Proof of Theorem 2.1.2

and define the lower pressure:

P(s) := lim inf

k→∞

1

k logX

|i|=k

φ^s(i). (2.1.5)

Theorem 2.1.2. Let 0 ≤ s ≤ n. If F1, ..., Fl are contractive maps in form (2.1.1) and Fi ∈C^1+ε for every 1≤i≤l then

P(s) =P(s).

The proof of Theorem 2.1.2 is based on [B4] which uses the technique of [FM]. The result of the chapter was part of author’s Master Thesis.

The norms on Φ^minduce norms on the space of linear mappingsL(Φ^m,Φ^m) in the usual way by

eT= sup

w∈Φ^m,w6=0

eT w

kwk . Then with respect to the normk.k

eT=φ^m(T) (2.2.1)

and with respect to the k.k_∞

eT

∞= maxT^(m):T^(m) is anm×m minor ofT , (2.2.2) where T^(m) = T _s^r₁¹_,...,s^,...rm

m

is the determinant of that m × m minor of n × n matrix T which is determined by the elements of T in the rows 1 ≤ r1 < ... < rm ≤ n and columns 1 ≤ s1 < ... < sm ≤ n.

The space of linear mappings L(Φ^m,Φ^m) is of finite dimension _mⁿ2

. Since any two norms on a finite dimensional normed space are equivalent, there are constants 0< c1 < c2 <∞ depending only on n and m such that

c1

eT

∞ ≤ eT≤c2

eT

∞. (2.2.3)

Now we notice several lemmas relating to minors of matrices. We will need some well-known lemmas.

Lemma 2.2.1. Let xi ≥0, i= 1, ..., m and p∈R⁺.

1. If p >1, then (x^p₁+...+x^p_m)≤(x1+...+xm)^p ≤m^p−1(x^p₁+...+x^p_m) 2. If0< p≤1, then m^p−1(x^p₁+...+x^p_m)≤(x1+...+xm)^p ≤(x^p₁+...+x^p_m).

Lemma 2.2.2. Let an be a sequence of real numbers such that an+m ≤an+ am. Then there existslimn→∞an

n and it equals to infn an

n.

We first look at the expansion ofm×mminors of the product ofkmatrices A=A1A2· · ·Ak, where for i= 1, ..., k

Ai =







aⁱ₁₁ aⁱ₁₂ ... aⁱ_1n aⁱ₂₁ aⁱ₂₂ ... aⁱ_2n ... ... . .. ... aⁱ_n1 aⁱ_n2 ... aⁱ_nn





.

Lemma 2.2.3. For 1 ≤ m ≤ n, the m×m minors of A = A1· · ·Ak have formal expansions in terms of the entries of the Ai of the form

A

r1, ...rm

s₁, ..., sm

= X

c₁,...,c_k

±a¹_1(c1)· · ·a¹_m(c1)a²_1(c2)· · ·a²_m(c2)· · ·a^k_1(ck)· · ·a^k_m(ck) such that for each i = 1, ..., k, the aⁱ_1(ci)· · ·aⁱ_m(ci) are distinct entries aⁱ_rs of Ai. In particular, for each i, 1(ci), ..., m(ci) denote pairs (r, s) corresponding to entries in m different rows and columns of the ith matrixAi, and the sum is over all such entry combinations (c1, ..., ck) with appropriate sign ±.

The proof of this Lemma can be found in [FM, Lemmma 2.2]. Now we consider lower triangular matrices. For i= 1, ..., k, let

Ui =







uⁱ₁ 0 ... 0 uⁱ₂₁ uⁱ₂ ... 0 ... ... ... ...

uⁱ_n1 uⁱ_n2 ... uⁱ_n





.

We consider the product

U =U1· · ·Uk=







u1 0 ... 0

u21 u2 ... 0 ... ... . .. ... un1 un2 ... un





.

We note that

urs = X

r≥r1≥...≥rk−1≥s

u¹_rr1u²_r1r2· · ·u^k_rk−1s 1≤r ≤s≤n (2.2.4) since all other products are 0.

Lemma 2.2.4. With the notation as above, let U1, ..., Uk be lower triangular matrices and U =U1· · ·Uk. Then

1. If r < s, urs= 0

2. If r=s, urs≡ur =u¹_r· · ·u^k_r

3. If r > s, then the sum (2.2.4) for urs has at most k^r−s ≤ kⁿ⁻¹ non-zero terms. Moreover, each non-non-zero summand u¹_rr1u²_r1r2· · ·u^k_rk−1s has at mostn−1 non-diagonal terms in the product, i.e. terms withr6=r1

or ri 6=r_i+1 or rk−1 6=s.

The proof can also be found in [FM, Lemma 2.3] for upper-triangular matrices. Now we extend the estimate of Lemma 2.2.4 to minors.

Lemma 2.2.5. Let U1, ..., Uk and U be lower triangular matrices as above.

Then each m×m minor of U has an expansion of the form U

r₁, ...rm

s1, ..., sm

= X

c1,...,ck

±u¹_1(c1)u²_1(c2)· · ·u^k_1(ck)· · ·u¹_m(c1)u²_m(c2)· · ·u^k_m(ck) where 1(ci), ..., m(ci) are as in Lemma 2.2.3 and

1. there are at most m!k^m(n−1) terms in the sum which are non-zero, 2. each summand contains at most(n−1)^m non-diagonal elements in the

product.

The proof is equivalent to the proof of [FM, Lemma 2.4]. Before we prove Theorem 2.1.2, we define two sums.

H(s, r) = max

j1,...,jm−1

j₁^′,...,j_m^′

X

|i|=r

(dj1j1(i)· · ·djm−1jm−1(i))^m−s(dj₁^′j^′₁(i)· · ·dj_m^′ j_m^′ (i))^s−m+1 (2.2.5) where m−1< s≤m and djj(i) = infx|xjj(i, x)|. Moreover

T(s, r) = max

j1,...,jm−1

j₁^′,...,j_m^′

X

|i|=r

(tj₁j₁(i)· · ·tjm−1jm−1(i))^m−s(tj₁^′j₁^′(i)· · ·tj_m^′ j^′_m(i))^s−m+1 (2.2.6) where m −1 < s ≤ m and tjj(i) = sup_x|xjj(i, x)|. It is easy to see from Proposition 2.1.1 and the definition of the two sums that

H(s, r)≤T(s, r)≤C^sH(s, r). (2.2.7) Lemma 2.2.6. For every positive integers r, z, T(s, r+z)≤T(s, r)T(s, z).

Moreover limr→∞ logT(s,r)

r exists and equal with infrlogT(s,r)

r .

Proof of Lemma 2.2.6. From the definition ofT(s, r) it follows T(s, r+z) = max

j1,...,jm−1

j₁^′,...,j_m^′

X

|i|=r+z

(tj1j1(i)· · ·tjm−1jm−1(i))^m−s(tj^′₁j₁^′(i)· · ·tj^′_mj_m^′ (i))^s−m+1≤

≤ max

j1,...,jm−1

j^′₁,...,j_m^′

(X

|i|=r

X

|h|=z

((tj₁j₁(i)tj₁j₁(h)· · ·tjm−1jm−1(i)tjm−1jm−1(h))^m−s×

×(tj^′₁j₁^′(i)tj^′₁j₁^′(h)· · ·tj_m^′ j_m^′ (i)tj_m^′ j^′_m(h))^s−m+1) =

= max

j1,...,jm−1

j₁^′,...,j^′_m

(X

|i|=r

(tj₁j₁(i)· · ·tjm−1jm−1(i))^m−s(tj^′₁j₁^′(i)· · ·tj^′_mj_m^′ (i))^s−m+1×

× X

|h|=z

(tj₁j₁(h)· · ·tjm−1jm−1(h))^m−s(tj^′₁j₁^′(h)· · ·tj_m^′ j_m^′ (h))^s−m+1))≤

≤T(s, r)T(s, z).

The existence of the limit follows from Lemma 2.2.2.

The proof of Theorem 2.1.2 follows the method of the proof of [FM, The-orem 2.5], but our theThe-orem is not a consequence of it. The most important difference is that the functions in [FM] are affine maps. So the derivatives in our case are not constant matrices. Moreover, in the proof of [FM, The-orem 2.5], the singular value functions and the minors of the derivative ma-trices were compared. During the proof of Theorem 2.1.2 we will do this as well, however, we have to introduce in the proof a new IFS, which will be the r-th iteration of the original IFS, to take separation between the growth rate of the non-zero and the non-diagonal terms of the minors of the derivative matrices.

To control the consequences of the phenomenon of not constant matrices, we have to state the following lemma.

Lemma 2.2.7. LetX be a compact subset ofRⁿand let{fi}be finitely many continuous, real valued functions. Then

sup

x∈X

maxi fi(x) = max

i sup

x∈X

fi(x).

Proof of Lemma 2.2.7. SinceXis compact, we havex_i ∈Xsuch thatfi(x_i) = sup_xfi(x). Therefore

sup

x

maxi fi(x)≤max

i sup

x

fi(x) = max

i fi(x_i) = max

i,j fi(x_j) = max

j max

i fi(x_j)

≤sup

x

maxi fi(x), which was to be proved.

Proof of Theorem 2.1.2. Let

{Gh}^lh=1^r ={Fi1...ir}^l,...,li₁=1,...,ir=1. (2.2.8) In this case an index his equivalent to a i∈ {1, ..., l}^r finite sequence, length r. Let us define

φ^′s(h) = sup

x

φ^s(DxGh), φ^′s(h) = inf

x φ^s(DxGh)

for h∈ {1, ..., l^r}^∗, corresponding to IFS {Gh}^lh=1^r , see (2.1.3).

It is easy to see that X

|i|=kr

φ^s(DxFi) = X

|h|=k

φ^s(DxGh), (2.2.9) where i ∈ {1, ..., l}^kr and h ∈ {1, ..., l^r}^k. The elements of DxGh, denoted by yij(h, x), are equal to xij(i, x) for an appropriate finite sequence i with length r. It is very simple to see that

φ^s(DxGh) = (φ^m−1(DxGh))^m−s(φ^m(DxGh))^s−m+1,

wherem−1< s≤m. By using relations (2.2.1), (2.2.2) and (2.2.3) it follows that

φ^m(DxGh)≥c2maxnDxG^(m)_h :DxG^(m)_h is anm×mminor ofDxGh

o. The maximum m × m minor of DxGh is at least the largest product of m distinct diagonal elements of DxGh, since such products are themselves minors of triangular matrices. Therefore

φ^′s(h)≥ c^s₂

infx

yj₁j₁(h, x)· · ·yjm−1jm−1(h, x)^m−s infx

yj₁^′j₁^′ (h, x)· · ·yj_m^′ j_m^′ (h, x)^s−m+1

for every j₁, ..., jm−1, j₁^′, ..., j_m^′ .

By the chain rule DxGh = DGh2...hk(x)Gh1DGh3...hk(x)Gh2· · ·DxGhk, yjj(h, x) = yjj(h1, Gh₂...hk(x))yjj(h2, Gh₃...hk(x))· · ·yjj(hk, x). It follows with the notation infx|yjj(h, x)| =d^′_jj(h) that

infx

y_j1j1(h, x)· · ·yjm−1jm−1(h, x)^m−sinf

x

y_j^′

1j^′₁(h, x)· · ·yj_m^′ j^′_m(h, x)^s−m+1 ≥

≥(d^′_j1j1(h1)· · ·d^′_j1j1(hk)d^′_j2j2(h1)· · ·d^′_jm−1jm−1(h1)· · ·d^′_jm−1jm−1(hk))^m−s×

×(d^′_j^′

1j^′₁(h1)· · ·d^′_j^′

1j₁^′(hk)d^′_j^′

2j₂^′(h1)· · ·d^′_j^′

mj^′_m(h1)· · ·d^′_j^′

mj^′_m(hk))^s−m+1.

The next inequality follows from the rearrangement of the product X

|h|=k

φ^′s(h)≥ c^s₂ X

|h|=k

(d^′_j1j1(h1)· · ·d^′_jm−1jm−1(h1))^m−s(d^′_j^′

1j₁^′(h1)· · ·d^′_jm^′ _jm^′ (h1))^s−m+1· · ·

· · ·(d^′_j1j1(hk)· · ·d^′_jm−1jm−1(hk))^m−s(d^′_j^′

1j^′₁(hk)· · ·d^′_j^′

mj^′_m(hk))^s−m+1 = c^s₂((d^′_j1j1(1)· · ·d^′_jm−1jm−1(1))^m−s(d^′_j^′

1j₁^′(1)· · ·d^′_j^′_mjm^′ (1))^s−m+1+· · ·

· · ·+ (d^′_j1j1(l^r)· · ·d^′_jm−1jm−1(l^r))^m−s(d^′_j^′

1j₁^′(l^r)· · ·d^′_j^′

mj_m^′ (l^r))^s−m+1)^k.

The inequality above is true for every j₁, ..., jm−1, j₁^′, ..., j_m^′ , therefore we ob-tain the maximum. From the definition of {Gh}^lh=1^r and H(s, r), see (2.2.5) and (2.2.8), it follows

X

|h|=k

φ^′s(h)≥c^s₂H(s, r)^k. (2.2.10)

By using relations (2.2.1), (2.2.2) and (2.2.3) it follows similarly that φ^m(DxGh)≤c1maxnDxG^(m)_h :DxG^(m)_h is anm×mminor ofDxGh

o.

Therefore X

|h|=k

φ^′s(i)≤

c²₁ X

|h|=k

sup

x

m−1×maxm−1 minor

DxG^(m_h ⁻¹⁾

m−s sup

x

m×maxmminor

DxG^(m)_h

s−m+1

. By Lemma 2.2.7, the order of the supremum and the maximum can be changed in this situation and we can estimate the sum with

C max

r1,...,rm−1

s₁,...,sm−1

_ max





r′ 1,...,r′

m

s^′₁,...,s^′_m





X

|h|=k

sup

x

DxG^(m_h ⁻¹⁾

^m−s sup

x

DxG^(m)_h

^s−m+1

where r1, ..., rm−1 are the rows and s1, ..., sm−1 are the columns of the (m−1)×(m−1) minor, and r₁^′, ..., r^′_m are the rows and s^′₁, ..., s^′_m are the

columns of m×m minor, moreover C =c²_{1 m}ⁿ2 n m−1

2

. By the chain rule DxGh=DG_h2...hk(x)Gh₁DG_h3...hk(x)Gh₂...DxGh_k, we obtain

DxGh

r1, ..., rm

s₁, ..., sm

= X

c₁,...,c_k

±y1(c1)(h1, Gh₂...h_k(x))...y1(ck)(hk, x)...ym(c1)(h1, Gh₂...h_k(x))×

×ym(c2)(h2, Gh₃...hk(x))...ym(ck)(hk, x).

(2.2.11)

Therefore sup

x

DxG^(m)_h ≤ X

c₁,...,ck

sup

x

y1(c1)(h1, x)...sup

x

y1(ck)(hk, x)...sup

x

ym(c1)(h1, x)×

×sup

x

y_m(c2)(h2, x)...sup

x

y_m(ck)(hk, x). (2.2.12)

Denote by t^′_kl(h) := sup_x|ykl(h, x)| the supremum. It follows from the in-equality (2.2.12) and the Lemma 2.2.1

X

|h|=k

sup

x

DxG^(m_h ⁻¹⁾

m−s

sup

x

DxG^(m)_h

s−m+1

≤ X

c1,...,ck

c^′₁,...,c^′_k

((t^′_1(c1)(1)...t^′_m−1(c1)(1))^m−s(t^′_1(c^′

1)(1)...t^′_m(c^′

1)(1))^s−m+1+ ...+ (t^′_1(c1)(l^r)...t^′_m−1(c1)(l^r))^m−s(t^′_1(c^′

1)(l^r)...t^′_m(c^′

1)(l^r))^s−m+1)× ...×((t^′_1(ck)(1)...t^′_m−1(ck)(1))^m−s(t^′_1(c^′

k)(1)...t^′_m(c^′

k)(1))^s−m+1+ ...+ (t^′_1(ck)(l^r)...t^′_m−1(ck)(l^r))^m−s(t^′_1(c^′

k)(l^r)...t^′_m(c^′

k)(l^r))^s−m+1).

(2.2.13)

Lemma 2.2.5 implies that each non-zero term of the sum above has at most 2(n − 1)^m = b of the indices 1(c₁), ..., m − 1(c₁), ...,1(ck), ..., m − 1(ck), 1(c^′₁), ..., m(c^′₁), ...,1(c^′_k), ..., m(c^′_k) that are non-diagonal terms. Thus, for each set of indices (c1, ..., ck, c^′₁, ..., c^′_k), we have at least k − b of these in-dices such that 1(cr), ..., m− 1(cr),1(c^′_r), ..., m(c^′_r) are all diagonal entries.

For such cr and c^′_r

((t^′_1(cr)(1)...t^′_m−1(cr)(1))^m−s(t^′_1(c^′_r)(1)...t^′_m(c^′_r)(1))^s−m+1+...

...+ (t^′_1(cr)(l^r)...t^′_m−1(c1)(l))^m−s(t^′_1(c^′

r)(l^r)...t^′_m(c^′

r)(l^r))^s−m+1)≤ max

{j₁,...,jm−1},{^{j1′,...,j′m}}((t^′_j1j1(1)...t^′_jm−1jm−1(1))^m−s(t^′_j^′

1(1)...t^′_j^′_m(1))^s−m+1+...

...+ (t^′_j1j1(l^r)...t^′_jm−1jm−1(l^r))^m−s(t^′_j^′

1(l^r)...t^′_j^′

mj_m^′ (l^r))^s−m+1) =T(s, r).

The last equality follows from the definition of {Gh}^lh=1^r and T(s, r). Hence from (2.2.13)

X

|h|=k

sup

x

DxG^(m_h ⁻¹⁾

m−s

sup

x

DxG^(m)_h

s−m+1

≤ X

c1,...,ck

c^′₁,...,c^′_k

T(s, r)^k−b(l^r)^b

≤c^′′k^ql^rbT(s, r)^k−b, (2.2.14)

where, using Lemma 2.2.5, c^′′ =m!(m−1)! andq= (2m−1)(n−1).

By using (2.2.7), (2.2.9), (2.2.10) and (2.2.14) X

|i|=kr

φ^s(i) = X

|h|=k

φ^′s(h)≤c^′′k^ql^rbT(s, r)^k−b ≤c^′′(C^s)^kk^ql^rbT(s, r)^−bH(s, r)^k ≤ c^′′′(C^s)^kk^ql^rbT(s, r)^−b X

|h|=k

φ^′s(h) =c^′′′k^ql^rbT(s, r)^−b X

|i|=kr

φ^s(i). (2.2.15) We take the logarithm of both sides of the inequality and we divide by kr, then

logP

|i|=krφ^s(i)

kr ≤

logc^′′′

kr +qlogk

kr +rblogl

kr + (kb) log(C^s)

kr + −blogT(s, r)

kr +logP

|i|=krφ^s(i) kr(2.2.16) is true for every positivek, rinteger. We take limit inferior of both sides. The limit exists in the left-hand side of the inequality and in the right-hand side the limit of every term exists and equals zero except the last term. Therefore

P(s)≤P(s)

While the opposite relation is trivial this completes the proof.

The next corollary is a consequence of the previous proof.

Corollary 2.2.8. For 0 ≤ s ≤ n. If F₁, ..., Fl contractive maps in form (2.1.1) and Fi ∈C^1+ε for every 1≤i≤l then

P(s) = lim

r→∞

1

rlog( max

j1,...,jm−1

j₁^′,...,j^′_m

X

|i|=r

|xj1j1(i, x)|...x_jm−1jm−1(i, x)^m−s×

× xj^′₁j₁^′ (i, x)...xj^′_mj_m^′ (i, x)^s−m+1) (2.2.17) for every x∈M.

Proof. It follows from inequality (2.2.7) that the limr→∞ logH(s,r)

r exists and

rlim→∞

logH(s, r)

r = lim

r→∞

logT(s, r)

r .

It is clear by (2.2.15) that limr→∞ logT(s,r)

r =P(s). Because of the definition H(s, r), T(s, r), this is exactly what we want to prove.

In document Bal´azsB´ar´any DimensionTheoryofNon-conformalAttractorsandOverlappingSelf-similarSets (Pldal 51-60)