The Milnor-Thurston determinant and the Ruelle transfer operator

0.17 MB·English

by  Hans Henrik Rugh

The Milnor-Thurston determinant and the Ruelle
transfer operator.
H.H. Rugh
5
1 January 5, 2015
0
2

n
a Abstract
J

1 Thetopological entropy h ofacontinuous piecewisemonotone intervalmap
top
measures the exponential growth in the number of monotonicity intervals for it-
]
S erates of the map. Milnor and Thurston showed that exp( h ) is the smallest
top
D −
zero of an analytic function, now coined the Milnor-Thurston determinant, that
.
h keeps track of relative positions of forward orbits of critical points. On the other
at hand exp(htop) equals the spectral radius of a Ruelle transfer operator L, associ-
m ated with the map. Iterates of L keep track of inverse orbits of the map. For no
[ obvious reason, a Fredholm determinant for the transfer operator has not only

1 the same leading zero as the M-T determinant but all peripheral (those lying in
v the unit disk) zeros are the same.
4
The purpose of this note is to show that on a suitable function space, the
9
2 dual of the Ruelle transfer operator has a regularized determinant, identical to
0 the Milnor-Thurston determinant, hereby providing a natural explanation forthe
0
above puzzle.
.
1
0
5
1 1 Introduction
:
v
i
X
In the 70s, Milnor and Thurston came up with an intriguing way of computing the
r
a topological entropy htop for a continuous piecewise monotone interval map f. They
invented a "kneading-matrix", M(t), a finite matrix-valued powerseries in an auxiliary
variable t. The matrix keeps track of relative positions of forward orbits of critical
points relative to the critical points themselves. The "Milnor-Thurston" determinant,
d (t) = detM(t), defines an analytic function in the unit disk and they showed that
MT
if h > 0 then t = exp( h ) (0,1) is a zero of d (t). The zero is ‘extremal’
top ∗ top MT
− ∈
in the sense that it is the smallest in absolute value. Their result is computationally
useful, as the effort in a direct computation of the number of monotonicity intervals of
fn grows like enhtop whereas computing M(t) to a similar accuracy only grows linearly
with n.
1
The above-mentioned extremal property was shown indirectly by proving that
d (t) is identical to a certain Lefschetz zeta-function, generated from the periodic
MT
orbits for f. The identity is shown in one (simple) case and using continuous deforma-
tions of the map it is shown that the identity persists under such deformations. This
part is rather puzzling, as no arguments indicate why one would expect such a relation.
Also in the 70s, Ruelle used transfer operators to define (generalized) Fredholm
determinants and zeta-functions. In our context, an operator L would act upon func-
tions of bounded variation. Iterates of L keep track of inverse orbits of points and its
spectral radius equals precisely 1/t = exp(h ). Baladi-Keller showed [BK], that an
∗ top
associated determinant d (t) = det(1 tL) is analytic in the unit disk and (at least for
R
−
expanding maps) it is easy to see that the zeros within the unit disk are the same as
for the Lefschetz zeta function whence also for d (t). This suggests a deep relation
MT
between the the M-T determinant and the Ruelle-determinant of L, whence with the
operator L itself.
Asteptowardsclarifyingthisrelationshipwassuggestedthroughastudyofthedual
operatorbyBaladiandRuelle[BR],butthefunctionalsetupmakesexplicit computions
hard and requires operator-weights to be globally continuous. Gouëzel [G], elaborating
the functional setup further, still requires weights to be continuous at periodic points.
Both of these results exclude partially the case studied by Milnor and Thurston.
The goal here is to show that the determinant of Milnor-Thurston may be obtained
by restricting L to a suitable small subspace X of Bounded Variation functions, and
calculate an elementary regularized determinant of the dual operator L′. We first
exhibit a natural isometry between the dual space X′ and a space X of uniformly
bounded functions of "point-germs". This gives an explicit representation for the dual
operator, L = S PS, in which S has spectral radius 1 and P is of finibte rank. When
−
λ > 1 (we call such a value a ’peripheral’ value) then λ S is invertible through a
| | −
von Neumbann series. The elementary formula,
λ (S PS) = 1+PS(λ S)−1 (λ S),
− − − −
shows that λ is a spectral value iff th(cid:0)e operator (1+ P(cid:1)S(λ S)−1) is non-invertible.
−
As P is of finite rank, this happens iff the following finite dimensional determinant
vanishes:
det 1+PS(λ S)−1 := det P λ−kSk . (1)
− imP !
k≥0
(cid:0) (cid:1) X
Finally, setting λ = 1/t this turns out to be the Milnor-Thurston determinant. An
advantage of the present approach is that we may exploit positivity of the Ruelle
operator (see Theorem 3.3 below) to bypass the use of the Lifschetz zeta-function in
the proof of the extremal property of the zero.
2
2 Step functions of bounded variation.
Fix a < b + . The variation of a function φ : (a,b) C is:
−∞ ≤ ≤ ∞ →
varφ = sup φ(x ) φ(x ) : a < x < ... < x < b . (2)
i i+1 1 N
| − |
( )
i
X
We say that φ is of bounded variation (BV) iff varφ < + . For a BV-function
∞
right and left limits always exist and following usual conventions we write φ(x+) =
lim φ(x + t) and φ(x−) = lim φ(x t) defined for x [a,b) and x (a,b],
t→0+ t→0+
− ∈ ∈
respectively. We assume no a priori relation between the values of φ at x−,x and x+
(see also Remark 3.2 below). We define the ’boundary’ value of φ to be
∂φ = φ(a+)+φ(b−) . (3)
In the following we write x = xǫ for either x+ or x− and call it a point germ with
base point x and direction ǫ = +1 or ǫ = 1. We order point germs intertwined with
−
basepointsbydeclaringthatwhenx < y thenx < x+ < y− < y. Anyrealsegment may
b
then be specified throughtwo point-germs by setting: u,v = x (a,b) : u < x < v .
h i { ∈ }
So e.g. u−,v− = [u,v) (for u < v), u−,u+ = u and u,u = .
h i h i { } h i ∅
b b b b
Definition 2.1. A map φ : (a,b) C is said to be a simple step function if there
→ b b
is a finite set C = C (a,b) so that φ is constant on each connented component of
φ
⊂
(a,b) C. We write = (a,b) for the space of simple step functions on (a,b).
\ S S
When a < x < x < x < b then 2 φ(x) varφ + φ(x ) + φ(x ) . Letting
0 N 0 N
| | ≤ | |
x a+ and x b− yields sup φ 1(varφ+ ∂φ ). One easily obtains :
0 → N → | | ≤ 2 | |
Proposition 2.1. The quantity φ = varφ+ ∂φ defines a norm on S. We let X
BV
k k | |
denote the completion of S under this norm and call X = SBV(a,b) the space of Step
functions of Bounded Variation on (a,b). One has sup φ 1 φ for φ X.
| | ≤ 2k kBV ∈
We define for every a+ u b− the following "base" step-function (which is
≤ ≤
everywhere non-zero):
b
1 +1 , u < x < b
σub(x) = 2 × 1 , a < x < u (4)
(cid:26) −
b
In particular, σa+(x) = − σb−(x) = 21 for all a < x < bb. In terms of characteristic
functions one has: σub(x) = 21 −χha+,ubi(x)+χhub,b−i(x) .
Let X′ be the Banach dua(cid:16)l of X, i.e. the space of l(cid:17)inear maps ℓ : X C for which
→
ℓ,φ C φ BV, φ X andsome constant C < + . The norm ℓ X′ of ℓ is defined
h i ≤ k k ∀ ∈ ∞ k k
as the smallest such constant. Acting with ℓ upon a base step function σ we obtain
ub
3
a representation of ℓ as a function on the set of point germs I = [a+,b−]. We call this
the σ-transform of h′ X′ and write
∈
b
ℓ(u) := ℓ,σ , u I. (5)
h ubi ∈
As is readily seen, σub has nobrmb one, so kℓk∞ b≤ kℓbkX′. Thus, ℓ is an element of the
space X = B([a+,b−]), the bounded functions on I equipped with the uniform norm
. Since ℓ,σ = ℓ,σ , we have ℓ(a+)+ℓ(b−) = 0 so ℓ in fact belongs to the
k·k∞ h b−i −h a+i
closed sbubspace X = H X : H(a+)+H(b−) = 0b . But more is true:
0
{ ∈ }
Proposition 2.2. The σ-transform ℓ X′ ℓ X is a Banach space isometry.
b b ∈ 7→ ∈ 0
Proof: The collection {σub : a+ ≤ u < b−} fbormsba base for S, the space of simple
step functions. In fact, given any simple step function φ there is a finite number of
point germs a+ = u < < u < b− so that:
0 N b
···
N N
1
b φ =bw021+ wiσubi = wiσubi. (6)
i=1 i=0
X X
We omit a term involving b− since σ = σ and the latter is already included
− b− a+
in the sum. The decomposition is unique if every w = 0, i > 0. In terms of this
i
6
decomposition:
N
varφ = w and ∂φ = w . (7)
i 0
| |
i=1
X
To see this, note that the variation is bounded by the right hand sum and as is readily
checked, the supremum in the definition (2) may be realized by choosing a mesh that
verifies: a < x < u < x < u < ... < u < x < b. The norm of φ is given by:
1 1 2 2 N N+1
N
b b φ b= w . (8)
BV i
k k | |
i=0
X
Given a simple step function φ it may be decomposed as in (6) and using the
∈ S
expression (8) for the norm we see that:
N
ℓ,φ = w ℓ(u ) (9)
i i
h i
i=0
X
N N b
ℓ,φ = w ℓ(u ) w ℓ = φ ℓ . (10)
i i i ∞ BV ∞
|h i| | | | | ≤ | | k k k k k k
i=0 i=0
X X
Now, this extends by continuity to ebvery φ in X, thebcompletion ofb . Conversely, any
S
function H X gives rise to a linear functional on of the same norm, by setting
0
∈ S
ℓ,φ := N w H(u ) and extending by continuity ℓ becomes a linear functional
h i i=0 i i
ℓ X′ of the sabme norm.
∈ P
b
4
a a u b b a fa fu fb b
k k
k k
Figure 1: Graph of σu− and Lkσu− for ak < u < bk and with sk = +1.
Remarks 2.3. The space SBV(a,b) is isomorphic to C ℓ1((a,b)) (summable functions
⊕
on the uncountable set of point germs + the boundary value). Not surprisingly, X′ is
isomorphic to C ℓ∞((a,b)).
⊕
Proposition 2.4. Given a bounded linear operator A L(X) we obtain a representa-
∈
tion A L(X ) through:
0
∈
b b Aℓ (u) := A′ℓ,σ = ℓ,Aσ , u [a+,b−]. (11)
h ubi h ubi ∈
(cid:16) (cid:17)
bb b b
The operators A and A′ (whence A) have the same spectrum [Kato, III, Thm 6.22].
λ is an isolated eigenvalue of finite albraic multiplicity for A iff it is for A′ (whence
b
for A). In this case, algebraic (and geometric) multiplicites are the same for A, A′ and
A. [Kato, III, Remark 6.23].
b
b In the following we shall study a family of operators where the representation of
the dual operator takes a particular simple form.
3 A Ruelle transfer operator
Let = I = (a ,b ) (a,b) (below, k will serve as an index) and let f : I (a,b)
k k k k k
∅ 6 ⊂ →
be a continuous, strictly monotone map. We write s = s(f ) 1 for the sign
k k
∈ {± }
of monotonocity. Also let g C (the topological entropy corresponds to the case
k
∈
g 1). Given a triple (I ,f ,g ) we associate a bounded linear operator, L acting
k k k k k
≡
upon φ X = SBV(a,b) and defined as follows:
∈
L φ(y) = g φ f−1(y) χ (y) , y (a,b). (12)
k k ◦ k fkIk ∈
Acting with L e.g. upon a characteristic function χ of J (a,b) simplifies to:
k J ⊂
L χ (y) = g χ (y). (13)
k J k fk(Ik∩J)
We extend f to a map of point germs u = uǫ [a+,b−] by setting f (uǫ) = (u′)ǫ′,
k ∈ k k k
where u′ = f (uǫ) and ǫ′ = s ǫ (the direction of the point germ is reversed precisely
k k
b
b
5
when f reverses orientation). By the results of the previous section, the action of L
k k
on an SBV-function is uniquely determined by its action upon a base step function.
When a < u < b the action of L upon σ yields a simple step function with at most
k k k ub
3 jumps (see Figure 1) which may therefore be written as the sum of (at most) 3 base
step functions. When u is outside of the interval [a+,b−], (at most) 2 such base step
b k k
functions suffice. Taking into account the reversal of orientation when s = 1 we may
k
−
summarize as follows :
b
1 +σfbka+k −σfbkb−k, a < u < ak
Lkσub = 2 skgk × −σfbka+k −σfbkb−k +2σfbkub ak <bu < bk . (14)

 −σfbka+k +σfbkb−k, bk < ub< b

The term σfbkub only occurs when ak < u < bk while both σbfbka+k and σfbkb−k always
appear, but with a sign depending upon the position of u relative to a and b , respec-
k k
tively. We may collect terms and write: b
b
Lkσub = skgk χ(ak,bk)(u) σfbkub −σak(u) σfbka+k +σbk(u) σfbkb−k . (15)
h i
Here, σ (u) = σ (a ) takes the value +1/2 when u > a and 1/2 when u < a
ak − ub k b b k b − k
(and similarly for σ (u)). The essential point is that the RHS is again a sum of
bk
elementary step functions. Thus, acting with a linear functional on both sides we
b b b
obtain a representation for the dual of the Ruelle operator:
b
L ℓ(u) := L′ℓ,σ = ℓ,L σ (16)
k h k ubi h k ubi
= s g χ (u)ℓ(f u) σ (u)ℓ(f a+)+σ (u)ℓ(f b−) . (17)
cbb k k (ak,bk) k − ak k k bk k k
h i
We extend this to any H X by sbetbtinbgb b b b b b b
∈
L H(u) = s g χ (u)H(f u) σ (u)H(f a+)+σ (u)H(f b−) . (18)
k k k (ak,bk)b k − ak k k bk k k
h i
The cconstabnt function 1 is inbthe kberbnel of Lkb, sincebfor every ub [a+b,b−]:
∈
Lk1(u) = skgk χ(ak,bk)(u)c−σak(u)+σbk(u) b≡ 0. (19)
h i
c
One also has L Hb(a+) = 1s g Hb(f a+) bH(f b−)b = L H(b−). Conse-
k 2 k k k k − k k − k
quently, L H(a+) + L H(b−) = 0, s(cid:16)o L not only preser(cid:17)ves X but maps X into
k k k 0
c b b c
X . In particular, any (non-zero) spectral property does not change by considering the
0
extended ocperator. Incorder to simplifycexpressions, we introdubce the followinbg two
obperators acting upon H X:
∈
S H(u) = s g χ (u)H(f u), (20)
k b k k (ak,bk) k
P H(u) = σ (u)H(a+) σ (u)H(b−). (21)
k ak k − bbk k
b b b
b b 6 b
In terms of these, our dual representation may be written as
L = S P S . (22)
k k k k
−
The weighted composition operator S is (just) a bounded linear weighted composition
ck
operator, but the image of each P is spanned by the elements σ and σ so P is
k ak bk k
of rank two. Our representation of the dual Ruelle operator is therefore a finite rank
perturbation of S .
k
As the reader may verify, neither S nor P S need preserve X (although their
k k k 0
difference does). It is, however, somewhat easier to work with S and P acting upon
k k
X, rather than making identifications to restrict their action to X .b
0
b b
3.1 A dynamical system
We consider here the situation when I = (a ,b ) = (c ,c ) with a = c < c < ... <
k k k k k+1 0 1
c = b. The intervals are disjoint and their union fill out (a,b) apart from the cutting
d+1
points c ,...,c .
1 d
We associate as in the previous section a map, a weight and an operator to each
interval. We write (I ,f ,g ), k = 1,...,d for the collection of triples and denote by
k k k
L = L the Ruelle transfer operator associated to this collection.
k k
BPy our choice of cutting points, we may extend the collection of fk’s to a unique
map f defined upon the entire set of point germs [a+,b−] (even if f need not extend
to a continuous map on I !). Signs and weights also extend to functbions of [a+,b−].
Whenbu I = [c+,c− ]. we write s(u) = s , g(u) = g and we have the expression
∈ k k k−1 k k
for the corresponding sum of composition operators:
b b b b
SH(u) = S H(u) = s g χ (u)H f (u) = (sg)(u)H(f u).
k k k (ak,bk) ◦ k
k k
X X
b b
b b b b b b
Let us also write g(n)(u) = g fn−1(u) g(u) (and similarly for s(n)) for the product
◦ ···
of weights (or signs) along the orbit of u. Then SnH(u) = (sg)(n)(u)H(fnu) and the
spectral radius of S isbgiven byb b b
b
b b b b
ρ = lim Sn 1/n = lim g(n) 1/n.
∞ n k k∞ n k k∞
From the previous section we see that the dual of the Ruelle transfer operator, L′ =
L′, is a finite rank perturbation of S so the essential spectral radius of L′ (whence
k k
of L) is not greater than ρ . If ρ(L) > ρ then L is quasi-compact ([DS, VIII.8]) and
∞ ∞
P
any spectral value λ with λ > ρ (we call such a value a peripheral spectral value)
∞
| |
must be an isolated eigenvalue of L of finite algebraic multiplicity.
A function of the form S H has support in I so we have P S = 0 whenever
k k m k
k = m. Whence, P S = P S, where P = P . Furthermore, we may recollect
6 k k k k
P P
7
terms to get:
d d
PH(u) = σ (u)H(c+) σ (u)H(c− ) = σ (u)∆ H + σ (u)∆ H ,
ck k − ck+1 k+1 a a cj cj
Xk=0h i Xj=1
b b b b b
in which we have used σ (u) = σ (u) = 1 and defined
b − a
∆ H = H(c+) H(c−), j 1 , (23)
bcj b j − j ≥
∆ H = H(a+)+H(b−) . (24)
a
The image of P is d + 1 dimensional and as ∆cσc′ = δc,c′, P is also a projection.
If λ > ρ then λ S is invertible through a von Neumann series so λ L =
∞
| | − −
(1+PS(λ S)−1)(λ S). Equivalently, writing t = 1/λ:
− −
b
1 tL = G(t)(1 tS), (25)
− −
where
b
G(t) = 1+t PS(1 tS)−1 = (1 P)+P(1 tS)−1. (26)
− − −
We conclude that λ L is non-invertible (i.e. λ = 1/t is a spectral value) iff the
−
operator G(t) is non-invertible.
b
Since (1 P)G(t) = (1 P) it suffices to look at the restriction of G(t) to the image
− −
of P which is spanned by σ = σ and σ ,...,σ . Acting upon these d+1 functions
c0 a c1 cd
we get the following (d+1) (d+1) matrix representation
×
d
G(t) σ = P(1 tS)−1σ = σ M (t)
ck − ck cj jk
j=0
X
where
M (t) = ∆ (1 tS)−1σ = ∆ Θ ( ,t) (27)
jk cj − ck cj ck ·
(cid:16) (cid:17)
and
Θ (x,t) = (1 tS)−1σ (x) = tn(sg)n(x)σ (fnx), (28)
ck − ck ck
(cid:16) (cid:17) Xt≥0
b
isequivalent tothebso-calledkneadingcoordibnateofMilnor-Thburston. bOurmatrixM(t)
is of size (ℓ+1 (ℓ+1) and is one version of the Milnor-Thurston kneading matrix.
)
×
The original [MT] is, ℓ (ℓ+1) dimensional) but the associated M-T determinant is
×
identical to the determinant of M(t) as shown in [RT, Appendix B].
The value λ = 1/t is an eigenvalue iff d (t) := detM(t) = 0. Moreover, the order
MT
of the zero of the determinant equals the algebraic multiplicity of the eigenvalue. This
last property is known as the Weinstein Aronszajn formula (while not difficult to show
it is somewhat lengthier, so we refer to e.g. [Kato, Ch. IV.6] or [Ru, Lemma 2.8] for a
proof). In summary we have shown:
8
Theorem 3.1. Let σ (L) be the spectrum of the Ruelle transfer operator acting upon
sp
X = SBV(a,b). The peripheral spectrum, z σ (L) : z > ρ consists of isolated
sp ∞
{ ∈ | | }
eigenvalues of finite multiplicity only. Furthermore, λ is a peripheral eigenvalue iff t =
1/λ is a zero of the Milnor Thurston determinant d (t) and the algebraic multiplicity
MT
of λ equals the order of the zero of t.
Remarks 3.2. In the above we have used a representation of a simple step function
φ in which the value at each point x (a,b) is specified independently of the limit-
∈
values at x+ and x−. Other possibilities are to assign the value 1 (φ(x−)+φ(x−))
2
at x or to declare two functions equivalent if they agree except at a countable set.
The dual space of the completion in both of the latter cases may be identified with
H ℓ∞([a,b]) : H(a)+H(b) = 0 . This is, however, inconvenient when the map is
{ ∈ }
discontinuous and the forward orbits of c+ and c− are distinct. With our approach the
j j
distinction is automatic as it is build into the space of point-germs.
3.2 Positivity and leading zero
The most interesting situation is when all weights are non-negative and the Ruelle
transfer operator is positive, i.e. preserves the cone of non-negative functions. We may
use this to give a direct simple proof of the extremal properties of the zero as alluded
to in the introduction. Let Z denote the collection of monotonicity intervals (also
n
known as cylinder sets) of fn. We define the partition function by
(n)
Ω = g (29)
n |α
αX∈Zn
The particular case g 1 for every k, yields the topological entropy through h =
k top
≡
1/n
loglim Ω .
n n
Theorem 3.3. Suppose that all g 0. Then we have
k
≥
ρ := limΩ1/n = lim Ln1 1/n = ρ (L). (30)
1 n n n k k∞ sp
Moreover, if ρ (L) > ρ then ρ (L) is an isolated eigenvalue of L of finite multiplicity
sp ∞ sp
and t∗ = 1/ρ (L) > 0 is the smallest zero (in absolute value) of the Milnor-Thurston
sp
determinant d (t).
MT
Proof: As Ln1(y) Ω for all n and y we have ρ lim Ln1 1/n ρ .
n ∞ n 1
≤ ≤ k k ≤
Consider thesetZ of(non-empty) monotonicityintervalsforfn. Anyopeninterval
n
α Z may be written in terms of point germs as α = u,v where each directed end
n
∈ h i
point is either a cutting point itself or it maps into a cutting point within at most n 1
−
iterations. Let 0 k < n be the smallest such number for which fk(x) = c where c is
b b
≤
either c− or c+ for some 1 j d. Now, the partition sum is the sum of g(n) over all
j j ≤ ≤ |α
b
b b b
9
α Z . We get twice the value if we instead sum over all interval endpoints. Using
n
∈
that g(n)(x) = g(k)(x)g(n−k)(c ) and rewriting the sum in view of the previous remarks
j
we get the identity (the boundary points a+ and b− need a special treatment):
b b b
d n−1
2Ω = g(n)(x) = g(n)(a+)+g(n)(b−)+ Lk1(c ) g(n−k)(c). (31)
n j
hub,Xvbi∈Znxb∈X{ub,vb} Xj=1 Xk=0 bcX=c±j
b b
We deduce the inequality 2Ω 2 g(n) +d n−1 Lk1 g(n−k) and then:
n ≤ k k∞ k=0k k∞k k∞
ρ = lim Ω1/n lim Ln1 1/nP Ln1 1/n ρ (L).
1 n→∞ n ≤ n→∞k k∞ ≤ k kX ≤ sp
If ρ (L) = ρ we are through. So suppose that ρ (L) > ρ . As any peripheral
sp ∞ sp ∞
spectral value is en eigenvalue of finite multiplicity, there must be λ C and a non-
∈
trivial φ X with λφ = Lφ and λ = ρ (L). Taking absolute values we get by
sp
∈ | |
positivity of the operator
λ φ = g φ f−1 χ g φ f−1 χ = L φ . (32)
| | | | (cid:12) k ◦ k fkIk(cid:12) ≤ k | |◦ k fkIk | |
(cid:12)Xk (cid:12) Xk
(cid:12) (cid:12)
Similarly λ n φ L(cid:12)(cid:12)n φ Ln1 φ ∞, fro(cid:12)(cid:12)m which we deduce that
| | | | ≤ | | ≤ k k
ρ (L) = λ lim Ln1 1/n ρ ρ (L).
sp 1 sp
| | ≤ n k k ≤ ≤
For the last assertion, consider the above φ and λ. When 0 < t < 1/ρ (L) we
sp
get (1 tL)−1 φ = tkLk φ tk λ k φ = (1 t ρ (L))−1 φ which diverges as
sp
t (1/−ρ (L))|−.| This impli|es|t≥hat (ρ |(L| )|1| L) m−ust be non-in|v|ertible and whence
sp sp
→ P P −
that ρ (L) is an eigenvalue of L. Then t = 1/ρ (L) is a zero of d (t), and the
sp ∗ sp MT
smallest such since there can be no spectral value larger than ρ (L).
sp
Examples 3.4. Consider the two triples of maps (with unit weights):
I = (0, 1),f (x) = 2x,g = 1 and I = (1,1),f (x) = x 1,g = 1.
1 2 1 1 2 2 2 − 2 2
f maps the four directed boundary points 0+, 1−, 1+ and 1− to 0+,1−,0+ and 1−,
2 2 2
respectively. Θ (1+,t) is calculated by looking at the forward orbit of 1+ relative to 1.
1
2 2 2
Fbor example, a2s 1+ > 1 (which yields a plus sign) and f(1+) = 0+ = f(0+) < 1 (so a
2 2 2 2
minus sign) we get
b b
1+ 1 11 2t
Θ ( ,t) = 1 t t2 t3 = − . (33)
1
2 2 2 − − − −··· 2 1 t
−
(cid:0) (cid:1)
10