pairwise disjoint sets. (Note that, since f is conformal on
n
local stable manifolds, we can assume that the sets U (z) are in fact balls, and we denote the radii of these
n
by r(n,z); also, we write 5U (z) to denote the ball of radius 5r(n,z) centred at the centre of U (z)). The
n n
next step is to ”inflate” this cover, that is, to enlarge it to a ”richer” cover of W. For this, we consider
an (n−1)-preimage of w in Λ which we denote by w(n−1), for each point w ∈ W. Let us assume that
w(n−1)∈V and hence, that w(n−1) has at least d 1-preimages in Λ. Now, since E is (n,ε)-spanning,
1 1 n
for each point ξ ∈ Λ, there exists at least one point y ∈ E such that ξ ∈ B (y,ε). However, we cannot
n n
have two 1-preimages of some w(n−1) belonging to different Bowen balls B (y,ε) and B (y′,ε) such that
n n
y and y′ are both in G(0). This is an immediate consequence of the fact that {U (z):z ∈G(0)} consists of
n
pairwise disjoint sets.
Therefore, by way of successive eliminations, we can find d pairwise disjoint families, denoted by
1
F(1,d ;1),...,F(1,d ;d ),suchthat{5U (z):z ∈F(1,d ;i)}isacoveroftheset{w∈W :w(n−1)∈V },
1 1 1 n 2 1
for each 1 ≤ i ≤ d . Obviously, for w(n−1) ∈ V we can proceed in a similar way, which then gives rise
1 2
to d mutually disjointfamilies F(1,d ;1),...,F(1,d ;d ) for which we have that {5U (z):z ∈F(1,d ;j)}
2 2 2 2 n 2
is a cover of {w ∈ W : w(n−1) ∈ V }, for each 1 ≤ j ≤ d . Note that, since d(V ,V ) > 0, we have that
2 2 1 2
F(1,d ;i)∩F(1,d ;j) = ∅, for all i and j, and that by construction we have that the so obtained disjoint
1 2
families are all contained in E . Next, define
n
2
F(1):= ∪ ∪ F(1,d ,j)
i
i=11≤j≤di
and let G(1,d ) be given, for k ∈{1,2},by
k
(diamU (z))t (diamU (z))t
n n
=min :i∈{1,...,d } .
∆(f2(z))...∆(fn(z))  ∆(f2(z))...∆(fn(z)) k 
z∈GX(1,dk) z∈FX(1,dk;i) 
 
6 EUGENMIHAILESCUANDBERNDSTRATMANN
For G(1):=G(1,d )∪G(1,d ), we then obtain, by adding the sums over G(1,d ) and G(1,d ),
1 2 1 2
(diamU (z))t (diamU (z))t
n n
(2) ≤ .
∆(f2(z))...∆(fn(z)) ∆(f(z))...∆(fn(z))
z∈XG(1) z∈XF(1)
Notethatherewehaveusedthetrivialfactthatforeachx∈Λwehavethat 1/∆(x)=1. Also,
y∈Λ,f(y)=x
note that the sum over the family G(1) on the left hand side of the inequality in (2) is smaller than the sum
P
overthe largerfamily F(1) on the righthandside. However,andthis is the crucialpoint, the summands on
the right hand side have one more factor in their denominator than the summands on the left hand side.
Let us now bring the argument to its next level by enlarging the family F(1) as follows. Recall that for
each w ∈ W we have fixed an (n−1)-preimage w(n−1) ∈ Λ. We now define w(n−2) := f(w(n−1))
and consider not only w(n −1) but also the other 1-preimages of w(n −2) in Λ. Subsequently, we will
then take the 1-preimages of these 1-preimages of w(n−2) and obtain new covers of W. Indeed similarly
as before, if w(n − 2) ∈ V then we can construct, by succesive eliminations, pairwise disjoint families
1
F(2,d ;1),...,F(2,d ;d ) by selecting the 1-preimagesof the i-thpreimage of w(n−2), for each1≤i≤d .
1 1 1 1
In fact one of these families is F(1). As in the first step, the sets {5U (z): z ∈F(2,d ;i)} cover {w ∈W :
n 1
w(n−2) ∈ V }, for each i. Let us remark that the procedure of successive elimination works, since if we
1
take for instance the family F(2,d ;1), then for an arbitrary w ∈W we cannot have two 1-preimagesy and
1
y′ of w(n−2) and 1-preimages ξ of y and ξ′ of y′ such that ξ and ξ′ are both contained in either B (z,ε)
n
or B (z′,ε), for some z,z′ ∈ F(2,d ;1). Indeed, since f2(B (z,ε))∩f2(B (z′,ε)) 6= ∅, in this situation it
n 1 n n
wouldfollowthatU (z)∩U (z′)6=∅andhencewewouldhaveacontradiction. Thisimplies thatthereexist
n n
d disjoint families F(2,d ;i) corresponding to the d 1-preimages of w(n−2)∈V .
1 1 1 1
Clearly,we can proceed analogouslyin the case in which w(n−2)∈V , which then gives rise to pairwise
2
disjoint families F(2,d ;1),...,F(2,d ;d ) for which {5U (z) : z ∈ F(2,d ;j)} covers {w ∈ W : w(n−2) ∈
2 2 2 n 2
V }, for each j. Note that we cannot have repetitions of points from E when taking the union of the
2 n
collectionsF(2,d ;j)overalli∈{1,2}and1≤j ≤d . Indeed,ifwewouldhavetwo1-preimagesy,y′ ∈Λof
i i
somew(n−2)andtwo1-preimagesξ,ξ′ ∈Λofy,andy′ respectively,sothatξ ∈B (z,ε)andξ′ ∈B (z′,ε),
n n
for some z,z′ ∈ F(2,d ;i), then it would follow that U (z) ∩ U (z′) 6= ∅, which gives a contradiction.
1 n n
Moreover, by construction we have that F(2,d ;i)∩F(2,d ;j) = ∅, for all i and j. This follows, since if
1 2
f2(z) ∈ V , for some z ∈ F(2,d ;i), and if at the same time f2(z′) ∈ V , for some z′ ∈ F(2,d ;j), then it
1 1 2 2
would follow that V ∩V 6=∅ and hence, we would get a contradiction.
1 2
Now,asinthe firststep, foreachi∈{1,2}and1≤j ≤d there existsafamily G(2,d ,j)in{F(2,d ;ℓ):
i i k
k ∈{1,2},1≤ℓ≤d } satisfying
j
(diamU (z))t (diamU (z))t
n n
(3) ≤ .
∆(f2(z))...∆(fn(z)) ∆(f(z))∆(f2(z))...∆(fn(z))
z∈GX(2,di,j) z∈FX(2,di;j)
Among these soobtainedfamilies G(2,d ;j)we nowchooseforeachi∈{1,2}aparticularfamily, whichwill
i
be denoted by G(2,d ), such that
i
(diamU (z))t (diamU (z))t
n n
(4) =min :j ∈{1,...,d } .
∆(f3(z))...∆(fn(z))  ∆(f3(z))...∆(fn(z)) i 
z∈GX(2,di) z∈GX(2,di;j) 
Combining (3) and (4), we now have for each i∈{1,2} that
 
(diamU (z))t (diamU (z))t
n n
≤
∆(f3(z))...∆(fn(z)) ∆(f2(z))...∆(fn(z))
z∈GX(2,di) z∈S1≤j≤XdiG(2,di;j)
(diamU (z))t
n
≤ .
∆(f(z))∆(f2(z))...∆(fn(z))
z∈S1≤j≤XdiF(2,di;j)
UPPER ESTIMATES FOR STABLE DIMENSIONS OF FRACTAL SETS WITH VARIABLE NUMBERS OF FOLDINGS 7
Therefore, by defining
F(2):= F(2,d ;j) and G(2):=G(2,d )∪G(2,d ),
i 1 2
i∈[{1,2}1≤[j≤di
we have now shown that
(diamU (z))t (diamU (z))t
n n
(5) ≤ .
∆(f3(z))...∆(fn(z)) ∆(f(z))...∆(fn(z))
z∈XG(2) z∈XF(2)
Continuing the above procedure assume we have constructed a family F(k)⊂E and a subfamily G(k),
n
so that the sets (U (z)) 5-cover W and
n z∈G(k)
(diamU (z))t (diamU (z))t
n n
≤ .
∆(fk+1(z))...∆(fn(z)) ∆(f(z))...∆(fn(z))
z∈XG(k) z∈XF(k)
For each w ∈ W, we then take the k-th iterate of w(n − 1) and denote it by w(n − k − 1); this is an
(n−k−1)-preimage of w in Λ. Now, if w(n−k−1) ∈ V then it has d 1-preimages in Λ and to each
1 1
of these we can apply the same procedure from step k. In this way we obtain by succesive eliminations d
1
mutually disjoint families F(k+1,d ;i),1≤i≤d and inside each of these a subfamily G(k+1,d ;i) such
1 1 1
that
(diamU (z))t (diamU (z))t
n n
≤ .
∆(fk+1(z))...∆(fn(z)) ∆(f(z))...∆(fn(z))
z∈G(kX+1,d1;i) z∈F(Xk+1,d1;i)
The succesive elimination procedure works, since we cannot have two differerent 1-preimages y and y′ of
w(n−k−1) having (n−k)-preimages ξ ∈Λ and ξ′ ∈Λ respectively, such that ξ ∈B (z,ε),ξ′ ∈B (z′,ε),
n n
for some z,z′ ∈ F(k+1,d ;i). Indeed, it would then follow that the family {U (z) : z ∈ F(k+1,d ;i)}
1 n 1
does not consist of pairwise disjoint sets, which clearly is a contradiction. Moreover, since V ∩V = ∅, we
1 2
must have F(k+1,d ;i)∩F(k+1,d ;j)= ∅. Hence, there is no repetition of elements, when we consider
1 2
the union
F(k+1):= ∪ F(k+1,d ;j)∪ ∪ F(k+1,d ;j).
1 2
1≤j≤d1 1≤j≤d2
Now among the collections G(k+1,d ;i), for 1 ≤ i ≤ d , let us consider the one which gives rise to the
1 1
least sum (diamUn(z))t . Denote this minimizing collection by G(k+1,d ). Similarly,
z∈G(k+1,d1;i) ∆(fk+2(z))...∆(fn(z)) 1
we obtain the collection G(k+1,d ). We now have that
P 2
(diamU (z))t (diamU (z))t
n n
≤
∆(fk+2(z))...∆(fn(z)) ∆(fk+1(z))...∆(fn(z))
z∈G(Xk+1,d1) z∈1≤i∪≤d1XG(k+1,d1;i)
(6)
(diamU (z))t
n
≤ .
∆(f(z))...∆(fn(z))
z∈1≤i∪≤d1XF(k+1,d1;i)
Of course,we can proceed similarly for G(k+1,d ). With G(k+1):=G(k+1,d )∪G(k+1,d )., it follows
2 1 2
from above that
(diamU (z))t (diamU (z))t
n n
≤ .
∆(fk+2(z))...∆(fn(z)) ∆(f(z))...∆(fn(z))
z∈GX(k+1) z∈1≤i∪≤Xd1F(k+1)
Therefore, we obtain by finite induction a union F(n) of families in E , as well as one particular family
n
G(n) such that {5U (z):z ∈G(n)} covers the set W and has the property that
n
(diamU (z))t
(diamU (z))t ≤ n .
n ∆(f(z))...∆(fn(z))
z∈XG(n) z∈XF(n)
8 EUGENMIHAILESCUANDBERNDSTRATMANN
By combining this with the observation in (1) at the start of the proof, this shows that
(diamU (z))t <1.
n
z∈XG(n)
Since {5U (z):z ∈G(n)} is a covering of the set W =Ws(x)∩Λ, we can now conclude that
n r
δs(x)≤t<t .
ω
Inthemoregeneralcaseinwhichω isacontinuousfunctiononΛwiththepropertythatω(x)≤∆(x), for
all x∈Λ, we proceedas follows. Firstnote that, by continuity ofω,we have thatthere exists anincreasing,
positive function ρ on Λ such that ρ(ε) decreases to zero for ε tending to zero from above, and such that if
d(y,z)≤ε, then
|ω(y)−ω(z)|≤ρ(ε)
Sinceify ∈B (z,ε)thenfi(y)∈B(fiz,ε),thelatterimpliesthatify ∈B (z,ε)then|ω(fi(y))−ω(fi(z))|≤
n n
ρ(ε). Hence, since ∆(x)≥ω(x) for all x∈Λ, it follows that for each 0≤i≤n−1 we have
∆(fi(y))≥ω(fi(y))≥ω(fi(z))−ρ(ε).
Now in order to proceed, let us define the ε-pressure function P , for some arbitrary potential function ψ,
ε
by
n−1
1
P (ψ):=liminf loginf exp ψ(fk(x)) :E is a (n,ε)-spanning set for Λ
ε
n→∞ n ( ! )
x∈E k=0
X X
andlett denotetheuniquezeroofP (tΦs−log(ω−ρ(ε))). Thenlett>t befixedandnotethattheabove
ε ε ε
proofgoesthroughinthesamewayifinthesumsappearingthere,wereplacethefunction∆bythefunction
ω−ρ(ε). Indeed, this follows since for all 0≤i≤n−1 we have that ∆(fiy)≥ω(fi(y))≥ω(fi(z))−ρ(ε),
for each y ∈B (z,ε) and for some arbitrary fixed element z contained in some minimal (n,ε)-spanning set
n
E for Λ. In this way, the above inductive procedure gives rise to a family F(n) ⊂ E and to a particular
n n
family G(n) such that {5U (z):z ∈G(n)} covers the set W and such that
n
(diamU (z))t
(diamU (z))t ≤ n <1.
n (ω(f(z))−ρ(ε))...(ω(fn(z))−ρ(ε))
z∈XG(n) z∈XF(n)
Now, for η > 0 sufficiently small and 0 < ε < η, let τ refer to the unique zero of the pressure function
ε,η
P (tΦs−log(ω−ρ(η))) and let τ denote the unique zero of the pressure function P(tΦs−log(ω−ρ(η))).
ε η
Since limP (ψ)=P(ψ)for eachcontinuousfunctionψ, itfollowsthatlim τ =τ . Onthe otherhand,
ε ε→0 ε,η η
ε→0
note that for 0<ε <η we have that ρ(ε)<ρ(η) and therefore, tΦs−log(ω−ρ(ε))≤ tΦs−log(ω−ρ(η)).
This implies that τ ≤τ . Now, consider some arbitrary fixed t>τ . For ε>0 sufficiently small, we then
ε ε,η η
have that t > τ ≥ τ . Hence, from the above we have that for t in this range and for n sufficiently large,
ε,η ε
there exists a cover {5U (z):z ∈G(n)} of W such that
n
(diamU (z))t <1.
n
z∈XG(n)
This shows that t ≥ dim (W) and therefore, since t > τ was chosen to be arbitrary, it follows that
H η
τ ≥ dim (W). Finally, observe that the continuity of the pressure function implies that lim τ = t ,
η H η→0 η ω
and this then allows to deduce the desired inequality
dim (W)≤t .
H ω
(cid:3)
UPPER ESTIMATES FOR STABLE DIMENSIONS OF FRACTAL SETS WITH VARIABLE NUMBERS OF FOLDINGS 9
3. Proofs of Proposition 1 and Proposition 2.
Proof of Proposition 1.
Recall that here we assume that d is the minimal value of ∆ on Λ. Then note that, since ∆ is upper
semi-continuous on Λ and takes on only integer values, it follows that if ∆(x) = d for some x∈ Λ, then we
have that the preimage counting function ∆ must be equal to d on some open neighbourhood of x. This
implies that the set
A:={x∈Λ:∆(x)=d}
has to be open in Λ. In order to show that A is dense in Λ, assume to the contrary that there exists a
non-empty open set V ⊂ Λ such that ∆(x) ≥ d+1, for all x ∈ V. In this situation we can then find a
Lipschitz continuous function ψ on Λ such that d≤ψ(x)≤∆(x), for all x∈Λ, and such that ψ ≡d+1 on
some open ball contained in V.
Now note that Theorem 1 implies that t ≥ δs(x), for all x ∈ Λ. Also, since ψ(x) ≥ d for all x ∈ Λ, we
ψ
have that t ≤t . Therefore, if for some x∈Λ we have that t =δs(x), then it follows that
ψ d d
t =t =δs(x).
d ψ
Let us now consider the unique equilibrium measure µ for the Ho¨lder continuous potential t Φs −logψ
ψ d
(note that existence anduniqueness ofµ is guaranteed,since f is hyperbolic on Λ (see [7] and[11])). Also,
ψ
since µ is a f-invariant probability measure for which the Variational Principle holds for the potential
ψ
t Φs−logd, we have that
d
0=P(t Φs−logd)=P(t Φs−logψ)=h + (t Φs−logψ) dµ ≥h + (t Φs−logd) dµ .
d d µψ d ψ µψ d ψ
ZΛ ZΛ
This shows that
logψ dµ ≤ logd dµ .
ψ ψ
ZΛ ZΛ
However, recall that logψ(y) > logd, for all y in some open ball contained in V. Moreover, since µ is an
ψ
equilibrium measure,we have that µ is positive on Bowenballs andhence, it is positive onany open set in
ψ
Λ. Clearly, this gives a contradiction and therefore, it follows that ∆≡d on a dense open set in Λ.
Inordertoshowthatif∆≡donanopendense setthenitfollowsthatδs(y)=t forally ∈Λ,wedefine
d
the set
A :={x∈Λ:x has precisely dn n-preimages y and ∆(fj(y ))=d, for all 0≤j ≤n and 1≤i≤dn}.
n i i
TheaimistoshowthatA isopenanddenseinΛ,foreachn∈N. Forthis,wefirstshowthatA isopenin
n 1
Λ. Bydefinition,wehavethatifx∈A thenx∈Aandhence,xhaspreciselyd1-preimagesx ,...,x ∈A.
1 1 d
Now, let y be a point close to x. Since A is open, we can assume without loss of generality that y ∈A and
hence, y has precisely d preimages y ,...,y ∈ Λ. Since d is the least value ∆ can attain on Λ and since f
1 d
has no critical points in Λ, we have that each of the y is close to one of the x . Since A is open and since
i j
the x are contained in A, it follows that y ∈ A, for all 1 ≤ i ≤ d provided y is close enough to x. This
j i
shows that y ∈A and hence it follows that A is open in Λ.
1 1
Inorderto showthatA is dense inΛ, considersomeopen setV inΛ. Since A is dense inΛ, thereexists
1
some point y ∈A∩V, which must have precisely d 1-preimages y ,...,y ∈Λ. Now, let B ⊂A be a small
1 d
ballcentredaty. For each1≤i≤d, choosea sufficiently smallball B centredat y suchthat the resulting
i i
family of balls is pairwise disjoint and such that f is injective on B and on B ⊂f(B ). The aim is to show
i i
thatB∩f(B ∩A)isopenanddenseinB. Indeed,ifz ∈B∩f(B ∩A), thenz hasa1-preimagez ∈B ∩A.
i i i i
Now, if z′ is close enough to z, then z′ belongs to A and hence, z′ has a 1-preimage z′ ∈B which lies close
i i
to z . Since z ∈A and since A is open, it follows that z′ ∈A. This gives that B∩f(B ∩A) is open in B.
i i i i
Also, if there were a non-empty open set B′ ⊂B such that B′∩f(B ∩A)=∅, then B ∩f−1(B′) would be
i i
10 EUGENMIHAILESCUANDBERNDSTRATMANN
open and non-empty. Clearly, this contradicts the fact that A dense in Λ. This shows that B ∩f(B ∩A)
i
mustbe openanddense inB, forall1≤i≤d. Since a finite intersectionofopenanddense subsetsis again
open and dense, it now follows that A has to be open and dense in Λ.
1
Clearly,the samemethods as inthe previousargumentcanbe usedto proveby wayofinductionthat A
n
is open and dense in Λ, for each n ∈ N. Therefore, we now have that for each n ∈ N, there exists an open
dense set A such that for every y ∈ A there exist exactly dn n-preimages y ,...,y ∈ Λ of y such that
n n 1 dn
∆(fiy )=d, for each0≤i≤n and 1≤j ≤dn . This showsthat in the proofs ofTheorem1 andthe main
j
theorem of [12] one can work exclusively with points from A . Indeed since A is open and dense in
n∈N n n
Λ, it follows that for every z ∈E we can take a point z′ sufficiently close to z such that fn(z′)∈A ; thus
n S n
we obtain a set E′ with the same number of elements as E which is again(n,ε)-spanning and can be used
n n
in the condition on the pressure in order to obtain good covers of Ws(x)∩Λ. Then all the iterates up to
r
order n of any z′ ∈E′ will have exactly d 1-preimages in Λ and then we obtain δs(x)=t , for all x∈Λ.
n d
(cid:3)
Proof of Proposition 2.
(a) In the sequel let x ∈ Λ be fixed and put W := Ws(x)∩Λ. As in the proof of Theorem 1, for each
r
ε>0 sufficiently small, there exists n ∈N and a minimal (n ,ε)-spanning set E for Λ such that for each
0 0 n0
t>t sufficiently large we have, for some fixed β >0,
ω
|Dfn0(z)|t
(7) s <e−βn0 <1/2.
ω(f(z)...ω(fn0z)
z∈XEn0
Let us assume E =: {e ,...,e }. As before, define U (z) := fn(B (z,ε))∩Ws(x), for n ∈ N and
n0 1 m0 n n r
z ∈Λ. The aim is to construct a covering of W which consists of sets of comparable diameter. For this, let
{|Dfsn0(z)|:z ∈En0}=:{δ1,...,δm0} and then define for n∈N the value χ(n) by
n
χ(n):=inf δ :1≤j ≤m .
ji i 0
( )
i=1
Y
Now,foreachw∈Λandforeachnn0-preimagew(−nn0)∈Λofw,wehavethatfjn0(w(−nn0))∈Bn0(zj,ε),
for each 0≤j ≤n−1. From this we deduce that |Dfsnn0(w(−nn0))|≥χ(n). Next observe that in general,
given any full prehistory wˆ = (w,w ,...) ∈ Λˆ of some element w ∈ Λ, there exists k(wˆ,n) ∈ N such that
−1
|Dfk(wˆ,n)n0(w )| is comparable to χ(n), that is,
s −k(wˆ,n)n0
C−1·χ(n)<|Dfk(wˆ,n)n0(w )|<C ·χ(n),
0 s −k(wˆ,n)n0 0
where we have put C0 := supz∈Λ·|Dfsn0(z)|. This shows that for w ∈ W we have that the diameter
diamU (w )iscomparabletoχ(n), wherethe comparabilityconstantdoesdependneitheron
k(wˆ,n)n0 −k(wˆ,n)n0
w nor on n. Hence, the sets U (w ) provide a covering of W and their diameters are all of
k(wˆ,n)n0 −k(wˆ,n)n0
size comparable to χ(n). For later use, let us remark that one can choose a point z (wˆ) ∈ E such
k(wˆ,n) n0
that w−n0k(wˆ,n) ∈Bn0(zk(wˆ,n)(wˆ),ε) and similarly, points zk(wˆ,n)−j(wˆ)∈En0 such that fn0j(w−n0k(wˆ,n))∈
B (z (wˆ),ε), for each 1 ≤j < k(wˆ,n). Then recalling that E =:{e ,...,e }, the inequality in
n0 k(wˆ,n)−j n0 1 m0
(7) reads:
m0 δt 1
i < .
ω(f(ei))...ω(fn0(ei)) 2
i=1
X