LYAPUNOV EXPONENTS FOR RANDOM
MATRIX PRODUCTS USING DETERMINANTS
Mark Pollicott
University of Warwick
Abstract. In this article we study the Lyapunov exponent for random matrix products and express them in terms of associated complex functions. This leads to new
formulae for the Lyapunov exponents and to an efficient method for their computation.
0. Introduction
In this article we study Lyapunov exponents for random variables taking values in
a finite set {A1 , · · · , Ak } of non-singular d × d real matrices, with d ≥ 2, by relating
them to associated complex functions. in particular, we present both a new formula
for the Lyapunov exponents, in terms of finite products of the matrices, and a new
method for their computation, which proves remarkably efficient in practice.
For any n ≥ 1, we can consider products Ai1 · · · Ain from this finite set of ma+
trices. Let (p1 , · · · , pk ) be a probability vector, and let µ = (p1 , · · · , pk )Z be the
+
associated Bernoulli measure on the space of sequences Σ = {1, · · · , k}Z . The
Lyapunov exponent λ is given by the limit
Z
1
(0.1)
λ = lim
log ||Ai1 · · · Ain ||dµ(i)
n→+∞ n
where i = (in )∞
n=0 ∈ Σ. By a famous result of Kesten and Furstenberg from 1960
there is the following pointwise version [6].
Theorem (Kesten-Furstenberg). For a.e. (µ) i ∈ Σ one has
λ = lim
n→+∞
1
log ||Ai1 · · · Ain ||.
n
(0.2)
The Lyapunov exponent plays an important in a number of different contexts
including the study of the Ising model, Schrödinger equations, and the Hausdorff
dimension of measures [1], [2]. Recently there has been renewed interest because of
their usefulness in the study of the entropy rates of Hidden Markov Models.
The numerical computations in this article were carried out using Mathematica running on a
MacIntosch MacBook Pro. I am grateful to P. Cvitanovic, K. Khanin, B. Marcus, Y. Peres, A.
Quas for useful comments.
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1
2
It is a fundamental problem to find both an explicit expression for λ and a useful
method of accurate approximation. In [13, p.897] Kingman comments: “Pride of
place among the unsolved problems of subadditive ergodic theory must go to the
calculation of [the Lyapunov exponent]”. Unfortunately, there are few analytic
techniques available to study Lyapunov exponents. Traditionally, they have been
approximated using classical methods (such as Monte Carlo approximations, weak
disorder expansions and microcanonical ensembles cf. [2], [14], [16]). In a few very
special cases, it is also possible to determine explicitly the associated Furstenberg
measure on RP d , from which the Lyapunov exponent can be deduced, cf., [15] for
examples and references. We will return again to this point in §6.
Henceforth, we shall make the following standing assumption.
Assumption 1. All of the entries of Aj are strictly positive, for each j = 1, · · · , k.
If fact, it would suffice to assume the weaker condition that the matrices preserve
a positive cone. Under this hypothesis, Peres [18], Hennion [10] and Ruelle [20]
investigated the analytic dependence of the Lyapunov exponent.
The purpose of this note is to describe a new approach to studying λ. The
following theorem can be deduced from Theorem 2 in §1.
Theorem 1. There is an algorithm giving approximations λn to λ defined in terms
of the maximal eigenvectors and the eigenvalues of products of at most n of the
matrices.
Moreover,
there exist are C > 0 and K > 0 such that |λ − λn | ≤
1
K exp −Cn1+ d−1 .
Expressions for λn appear later in the paper cf. §1 and §5. It is also possible to
effectively estimate the constants K and C, cf. §8.
Previous approaches to computing λ were exponential in n, but were also exponential in the computational time [1], [2], [7], [12], [14], [18]. The faster convergence
guaranteed by Theorem 1 suggests
that the time required
by the algorithm to get
1
1+ d−1
. However, the practical use
an error of size ǫ > 0 is only O exp C(log |ǫ|)
of the algorithm is perhaps best illustrated by examples.
Example 1. Let us consider the matrices
2 1
3
A1 =
and A2 =
1 1
2
1
1
then with n = 9 this gives an approximation to λ of
λ9 = 1.1433110351029492458432518536555882994025 · · ·
We return to the question of the accuracy of this estimate in §7 and §8.
The basic idea of our approach has a very familiar analogy. Given a single square
matrix A, the rate of growth of the norm ||An || of the nth power of A is given by the
logarithm of the spectral radius of the matrix A. This, in turn, is determined by the
zeros of the characteristic polynomial det(zI −A). The appropriate generalization of
these functions are determinants of transfer operators, which were originally studied
in seminal work of Ruelle [19], in the particular context of Anosov flows. This work
was, in turn, inspired by earlier work of Grothendeick [8], [9]. The precise definition
in the present setting appears in §4. In a nutshell, it is the real analyticity of the
3
standard projective action of the matrices which allows us to invoke these powerful
results on determinants and thus provides both a new description for the Lyapunov
exponent λ and the surprisingly rapid algorithm for computating λ. The general
principal of “cycle expansions” is one which has been pioneered by P. Cvitanovic.
In section 1 we present the statements of the main theorem (Theorem 2). In
section 2 we describe the basic setting for the projective actions. In section 3
we describe the associated transfer operators, and in section 4 we consider their
determinants. In section 5 we present the proof of Theorem 2, and in section 6 we
describe a natural generalization (Proposition 6.1) of this theorem. In section 7 we
present two examples which illustrate the speed of convergence of this algorithm for
estimating Lyapunov exponents, and in the final section we explain how to estimate
the error.
1. The statement of the main theorem
In this section, we present the statement of the main theorem in this paper,
Theorem 2. We first observe that, without loss of generality, we can assume that
det Ai = ±1, for i = 1, · · · , k. More precisely, by the Birkhoff ergodic theorem we
have that for a.e. (µ) i ∈ Σ,
k
X
1
α := lim
pi log | det Ai |.
log | det(Ai1 · · · Ain )| =
n→+∞ n
j=1
(1.1)
If we replace Ai by Ai = | det A1 i |1/k Ai then we see from the Furstenberg-Kesten
theorem that the original Lyapunov exponent λ is related to the new Lyapunov
exponent λ for {A1 , · · · , Ak } by λ = λ + α. Thus we lose no generality by making
the following assumption.
Assumption 2. Each of the matrices Aj satisfies det Aj = ±1, for j = 1, · · · , k.
In order to state the theorem, we first need to define certain functions d(j) (z, t),
for j = 1, · · · , k.
Definition. Given a finite string i = (i1 , i, · · · , in ) ∈ {1, · · · , k}n we denote its
length |i| = n. Let pi = pi1 · · · pin . Let λi and xi denote the maximal simple positive
eigenvalue and associated eigenvector for Ai := Ai1 · · · Ain . Let σi = (i2 , · · · , in , i1 )
denote a cyclic permutation. We define a determinant function by

∞
X
pi
zn X
(j)

d (z, t) = exp −
n
(1 − λ−2
i )
n=1
|i|=n
n−1
Y
l=0
!t 
||Aj xσl i ||
,
||xσl i ||
(1.2)
where for each t ∈ R the infinite series converges to an analytic function for |z|
sufficiently small.
The following theorem gives an explicit expression for the Lyapunov exponent
in terms of a rapidly convergent series whose terms come from the information on
finite matrix products.
4
Theorem 2. We can write
λ=
∂d(j)
∂t
∂d(j)
j=1 ∂z
k
X
where the determinant d(j) (z, t) = 1 +
(j)
(j)
P∞
(1, 0)
(1, 0)
,
(j)
n=1
an z n is an entire function. Moreover:
(1) the coefficients an = an (t) can be written explicitly in terms of the maximal eigenvectors and the eigenvalues of the matrices Ai1 · · · Ail for i1 , · · · , il ∈
{1, · · · , k}, with 1 ≤ l ≤ n; and
1
(j)
(2) there are K0 > 0 and C > 0 such that |an | ≤ K0 exp −Cn1+ d−1 .
Of course, we can calculate the coefficients an explicitly by expanding (1.2) as a
power series in z. We will return to this point again in §5. We can now write
λ=
k
X
j=1
(j)
(j)
∂a(j)
(j)
P∞
(j)
n=1 cn
P∞ (j)
n=1 bn
!
where bn = nan (0), cn = ∂tn (0), for j = 1, · · · , k. We can now deduce
Theorem 1 from Theorem 2, by setting
λn =
k
X
j=1
Pn
(j)
i=1 ci
(j)
i=1 bi
Pn
!
.
Theorem 2 (2) allows us to deduce that λn to λ faster then any exponential.
2. The projective action
We now recall some preliminary results on real projective space and the projective action of the matrices. Let RP d denote the usual real (d − 1)-dimensional
projective space, i.e., RP d = (Rd − {(0, · · · , 0)})/ ∼, where v ∼ w if there exists
β 6= 0 such that βv = w. Let ∆ ⊂ RP k denote the open set corresponding to the
positive open quadrant Q = {(x1 , · · · , xd ) ∈ Rd : x1 , · · · , xd > 0}. It is sometimes
notationally convenient to choose representative vectors x ∈ Rd for x ∈ P Rd .
Since the linear action preserves lines which pass through the original we have
associated a well defined projective action Aj : P Rd → P Rd given by
Aj (x) =
Aj x
,
||Aj x||
(2.1)
for j = 1, · · · , k.
The next lemma summarizes some simple properties of the projective actions.
Lemma 2.1. For j = 1, . . . , k:
(1) The closure of the image Aj ∆ satisfies closure(Aj ∆) ⊂ ∆; and
(2) Aj : ∆ → ∆ is real analytic (i.e., it has an extension as a complex analytic
function to a neighbourhood U ⊃ ∆ in the complexification).
5
Proof. By the standing assumption that the entries of each Ai are strictly positive, we have that closure(Aj (Q)) ⊂ Q ∪ {0}, and part (1) of the Lemma follows
immediately, cf [4],[22]. Part (2) follows by a simple explicit computation. In the sequel we will need to consider the associated functions fj : RP d → R
defined by
||Aj x||
,
fj (x) = log
||x||
for j = 1, · · · , k.
3. Transfer operators
In this section we shall describe the connection between Lyapunov exponents
and transfer operators. We first choose a neighbourhood U in the complexification
of ∆, as in Part (1) of Lemma 2.1 . Let C ω (U ) denote the Banach space of analytic
function on U , with respect to the supremum norm khk = sup{|h(z)| : z ∈ U }.
Given a probability vector (p1 , · · · , pk ) we can define a bounded linear operator
L : C ω (U ) → C ω (U ) by
k
X
pj w(Aj x).
Lw(z) =
j=1
Such operators are generally referred to as transfer operators, and were introduced
in this context by Ruelle. Furthermore, Ruelle [19] showed how earlier work of
Grothendieck [7], [8] on nuclear operators could be applied in this setting.
Definition. A bounded linear operator L : B → B on a Banach space B is called
nuclear if there exist un ∈ B, ln ∈ B ∗ (with ||un || = 1 and ||ln || = 1) and
P
∞
n=0 |ρn | < +∞ such that
L(v) =
∞
X
ρn ln (v)un ,
n=0
for all v ∈ B.
Moreover, L is said to be of order zero if for any β > 0 we have that
+∞.
P∞
n=1
|ρn |β <
Lemma 3.1.
(1) The constant function 1 is the eigenvector associated to the (simple) maximal positive eigenvalue 1;
(2) This is a nuclear operator of order zero (and thus, in particular, a compact
operator);
(3) There is an eigenprojection µ associated to the eigenvalue 1, i.e., L∗ µ = µ;
(4) The limits
(3.1)
lim Ln fj (x) = µ(fj )
n→+∞
exist for j = 1, · · · , k and are independent of x ∈ ∆.
Proof. For part (1), we observe from the definitions that L1 = 1 and simplicity
follows easily, cf. [17]. For part (2) we see by [19, p. 234] that L is a nuclear
operator. For part (3) we observe that by a standard argument 1 is a simple
eigenvalue and, since the operator is compact, 1 is also a simple eigenvalue for the
6
dual operator and thus there is an eigenmeasure Lµ = µ [17]. Finally, part (4)
follows easily from 1 being a simple maximal eigenvalue and the constant functions
being the corresponding eigenfunctions. Peres proved a version of Lemma 3.1 (4) that applies to more general Hölder
functions using the Birkhoff metric on cones [18, p.136] . The measure µ is precisely
the Furstenberg measure on RP k which is the weak star limit of the probability
measures
1 X
δAi1 ···Ain x for n ≥ 1,
µn = n
k i ,··· ,i
1
n
for any x ∈ Q [5].
In low dimensions one can see the nuclearity quite explicitly.
Example (The special case d = 2). When d = 2, the set ∆ can be identified with the
one dimensional interval [0, 1], say, by first projecting ∆ radially onto the standard
one dimensional simplex and then projecting onto the first coordinate. In these
coordinates we can choose a neighbourhood [0, 1] ⊂ U ⊂ C in the complexification
such that for any w ∈ C ω (U ) the function Lw is analytic in a neighbourhood of
the closure closure(U ). We can then use Cauchy’s theorem to write
1
Lw(z) =
2πi
Z
∂U
Z
∞
X
Lw(ξ)
Lw(ξ)
1
n
dξ =
z
dξ ,
n+1
ξ−z
2πi
ξ
∂U
n=0
where w ∈ C ω (U ).
Lemma 3.1 (4) and the spectral gap for L immediately gives that for any x,
k
X
pj µ(fj ) =
k
X
j=1
j=1
pj Ln fj (x) + O(ρn )
(3.2)
where 0 < ρ < 1 is a bound on the modulus of the next smallest eigenvalue for L.
The connection with the Lyapunov exponent comes from the next lemma.
Lemma 3.2. We have the following identity for the Lyapunov exponent:
λ=
k
X
pj µ(fj ).
(3.3)
j=1
Proof. This could be deduced from [5], where µ is the Furstenberg measure. However, for completeness we include a simple direct proof.
Since L contracts in the Birkhoff projective metric [18], the limits in (3.2) exist.
(Alternatively, we can also see this directly by Lemma 2.1.) Moreover, for any
x ∈ ∆ the formula (3.1) gives that: µ(fj ) = limm→+∞ Lm fj (x) , for j = 1, · · · , k,
and thus by taking averages
k
X
j=1
n−1
X
1
n→+∞ n
m=1
µ(fj ) = lim


k
X

Ln−m fj (x) .
j=1
7
We thus see that
k
X
pj µ(fj ) = lim
n→+∞
j=1
n−1
X
1
n m=1


k
X

pj Ln−m fj (x)
j=1


n−1
k
X
X
||Aj x|| 
1

pj Ln−m log
= lim
n→+∞ n
||x||
m=1
j=1
n−1
1 X
= lim
n→+∞ n
m=1
X
im ···in
−
(3.4)
pim · · · pin log ||Aim · · · Ain x||
X
im+1 ···in

pim+1 · · · pin log ||Aim+1 · · · Ain x||
1 X
pi1 · · · pin log ||Ai1 · · · Ain x||
= lim
n→+∞ n
i ···i
1
n
We can also write
X
i1 ···in
pi1 · · · pin log ||Ai1 · · · Ain x|| =
Z
log ||Ai1 · · · Ain x||dµ(i).
Comparing (0.1), (3.4) and (3.5) shows that (3.3) holds, as required.
(3.5)
Remark. An analogous result holds for Markov measures µ, after a suitable modification to the transfer operator cf. §6.
4. Determinants
In this section we present some useful properties of determinants associated to
families of transfer operators. More precisely, for any ǫ > 0, and each j = 1, · · · , k,
we can consider the parameterized family of transfer operators
Lj,t w(x) =
k
X
i=1
etfj (Ai x) pi w(Ai x), for |t| < ǫ,
where w ∈ C ω (U ). We require the following version of a result due to Ruelle [19]
(cf. also [3]).
Lemma 4.1. For each j = 1, · · · , k:
(1) The traces tr(Lnj,t), n ≥ 1 exist and can be explicitly written
tr(Lnj,t )
=
X
pi exp t
|i|=n
n
X
r=1
fi (xσr i )
!
1
;
det(I − DAi (xi ))
(2) The complex functions
!
∞
n
X
z
tr(Lnj,t ) .
d(j) (z, t) := det(I − zLj,t ) = exp −
n
n=1
(4.1)
8
are analytic for all z ∈ C and
P∞t ∈ R; n
(j)
(3) If we expand d (z, t) = 1+ n=1 an z where an = an (t), then for
any ǫ > 01 (j)
there exists there are K0 > 0 and C > 0 such that |an | ≤ K0 exp −Cn1+ k−1
whenever |t| < ǫ.
Proof. This lemma can be deduced from the original work of Ruelle [19] (cf. also
[3], [11]). The operators are nuclear, and thus of trace class, by [19, p.234]. For
part (1), we recall from [19, pp.235-236] that
tr(Lnj,t )
=
X
|i|=n
pi
Pn
fj (xσr i )
.
det I − DAi (xi )
exp t
r=1
We can then apply Lemma 1.2. Part (2) follows easily from part (3). For d = 2,
the proof of Part (3) can also be found in [19, p.236]. For d ≥ 3, the bounds in [19]
need slightly modifying, as described in [3] (cf. also [11]). Example (The special case d = 2). When d = 2, we have already observed that
we can identify ∆ with [0, 1] and choose an open set [0, 1] ⊂ U ⊂ C such that
Ai (U ) ⊂ U , for i = 1, · · · , k. If we could choose U = B(z0 , r) to be the open ball
and C > 0 such that Ai (B(z0 , r)) ⊂ B(z0 , e−C r), then we take this value for C
in Lemma 4.1. More generally, if we assume U is connected and simply connected
we can use the Riemann mapping theorem to map U to a ball, and then apply a
similar argument.
Lemma 4.2. We have the identity
det Ai
1−
λ2i
det(I − DAi (xi )) =
!
Proof. This is a straightforward computation, cf. [21]. 5. The proof of Theorem 2
To complete the proof of Theorem 2 we need the following lemma which, when
combined with Lemma 3.2, relates the determinant d(z, t) to the Lyapunov exponent λ.
(j)
Lemma 5.1. There exists an analytic family of simple zeros z = zt
(j)
providing |t| is sufficiently small, such that z0 = 1 and
(j)
∂zt
|t=0 = −
∂t
Z
for d(j) (z, t),
fj dµ,
(5.1)
for j = 1, · · · , k.
(j)
Proof. The proof uses a standard perturbation argument. The zero zt corresponds
(j)
(j) (j)
(j)
(j)
to an eigenvalue λt = (zt )−1 for Lj,t cf. [19]. Let Lj,t wt = λt wt be the
associated eigenvalue equation. By Lemma 2.1 and analytic perturbation theory,
9
(j)
(j)
both wt and λt
t = 0 gives
(j)
are analytic (with w0
(j)
∂wt
(j)
|t=0 + fj w0
∂t
Lj,0
(j)
!
(j)
(j) ∂wt
= λ0
(j)
(j)
= 1 and λ0
∂t
= 1). Differentiating at
(j)
|t=0 +
∂λt
(j)
|t=0 w0
∂t
(5.2)
(j)
and we recall that z0 = λ0 = 1 and w0 = 1. Integrating both sides with respect
(j)
R
∂λ
to µ (and recalling that L∗0 µ = µ) the identity (5.2) gives ∂tt |t=0 = fj dµ.
(j)
(j)
However, since zt = (λt )−1 this is equivalent to (5.1). Using Lemma 5.1 and the implicit function theorem we have that
Z
(j)
∂z
fj dµ = − t |t=0 =
∂t
∂d(j) (1,t)
|t=0
∂t
(j)
∂d (z,0))
|z=1
∂z
(5.3)
as required for Theorem 1.
P∞ (j)
Using the expansion d(j) (z, t) = 1 + n=1 an z n in Lemma 4.1 (3) we have the
following result.
Lemma 5.2. For each j = 1, · · · , k we can expand
∞
∞
X
X
∂d(j) (z, 0)
∂d(j) (1, t)
|t=0 =
cn(j) and
|z=1 =
bn(j)
∂t
∂z
n=1
n=2
(j)
(j)
(j)
∂an
∂t |t=0 tend to zero super exponentially,
1
(j)
(j)
such that |bn |, |cn | ≤ K1 exp −Cn1+ d−1
(j)
where bn = nan−1 and cn =
i.e., there
are K1 > 0 and C > 0
whenever
|t| < ǫ.
Proof. The identities follow explicitly by differentiating the expansion for d(z, t).
(j)
(j)
In particular, the bound on |bn | comes from that on |cn | using Cauchy’s Theorem. The convergence of the series leads to expressions for an , bn and cn which show
the explicit dependence on the products of matrices. We can use the Taylor expanP∞ k
sion exp(x) = k=0 xk! to write
!
!k
∞
∞
∞
n
n
X
X
X
z
1
z
−
tr(Lnj,t ) =
tr(Lnj,t )
d(j) (z, t) = exp −
n
k!
n
n=1
n=1
k=1
!
∞
nl kl
n1 k1
l
X
X
)
)
·
·
·
tr(L
tr(L
(−1)
j,t
j,t
=
zm
,
k1 ! · · · kl !
nk11 · · · nkl l
m=1
n1 k1 +···nl kl =m
for j = 1, · · · , k. Comparing coefficients we see that for each m ≥ 1,
nl kl
n1 k1
l
X
)
)
·
·
·
tr(L
tr(L
(−1)
j,t
j,t
(j)
.
am
=
kl
k1
k1 ! · · · kl !
n1 · · · nl
n1 k1 +···nl kl =m
(j)
(j)
Explicit expressions for bm and cm can be derived using Lemma 5.2. In particular,
(j)
(j)
(j)
we observe that an , bn and cn depend only on properties of the products of at
most n of the matrices.
10
6. Generalizations
(a) We first observe that method computation of the Lyapunov exponents can be
easily generalized to the computation of integrals of analytic functions with respect
to the Furstenberg measure. Given any real analytic function f : ∆ → R we have
that
Z
Z
1 X
f (xi )
f dµ = lim
f dµn = lim n
(6.1)
n→+∞
n→+∞ k
|i|=n
The convergence in (6.1) is exponential. However, in order to ensure faster convergence one can consider a different arrangement of the weights f (Ai x) analogous to
Theorem 1. More precisely, we define

!t 
n−1
∞
n
Y
Xz X
pi
.
exp f (xσl i )
D(z, t) = exp −
−2
n
(1
−
λ
)
i
n=1
l=0
|i|=n
The analogue of Theorem 2 is the following.
Proposition 6.1. We can write
Z
f dµ =
∂D
∂t
∂D
∂z
(1, 0)
(1, 0)
,
(6.2)
P∞
where the determinant D(z, t) = 1 + n=1 an (t)z n is an entire function. Moreover:
(1) the coefficients an = an (t) can be written explicitly in terms of the maximal
eigenvectors and the eigenvalues of the matrices Ai for
|i| ≤ n; and
1
(2) there are K2 > 0 and C > 0 such that |an | ≤ K2 exp −Cn1+ d−1 .
The proof of this proposition is completely analogous to that of Theorem 2.
(b) Secondly, we note that the results for Bernoulli measures easily generalize
to Markov measures, associated to a stochastic matrix P , say. The statement
of Lemma 3.2 can be extended to Markov measures, where the transfer operator is modified to L : C ω (U, Cd ) → C ω (U, Cd ) defined by (Lw)i (x1 , · · · , xd ) =
Pk
j=1 Pij wi (Aj x). The corresponding determinant then becomes

∞
X
Pi
zn X
(j)

d (z, t) = exp −
n
(1 − λ−2
i )
n=1
|i|=n
n−1
Y
l=0
||Aj xσl i ||
||xσl i ||
!t 
,
where Pi := P (i1 , i2 ) · · · P (in−1 , in )P (in , i1 ) for i = (i1 , · · · , in ). The statement
and proof of Theorem 2 then generalize.
7. Examples
In this section illustrate Theorem 2 with two simple examples (both with d = 2
and k = 2).
Example 1. Consider the matrices
2 1
3
A1 =
and A2 =
1 1
2
1
1
11
Z+
. Working directly from the definition (0.1) of λ we get a
and let µ = 12 , 21
sequence of approximations to λ. Alternatively, we can can get better (although
non-rigorous) estimates on λ by estimating n1 log ||Ai1 · · · Ain || using random products of matrices as in (0.2). By the Furstenberg-Kesten theorem we expect for
a typical sequence i this sequence converges to λ, although in practice we have
no way of knowing if a given sequence is typical. Numerical estimates on these
two approximations to λ are presented in Table 1.1 and, as expected, these simple
estimates do not appear to converge particularly quickly.
n
1
2
3
4
5
6
1
n
R
n
log ||Ai1 · · · Ain ||dµ(i)
1.163
1.151
1.148
1.147
1.146
1.145
100
500
1000
5000
10000
20000
1
n
log ||Ai1 · · · Ain ||
1.157
1.149
1.148
1.145
1.145
1.144
Table 1.1. Approximations to λ using (i) the definition (0.1); (ii) a
randomly chosen sequence and (0.2).
To get better estimates, the positivity of the matrices implies that we can also
get exponential rate of convergence to λ using the approximations
||A2 · ||
1
||A1 · ||
n
n
(7.1)
ρn =
L log
(x) + L log
(x)
2
|| · ||
|| · ||
for any fixed x ∈ ∆ (using (3.2) and (3.4)) cf. [18]. Numerical estimates on the first
nine of these approximations are presented in Table 1.2, illustrating the exponential
convergence.
n
ρn
1
2
3
4
5
6
7
8
9
1.1404473802
1.1430287664
1.1432808871
1.1433077799
1.1433106831
1.1433109970
1.1433110309
1.1433110346
1.1433110350
Table 1.2. Approximations to λ using ρn (using n iterates of the transfer operator and the initial point corresponding to x = (1, 1) ∈ Q).
12
Finally, we present in Table 1.3 numerical estimates for the first nine approximations to λ given by the algorithm in Theorem 2, illustrating the super exponential
convergence.
n
λn
1
2
3
4
5
6
7
8
9
1.1435949601546489930611282560219921476826
1.1432978985074534413937646485571388968329
1.1433110787994660471763348416564186089168
1.1433110350856466164727559958382071786676
1.1433110351029501308232209336496360457362
1.1433110351029492458371384694231808633421
1.1433110351029492458432518595030145277475
1.1433110351029492458432518536555875134112
1.1433110351029492458432518536555882994025
Table 1.3. Approximations to λ using λn (given by the algorithm in
Theorem 2)
In Figure 1 we plot the number of places to which the nth approximation λn
agrees with λn+1 . The super exponential convergence is reflected in the convexity
of the plot.
40
35
30
25
20
15
10
3
4
5
6
7
8
9
Figure 1. Plot of the number of decimal places to which λn agrees with
λn+1 .
13
The approximations in Tables 1.2 and 1.3 suggest that the estimate ρ9 is probably
accurate to 9 decimal places, whereas λ9 appears accurate to 32 decimal places.
Both estimates involve computations using the same 1022 matrices Ai1 · · · Ail , with
i1 , · · · , i9 ∈ {1, 2} and 1 ≤ l ≤ 9.
Example 2. Consider the matrices
10 1
9 1
A1 =
and A2 =
9 1
8 1
Z+
. In Table 2.1, we present numerical estimates on the first
and let µ = 12 , 21
nine of the approximations ρn coming from (7.1), illustrating again the exponential
convergence.
n
ρn
1
2
3
4
5
6
7
8
9
2.3406490176369251220
2.3409978245595632085
2.3410010693762054102
2.3410010995663970319
2.3410010998472908916
2.3410010998499043683
2.3410010998499286845
2.3410010998499289107
2.3410010998499289128
Table 2.1. Approximations to λ using ρn (using n iterates of the transfer operator and x = (1, 1) ∈ Q).
In Table 2.2, we give numerical estimates for the first nine of the approximations λn to λ given by the algorithm in Theorem 2, illustrating again the superexponential convergence.
n
λn
1
2
3
4
5
6
7
8
9
2.341004439933296719772529760910725271776314889929526146521359936127823993879943
2.341001098971159429528556454970844691040896647105606075784302295458470606105270
2.341001099849930351103674905118261282575383610954371455389408463437561113313503
2.341001099849928912869691647574932766880686312348274316277170054896209376480232
2.341001099849928912869710477313030753335154640635330757905600135584517519416889
2.341001099849928912869710477313028599757800886036333444280974287802196538005819
2.341001099849928912869710477313028599757800888289781051642495952865120788671359
2.341001099849928912869710477313028599757800888289781051642473759039904012072465
2.341001099849928912869710477313028599757800888289781051642473759039904012074561
Table 2.2. Approximations to λ using λn (given by the algorithm in
Theorem 2).
14
The approximations in Tables 2.1 and 2.2 suggest that the estimate ρ9 is probably accurate to 19 decimal places, whereas λ9 appears accurate to 91 decimal
places. Both estimates involve computations using the same matrices Ai1 · · · Ail ,
with i1 , · · · , i9 ∈ {1, 2} and 1 ≤ l ≤ 9.
Remark. Heuristically, the apparently faster convergence of Example 2, compared
to Example 1, can be understood in terms of the greater contraction of the projective action of the matrices (leading to a larger choice for C > 0).
8. Bounds on the terms an
In this final section we shall give some simple estimates on the possible choices
of C in Theorems 1 and 2, rigorous bounds on the terms an and thus more accurate
estimates on the approximations λn to λ. For simplicity, we concentrate on the
particular case of d = 2. Whereas the basic approach can be generalized to the case
of arbitrary d ≥ 2, the estimates are particularly explicit in this case. Consider the
2 × 2 matrices
ai bi
Ai = ci di for i = 1, · · · , k
with positive entries. We can associate the projective actions Ai : ∆ → ∆ on the
one dimensional simplex ∆ = {(λ, 1 − λ) : 0 < λ < 1} given explicitly by
di + λ(ci − di )
bi + λ(ai − bi )
.
,
Ai (λ, 1 − λ) =
(bi + di ) + λ(ai + ci − bi − di ) (bi + di ) + λ(ai + ci − bi − di )
By taking the first coordinate of ∆ we have a natural identification with the unit
interval [0, 1]. Using this coordinate the action of A becomes the linear fractional
map fi : [0, 1] → [0, 1] given by
fi (λ) =
λ(ai − bi ) + bi
.
λ(ai + ci − bi − di ) + (bi + di )
Let Λ ⊂ [0, 1] be the limit set of the maps f1 , · · · , fk , i.e., the smallest closed set
such that Λ = ∪ki=1 fi (Λ). Assume that we can choose z0 ∈ [0, 1] and r > 0 such
that Λ ⊂ D(z0 , r) := {z ∈ C : |z − z0 | < r} ⊂ C is an open neighbourhood such
that fi (D(z0 , r)) ⊂ D(z0 , r) and
z
z
log kAi
k/k
k
1−z
1−z
(8.1)
2
2
2
2
= log (ai z + bi (1 − z)) + (ci z + di (1 − z)) − log(z + (1 − z) )
is analytic and bounded for z ∈ D(z0 , r), for each i = 1, · · · , k. In particular, the
operators Lt : C ω (D(z0 , r)) → C ω (D(z0 , r)) are well defined, for all t.
Since each fi is a linear fractional map the preimages fi−1 (D(z0 , r)) are again
disks centered on the real line. Assume that we can choose R > r > 0 such
that D(z0 , R) ⊂ ∩ki=1 fi−1 (D(z0 , r)). Using Cauchy’s theorem, the transfer operator
Lt : C ω (D(z0 , r)) → C ω (D(z0 , r)) satisfies
1
Lt w(z) =
2πi
Z
Γ
Lt w(ξ)
dξ
(ξ − z)
15
where Γ = {ξ ∈ C : |z0 − ξ| = R}, and this can be expanded as
1
Lt w(z) =
2πi
Z
Γ
Z
∞
X
1
Lt w(ξ)dξ
1
Lt w(ξ)
n
dξ =
n+1 (z − z0 )
(ξ − z0 ) 1 − (z−z0 )
2πi
Γ (ξ − z0 )
n=0
(ξ−z0 )
P∞
n
giving the nuclear presentation Lt u(z) = n=0 λn ln (u)vn , where vn = (z − z0 ) ,
up to a suitable normalization. In particular, following [19] (cf. [8], [9], [3], [11])
we can write
X
n
λk1 . . . λkN det lip (viq ) p,q=1
aN = (−1)N
k1 <...<kN
where
N
det lip (viq ) p,q=1
li1 (vi1 ) . . .
..
=
.
li (vi ) . . .
1
N
liN (vi1 ) ..
.
li (vi ) N
N
denotes the determinant of the N × N matrix with entries lip (viq ), 1 ≤ p, q ≤ N ,
corresponding to functionals lip and functions viq . This gives a bound
|aN | ≤ K N N N/2
X
k1 <...<kN
r k1 +···+kN
R
,
from which we can easily deduce the following.
Proposition 8.1. We can bound
r N(N−1)/2
K N N N/2
|aN | ≤ Q∞
,
r n
) R
n=1 (1 − R
where K = sup|z− 12 |≤1 ||Lj,t||.
In particular, we can make K arbitrarily close to 1 by choosing |t| sufficiently
small.
√ √
Example 1 (revisited). In this example the limit set satisfies Λ ⊂ [1/ 3, ( 5−1)/2]
and we can choose, for example, z0 = 58 and r = 18 < R = 58 . Using Proposition 8.1
P∞
we can bound n=10 |an | ≤ 1.78 × 10−21 . Similar estimates on the series for λ in
§1 give a more accurate bound on the error |λ − λ9 | of 3.2 × 10−14 .
√
13)/34, (3+
Example
2
(revisited).
In
this
example
the
limit
set
satisfies
Λ
⊂
[(7+3
√
0 · 015 < R = 0 · 515.
2 5)/15] and we can choose, for example,
P∞ z0 = 0 · 515 and r =
−51
Using Proposition 8.1 we can bound n=10 |an | ≤ 1.06 × 10 . Similar estimates
on the series for λ in §1 give a more accurate bound on the error |λ − λ9 | of
1.4 × 10−32 .
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Mark Pollicott, Department of Mathematics, Warwick University, Coventry,
CV4 7AL, UK