Continuity of subadditive pressure for matrix cocycles and the dimension of a self-affine set

56 Pages·2016·0.46 MB·English

Continuity of subadditive pressure for
matrix cocycles and the dimension of
a self-a(cid:14)ne set
Master’s Thesis
Jussi Kormu
Department of Mathematical Sciences
University of Oulu
Spring 2016
Contents
Introduction 2
1 Preliminaries 3
1.1 Ergodic theory . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Symbolic dynamics . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Multilinear algebra . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Pressure functions and introduction of the continuity theo-
rems 13
2.1 Pressure for singular value function and dimension of a self-
a(cid:14)ne set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Generalised pressure for matrix cocycles . . . . . . . . . . . . 16
3 The cone condition 17
3.1 Cones and the cone condition . . . . . . . . . . . . . . . . . . 17
3.2 Block-diagonal matrices . . . . . . . . . . . . . . . . . . . . . 20
3.3 Space of splittings . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Subcocycles satisfying the cone condition 27
5 Subadditive thermodynamic formalism 36
5.1 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.2 Topological pressure and variational principle for subadditive
potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6 Proofs of the continuity results 40
1
Introduction
This master’s thesis is based on the article Non-conformal repellers and con-
tinuity of pressure for matrix cocycles by De-Jun Feng and Pablo Shmerkin
[12]. In the article Feng and Shmerkin prove that subadditive pressure
0 1
1 X
P(A;s) = nl!im1 n log@ ’s(Ai1(cid:1)(cid:1)(cid:1)Ain)A;
(i1;:::;in)2f1;:::;kgn
where ’s is the singular value function, is a continuous function of the linear
maps A = (A ;:::;A ): This, together with Kenneth Falconer’s dimension
1 k
formula 2.4 ([9, Theorem 5.3.]), implies that in some "typical" situation the
dimensionofaself-a(cid:14)nesetvariescontinuouslywiththede(cid:12)ninglinearmaps
A. As a matter of fact, Feng and Shmerkin prove a more general continuity
resultconsideringapressureP (A;s)formatrixcocycles.ContinuityofP((cid:1);(cid:1))
g
is then proved as a corollary.
InthisthesisIprovethecontinuityresultsofFengandShmerkinfollowing
the arguments of the original article. The proofs are given in greater detail
than the original ones. Also more background results and de(cid:12)nitions are
provided, so less preliminary knowledge is assumed on the reader’s part.
The proofs of the continuity theorems make heavy use of ergodic theory
and subadditive thermodynamic formalism. Although the necessary tools
on these areas are covered, there is no attempt to give any comprehensive
introduction to these subjects.
The necessary preliminaries considering ergodic theory, symbolic dynam-
ics, singular values and exterior algebra are given in Section 1. Section 2
introduces the pressure functions and their connection to the dimension of a
self-a(cid:14)ne set. Also the statements of the continuity theorems that are to be
proved, are given.
Matrix cocycles satisfying the cone condition play an important part in
theproofsofthecontinuitytheorems.Theconeconditionandrelatedlemmas
are covered in Section 3. In Section 4 a large subsystem satisfying the cone
condition is constructed for a locally constant matrix cocycle by utilising the
semi-invertible version of the multiplicative ergodic theorem.
Section 5 introduces the measure theoretic entropy and topological pres-
sure for a subadditive potential. These two key concepts of the subadditive
thermodynamic formalism are then linked together by the variational princi-
ple. Finally, in Section 6 the proofs of the continuity results are constructed
with the help of the tools presented in the previous sections.
2
1 Preliminaries
In this section we cover some basic de(cid:12)nitions of ergodic theory alongside
with a few core results, the most important of them being Kingman’s sub-
additive ergodic theorem, which we use repeatedly. The dynamical systems
that we work on are symbolic systems, which are introduced in Subsection
1.2. In Subsection 1.3 we recall some linear algebraic facts mostly considering
singular values. In the last subsection we introduce exterior algebras.
Firstly, we recall an important measure theoretic result, Egorov’s theo-
rem.
Theorem 1.1 (Egorov’s theorem). Let (X;B;(cid:22)) be a measure space and let
Y be a separable metric space. Suppose that f : A ! Y is a sequence of
n
measurable functions on a subset A (cid:26) X, with (cid:12)nite measure, such that f
n
converges to f for (cid:22)-almost every x 2 A: Then for every " > 0 there exists
a measurable set B such that (cid:22)(B) < " and f converges uniformly to f on
n
AnB.
Proof. See [6, Theorem 7.1.12, p. 72] which assumes the knowledge of the
proof [5, Theorem 2.2.1, p. 110].
1.1 Ergodic theory
Ergodic theory studies measure preserving dynamical systems.
De(cid:12)nition 1.2. A quartet (X;B;(cid:22);T) is called a measure preserving trans-
formation if (X;B;(cid:22)) is a measure space and T: X ! X is a measurable
transformation such that (cid:22)(T(cid:0)1(E)) = (cid:22)(E) for every E 2 B. Measure (cid:22) is
referred as a T-invariant measure.
If (X;B;(cid:22)) is a probability space, i.e. a measure space with (cid:22)(X) = 1,
then (X;B;(cid:22);T) is called a probability preserving transformation (ppt).
De(cid:12)nition1.3. Aprobabilitypreservingtransformation(X;B;(cid:22);T)iscalled
ergodic if for any measurable set E 2 B such that T(cid:0)1(E) = E either
(cid:22)(E) = 0 or (cid:22)(E) = 1. The measure (cid:22) and the map T are called an er-
godic measure and an ergodic transformation, respectively.
For every continuous transformation on a compact metric space, there
exists an ergodic measure.
Proposition 1.4. Let X be a compact metric space with the Borel sigma-
algebra B. Given any continuous transformation T: X ! X, there exists at
least one ergodic T-invariant probability measure (cid:22).
3
Proof. [20, Proposition 9.5, p. 93]
We use notation Tk = T (cid:14)(cid:1)(cid:1)(cid:1)(cid:14)T to denote the k fold composition of T
for an integer k (cid:21) 0. For E (cid:26) X, T(cid:0)k(E) denotes the preimage (Tk)(cid:0)1(E).
Proposition 1.5. Let (X;B;(cid:22);T) be a ppt. Following statements are equiv-
alent:
(1) (cid:22) is ergodic
(2) Every measurable L1(X;(cid:22))-function f: X ! R such that f (cid:14) T = f
(cid:22)-almost everywhere, is almost everywhere a constant.
(3) There exists a family A (cid:26) B such that A generates B, A\B 2 A for
every A;B 2 A and
n(cid:0)1
1 X
lim (cid:22)(T(cid:0)k(A)\B) = (cid:22)(A)(cid:22)(B) (1.1)
n!1 n
k=0
for any A;B 2 A.
(4) Equality (1.1) holds for every A;B 2 B.
Proof. Claims (1) and (2) are equivalent by [4, Propositions 2.10 and 2.11,
pp.44{45].Equivalenceofclaims(1),(3)and(4)isprovedin[17,Lemma2.2.2,
p. 30]
Theorem 1.6 (Birkho(cid:11)’s ergodic theorem). Let (X;B;(cid:22);T) be a ppt. If
f 2 L1(X;(cid:22)), then the limit
n(cid:0)1
1 X
f(cid:22)(x) := lim f(Tk(x))
n!1 n
k=0
exists for (cid:22)-almost every x 2 X, and in L1(X;(cid:22)). For (cid:22)-almost everywhere,
(cid:22) (cid:22)
f (cid:14)T = f. Moreover,
Z Z
(cid:22)
fd(cid:22) = fd(cid:22):
If in addition T is ergodic, then
Z
(cid:22)
f(x) = fd(cid:22)
for (cid:22)-almost every x 2 X.
4
Proof. See [17, Theorem 2.1.5, p. 23] for the proof of all but the last claim.
The last claim is then a direct corollary of Proposition 1.5.
Theorem 1.7 (Kingman’s subadditive ergodic theorem). Let (X;B;(cid:22);T) be
a ppt. Let ff g be a sequence of measurable functions such that f (x) (cid:20)
n n2N n+m
f (x)+f (Tn(x)) and f+ is integrable, where f+(x) = maxff(x);0g. Then
n m 1
(cid:22)
f =n converges almost everywhere to a function f: X ! [(cid:0)1;1) such that
n
(1) f(cid:22)+ is integrable
(cid:22) (cid:22)
(2) f (cid:14)T = f for (cid:22)-almost every x 2 X
(3) R f(cid:22)d(cid:22) = lim 1 R f d(cid:22) = inf 1 R f d(cid:22).
n!1 n n n2N n n
(cid:22)
If in addition T is ergodic, then f is constant (cid:22)-almost everywhere and (3)
can be rewritten as
Z Z
1
(cid:22) (cid:22)
f(x) = fd(cid:22) = lim f d(cid:22)
n
n!1 n
for (cid:22)-almost every x 2 X.
Proof. For the proof of (1), (2) and (3) see [3]. Note that even though the
claim (2) is not included in the statement of the theorem in [3], it is proved
duringtheproofsofothertwoclaims.Theergodiccaseisthendirectcorollary
of Proposition 1.5.
Theorem 1.8 (Khintchine’s recurrence theorem). Let (X;B;(cid:22);T) be a ppt.
For any " > 0 and any measurable set E 2 B, the set
F = fk 2 Z: (cid:22)(T(cid:0)k(E)\E) (cid:21) (cid:22)(E)2 (cid:0)"g
"
has bounded gaps, i.e. there is K 2 N such that F \fj;j+1;:::;j+K(cid:0)1g =6
"
;, for every j 2 Z.
Proof. See [19, Theorem 3.3, p. 37]
1.2 Symbolic dynamics
As a reference for the claims of this subsection see [7, Chapter 3] and [4,
Chapter 3.5]
5
De(cid:12)nition 1.9. For m > 1, we refer to the set A = f1;:::;mg as an
m
alphabet and its elements as symbols. Let (cid:6)+ := AN be the set of all one-
m m
sided in(cid:12)nite sequences of symbols of A and let (cid:6) := AZ be the set of
m m m
all two-sided in(cid:12)nite sequences. A (cid:12)nite sequence of symbols is called a word
and an in(cid:12)nite sequence x = (x ), one or two-sided, is said to contain the
i
word i = i ;i ;:::;i if for some j, i = x for l = 1;:::;k. Length of a
1 2 k l j+l
word i is denoted by jij. A word made of (cid:12)rst n symbols of a given sequence
x is denoted by xjn := x ;:::;x .
1 n
We de(cid:12)ne the shift map T: (cid:6)+ ! (cid:6)+ by setting T((x )1 ) = (x )1 .
m m i i=1 i+1 i=1
In the case of two-sided sequences the shift map is de(cid:12)ned as T: (cid:6) ! (cid:6),
T((x )1 ) = (x )1 . The pair ((cid:6)+;T) is called the full one-sided shift
i i=(cid:0)1 i+1 i=(cid:0)1 m
on alphabet A and the pair ((cid:6) ;T) is called the full two-sided shift.
m m
Full shift is a compact metric space with respect to the metric
d(x;y) = d((x );(y )) = 2(cid:0)minfjij:xi6=yig;
i i
where i 2 N in the case of one-sided shift or i 2 Z in the case of two-sided
shift. For a given word i = i ;:::;i 2 An and k 2 N, we de(cid:12)ne a cylinder
1 n m
set
[i] = f(x )1 2 (cid:6)+: x ;x ;:::;x = ig:
k i i=1 m k k+1 k+n(cid:0)1
The collection of all cylinder sets generates the Borel sigma-algebra of
(cid:6)+. Actually the cylinder sets with k = 1 are enough to generate B. In the
m
case of two-sided shift we de(cid:12)ne a cylinder set as
[i] = f(x )1 2 (cid:6) : x ;x ;:::;x = ig
k i i=(cid:0)1 m k k+1 k+n(cid:0)1
fork 2 Z.ThecollectionofallcylindersetsgeneratestheBorelsigma-algebra
of (cid:6) . We write [i] := [i] .
m 1
De(cid:12)nition 1.10. A subspace (X;Tj ) (cid:26) ((cid:6)+;T) is called a one-sided sub-
X m
shift of (cid:12)nite type if there exists an m(cid:2)m-matrix A with entries in f0;1g,
such that
X = fx 2 (cid:6)+: A = 1 for every i 2 Ng:
m xi;xi+1
The de(cid:12)nition of two-sided subshift of (cid:12)nite type is obtained by replacing (cid:6)+
m
with (cid:6) and N with Z. The matrix A is called a transition matrix. A word
m
w is allowed in X if there is a sequence x 2 X containing w. The set of all
allowed (cid:12)nite words in X is denoted by X(cid:3) and the set of all allowed words
of length n is denoted by X(cid:3).
n
Remark 1.11. Subshifts of (cid:12)nite type are often de(cid:12)ned more generally. Let
L be a (cid:12)nite list of forbidden words. If (X;Tj ) (cid:26) ((cid:6)+;T) is such that X
X m
6
is closed, T(X) (cid:26) X and no sequence in X contains a word from L, then
(X;Tj ) is called a subshift of (cid:12)nite type. De(cid:12)nition 1.10 is a special case
X
of this de(cid:12)nition, corresponding to the case where all forbidden words have
length 2. Nevertheless, every subshift of (cid:12)nite type (X;T ), de(cid:12)ned in the
X
general sense, is conjugate to some subshift of (cid:12)nite type (Y;T ), de(cid:12)ned in
Y
the sense of De(cid:12)nition 1.10 (see [24, p. 4-5]). Being conjugate means that
there exists a bijection (cid:30): X ! Y such that T (cid:14) (cid:30) = (cid:30) (cid:14) T . Thus, it is
Y X
justi(cid:12)ed to consider only subshifts of (cid:12)nite type given by De(cid:12)nition 1.10.
If (X;T) is a subshift of (cid:12)nite type, then X is a closed subset of the
full shift and hence a compact metric space. The shift map T: X ! X is
a continuous transformation. If B is the Borel sigma-algebra of X, then by
Proposition 1.4 there exists an ergodic T-invariant probability measure (cid:22).
1.3 Linear algebra
Let E = fe ;e ;:::e g be a set of linearly independent vectors on Rd. We
1 2 d
de(cid:12)ne an inner product ((cid:1)j(cid:1)) with respect to this base by requiring
(
0; if i 6= j
(e je ) = :
i j
1; if i = j
We call the basis E the canonical basis of the inner product space (Rd;((cid:1)j(cid:1))).
p
The inner product induces the norm jvj = (vjv) for every v 2 Rd.
In this thesis we work with linear maps from Rd to Rd. For this reason,
the following de(cid:12)nitions are given in the context of real square matrices. On
the set of matrices we use the operator norm kMk = sup jMvj, for
v2Rd;jvj=1
every M 2 Rd(cid:2)d.
De(cid:12)nition 1.12. A symmetric matrix M 2 Rd(cid:2)d with only non-negative
eigenvaluesiscalled positive semi-de(cid:12)nite.IfM hasonlypositiveeigenvalues,
then M is called positive de(cid:12)nite. An nth root of a positive semi-de(cid:12)nite
matrix M is a positive semi-de(cid:12)nite matrix M such that Mn = M. The nth
0 0
root of M is uniquely de(cid:12)ned and is denoted by M1=n ([16, Theorem 7.2.6,
p. 405]).
Lemma 1.13. For any matrix M 2 Rd(cid:2)d, the product MM(cid:3) is symmetric
and positive semi-de(cid:12)nite, where M(cid:3) denotes the transpose of M. If M is
non-singular, then MM(cid:3) is positive de(cid:12)nite.
Proof. The matrix MM(cid:3) is symmetric since (MM(cid:3))(cid:3) = (M(cid:3))(cid:3)M(cid:3) = MM(cid:3).
Let (cid:21) be an eigenvalue of MM(cid:3) and let v be a unit eigenvector for (cid:21). The
7
eigenvalue (cid:21) is non-negative since
jMvj2 = (MvjMv) = (Mv)(cid:3)(Mv) = v(cid:3)M(cid:3)Mv = (cid:21)v(cid:3)v = (cid:21):
Thus, all the eigenvalues of MM(cid:3) are non-negative. If M is non-singular,
then jMvj > 0, which implies that all the eigenvalues are positive.
De(cid:12)nition 1.14. Let M: Rd ! Rd be a linear map. The eigenvalues
(cid:11) (M) (cid:21) ::: (cid:21) (cid:11) (M) (cid:21) 0 of (MM(cid:3))1=2 listed with algebraic multiplici-
1 d
ties are called the singular values of M. Equivalently singular values can be
de(cid:12)ned to be the lengths of semi-axes of the ellipsoid M(B), where B (cid:26) Rd
is the unit ball. Throughout this thesis the singular values are indexed in
decreasing order, usually without speci(cid:12)cally mentioning it.
De(cid:12)nition 1.15. One can show that for every matrix M 2 Rd(cid:2)d there exists
a decomposition of the form M = UDV, where U;V are orthogonal matrices
and D is a diagonal matrix with diagonal elements being the singular values
of M (see [18, Theorem 2, p. 192]. Such decomposition is called a singular
value decomposition (SVD) of M. We can assume that the constructed SVD
issuchthatthediagonalmatrix D liststhesingularvaluesof M indecreasing
order, i.e. D = diag((cid:11) (M);:::;(cid:11) (M)).
1 d
De(cid:12)nition 1.16. A block matrix A 2 Rd(cid:2)d of the form
2 3
A 0 ::: 0
1
6 0 A ::: 0 7
2
A = 66 ... ... ... ... 77;
4 5
0 0 ::: A
n
where A 2 Rdi(cid:2)di, Pn d = d and the zeros are blocks of zeros, is called a
i i=1 i
block-diagonal matrix. We write
n
M
A = A :
i
i=1
The following lemma is a useful observation about how to construct a
SVD of a block diagonal matrix from SVDs of the diagonal blocks.
Lemma 1.17. Let A be a d (cid:2)d -square matrix and let U D V be its singular
i i i i i i
value decomposition, for i = 1;:::;n: A singular value decomposition of A =
Ln A is given by
i=1 i
! ! !
n n n
M M M
U D V :
i i i
i=1 i=1 i=1
8
Notethatthediagonalmatrix Ln D doesnotnecessarilylistthesingu-
i=1 i
lar values of A in decreasing order but this is the case if the singular values of
A are greater than or equal to the singular values of A for i = 1;:::;n(cid:0)1.
i i+1
The next proposition lists some facts about eigen- and singular values
needed in the later sections.
Proposition 1.18. Let M 2 Rd(cid:2)d and let (cid:21) (M);:::;(cid:21) (M) 2 C be the
1 d
eigenvalues of M listed with algebraic multiplicities and so that j(cid:21) (M)j >
i
j(cid:21) (M)j for every i. Also let (cid:11) (M) (cid:21) ::: (cid:21) (cid:11) (M) be the singular values
i+1 1 d
of M.
(1) jdetMj = Qd (cid:11) (M) and detM = Qd (cid:21) (M)
i=1 i i=1 i
(2) Maps M 7! detM, M 7! (cid:11) (M) and M 7! (cid:21) (M) are continuous. The
i i
set of matrices is concidered with operator norm induced by euclidean
norm.
(3) kMk = (cid:11) (M)
1
(4) For invertible M, kM(cid:0)1k = 1=(cid:11) (M)
d
(5) (cid:21) ((MM(cid:3))1=2n) = ((cid:21) (MM(cid:3)))1=2n for i = 1;:::;d, n 2 N.
i i
Proof. The (cid:12)rst equality of (1) is obtained by taking SVD of M and calculat-
ing the determinant. Let (cid:21) ;:::;(cid:21) be the eigenvalues of M. Recall that the
1 d
eigenvalues of M are the roots of characteristic polynomial p(t) = det(M (cid:0)
tI).Factoringthecharacteristicpolynomialas p(t) = ((cid:0)1)d(t(cid:0)(cid:21) ):::(t(cid:0)(cid:21) )
1 d
and setting t = 0 results the second equality.
For (2), note that the coe(cid:14)cients of characteristic polynomial depend
continuously on M and recall that the roots of polynomial depend contin-
uously on the polynomial coe(cid:14)cients. Thus, the eigenvalues of M depend
continuously on M. Now rest of (2) follows easily from (1) and the de(cid:12)nition
of singular values.
The truth of claim (3) is obvious if we adopt the alternative de(cid:12)nition of
singular values via lengths of semi-axes of transformed unit ball.
For (4), let (cid:11) be the least singular value of M and let v be a unit vector
d
such that jMvj = (cid:11) . As jM(cid:0)1Mv=(cid:11) j = 1=(cid:11) , we have kM(cid:0)1k (cid:21) 1=(cid:11) . On
d d d d
the other hand, it is impossible for kM(cid:0)1k to be strictly greater than 1=(cid:11) ,
d
since otherwise
(cid:12) (cid:12)
(cid:12) M(cid:0)1w (cid:12) 1
(cid:12)M (cid:12) = < (cid:11)
(cid:12) jM(cid:0)1wj(cid:12) jM(cid:0)1wj d
for some unit vector w 2 Rd. This would contradict the fact that (cid:11) =
d
min jMxj.
jxj=1
9