The Gibbs measure and coboundary condition

THE GIBBS MEASURE AND COBOUNDARY CONDITION

YOUNG-ONE KIM AND JUNGSEOB LEE

ABSTRACT. We investigate coboundary conditions for two func- tions defined on a mixing subshift of finite type to have the same Gibbs measure. Also we find conditions for a function to be a coboundary.

O. Introduction

We consider a dynamical system together with a real-valued contin- uous function which is called the energy distribution. The variational principle asserts that the supremum of all measure-theoretic pressures on such a system is equal to the topological pressure. IT the dynamical system is a subshift, then the topological pressure is attained as the maximum of measure-theoretic ones, and the maximizing measures are called the equilibrium states of the energy distribution. In the special case that the dynamical system is a mixing subshift of finite type and the energy distribution has summable variation, there is a unique equi- librium state, called the Gibbs measure of the energy distribution. A necessary and sufficient condition for two energy distributions (with summable variation) on a mixing subshift of finite type to have the same Gibbs measure is expressed in terms of the coboundary operator.

Specifically, if two energy distributions 4> and 7f; on a mixing subshift of finite type have summable variation, then their Gibbs measures are identical if and only if there are a constant K and a continuous function

f3 such that

Received January 19,1998. Revised Apri120, 1998.

1991 Mathematics Subject Classification: 54H20, 58F03.

Key words and phrases: subshifts, Ruelle operator, Gibbs measure.

This research was partially supported by KOSEF Grant 95-0701-02-01-3 and Ministry of Education through Research Fund BSRI-96-1441.

However, the known proof of this theorem

is

somewhat roundabout.

In this article, we investigate the relation between the Ruelle operators and the coboundary operator, and consequently present a direct and rather algebraic proof of the theorem. In fact, we will prove a slightly generalized version of the theorem. Then we will discuss conditions on a function 4> for which there

is

a function 13 such that 4>

=

813.

This paper

is

organized as follows. In §1, we introduce some notions such as mixing subshifts of finite type, the Ruelle operators and the coboundary operator; obtain some technical results which will be used in the sequel. In §2, we will give a simple proof of the theorem men- tioned in the previous paragraph. We conclude this paper with some discussions about conditions for a function to be a coboundary (§3).

1.

Preliminaries and Notations

Let X be a compact metric space and T : X

~

X a continuous surjective mapping. Let C(X) denote the Banach space· of all real- valued continuous functions on X with the supremum norm

11 . 11.

For

4>

E

C(X) let Pt/> denote the topological pressure of 4>. AT-invariant

Borel probability measure p.

is

called an equilibrium state of 4> if Pt/>

=

h(:p.) + L ^4>dp.,

where h(p.) is the measure-theoretic entropy of p..

The coboundary operator ar : C(X)

~

C(X) of T

is

defined by

Since

arf3

=

13 - 13

⁰

T (13

E

C(X)).

L ^arf3dp.

⁼

L ^{f3dp. -} L ¹³

⁰

^Tdp.

⁼

^0,

for all 13

E

C(X) and T-invariant Borel measures p. on X, it follows that for all 4>,13

E

C(X), 4> and 4>+813 have the same set of equilibrium states.

If T is a local homeomorphism, then for each 4>

E

C(X) the Ruelle operator LT,t/> : C(X)

~

C(X) is defined by

(LT,t/>J)(X)

=

L et/>(Y) fey) (f E C(X), x EX).

yET-l(x)

In this case the dual operator LT,</> of LT,</> is defined by

(f

E

C(X)).

(f,g

^E

C(X)).

for continuous real-valued functions f on X and signed Borel measures

J.L

on X.

LEMMA

1.1. Let X be a compact metric space and T : X - X a surjective local homeomorphism. Then for 4>, (3

E

C(X) we have the following.

(a) LT,</>+8r/3(f) = e-/3LT,</>(e/3f) (b) f LT,</>(g)

=

LT,</>((f

⁰

T) . g)

Proof. It follows directly from the definitions of Ruelle operators

and the coboundary operator. 0

REMARK.

The statement (a) in the above lemma can be rewritten in the form

LT,4>+8r/3

⁼

(L

1

,/3)-1

⁰

LT,</>

⁰

L

1

,/3' where I : X - X is the identity mapping.

LEMMA

1.2. Let X be a zero dimensional compact metric space and T : X - X a surjective local homeomorphism. Then for all

4>,,,p

^E

C(X), LT,</> = LT

^,1/J

implies 4>

= "p.

Proof. Let Yo

E

X be arbitrary. Since T is a local homeomorphism and X is zero dimensional, there is a clopen neighborhood U of Yo such that Tlu is one-to-one. Let hu denote the characteristic function of U. Then

LT,</>hu(Tyo) = L e4>(Y)hu (Y)

=

e</>(Yo).

yET-l (Tyo)

This proves our assertion. o

Throughout this paper, we will be interested in a special kind of

dynamical systems, namely the one-sided subshifts of finite type, which

are defined as follows. Let A be a finite set and M a 0-1, A x A matrix,

and assume that every column and row of M has a non-zero entry.

Define :EM and u : :EM - :EM by

:EM = {x =

(Xi)~O

E AN: M(Xi,Xi+l) = 1 for i = 0,1,2, ... }, and

u: :EM

:3

XOXIX2···

1---+

XIX2Xg···

E

:EM.

Let

d:

:EM x :EM - R be defined by

00

d(x, y)

=

L(1- 8(Xi, Yi»2- i

i=O

for x

=

XOXl ... , Y

=

YOYl···

E

:EM. Then d is a metric on :EM.

With this metric, :EM becomes a compact metric space and u a posi- tively expansive surjective local homeomorphism. The dynamical sys- tem (:EM, u) is called the one-sided subshift of finite type (or one-sided topological Markov shift) defined by M, and u is called the shift map.

If the matrix M is primitive, Le., there is a positive integer N such that every entry of M

^N

is positive, then the one-sided subshift (:EM,u) is said to be mixing.

Let (:EM, u) be a one-sided subshift of finite type. For </>

E

C (:E

M )

and k = 0, 1,2, ... , define the k-th variation of </> by

vark4J = sup{I</>(x) - 4J(y)I : x,

yE

:EM, XOXl··· xk = YOYl ... Yk}.

Since 4J is uniformly continuous, it follows that vark4J - ° ^{as k - 0.}

The function 4J is said to have summable variation if

00

L vark4J <

^00.

k=O

The set of functions with summable variation is denoted by S(:E

M ).

If there are a > ° ^{and b}

^E

(0,1) such that

vark4J~ abk (k=0,1,2, ... )

then </> is said to be Holder, and we denote by ll(:EM) the set of Holder

functions. Clearly C(:EM) c S(:E

_{M )} C

ll(:E

M ).

From here on, we will

use the simpler notations {) and L</J to denote the operators Ou and

L

_u,</>,

respectively.

THEOREM

1.3. Let (EM,u) be a

mixing

one-sided subsbift of finite type and {3

E

C(E M). ^Then {3

E

ll(EM) if and only if a{3

E

ll(E M).

Proof It is clear that if {3

E

ll(E M) ^{then a{3}

^E

ll(EM). Conversely, assume that a{3

E

ll(E M). For simplicity, write <p

=

a{3. Since <p

E

ll(E M), ^{there are a} > ° ^{and b}

^E

(0,1) such that (k = 0,1,2, ... ).

Since (EM, u) is mixing, there

is

a point

Z = ZOZl Z2 • ••

in

EM

such that the forward orbit r

=

{z,uz,u

²

z, ... } of

Z is

dense in EM.

Let kEN, x,

y

E r, and assume that

XOXl ••• Xk

=

YOYl .•• Yk.

We will show that

ab

k

1{3(x) - {3(y) I

~ 1-

bb,

from which our assertion follows because r is dense in

EM

and {3 is continuous.

Since x, Y

Er,

there are i,

j E

N such that x = ^u

^iZ

^{and Y} = ^u

^j

^z.

We may assume that

i

<

j.

From the assumption that

XOXl ••• Xk =

YOYl ••• Yk,

we have

ZiZi+l ••• Zi+k

=

ZjZj+l ••• Zj+k,

and hence there is a point

w

in

EM

such that

Then we have

j - i - l

L ^cP(ulw)

⁼

^0,

l=O

and

(1=0,1, ... ,j-i-1).

Therefore

j-i-1

= L

^<1>(

^{uHl z)}

l=O j-i-1

~ L 1<I>(uHlz) - <I>(ulw) I

l=O

o

j-i-1

~

L ^abi-

^{Hk- l}

l=O

<~bk.

- l - b

IT 8{3

E

S(E

_{M ),}

{3 may not necessarily lie in S(E

_{M )}

as we see in the folloWing example.

EXAMPLE

1.4. Let {3 : {O,

l}N-+ll~.

be defined by

00

{3(x) = ~ ^{(n :n} ₁ ^{)2 (x =} XOX1 X2'"

E {O,l}N).

IT x, Y

E

{O,

l}N

and XOX1 ... Xk

=

YOY1 ... Yk, then

00

{J(x)-{3(y)= L ~:;~~.

n=k+1 Hence

00 ¹ 100 ^{1 1}

vark{3 - ' " > dv - - -

- L....J (n + 1)2 -- (v + 1)2 -

k

+ 2 '

n=k+1 k+1

so that {3 is not in S( {O, 1}N). On the other hand,

~ 2n+1

18{3(x) - 8{3(y) I ~ L....J n ₂ (n + 1)2lxn - Ynl n=k+1

< _~ 2n+1

-- L....J n

²

(n + 1)2 n=k+1

rv ...,..---:__=_

1

(k+1)3'

whenever x,y E {O,1}N and

XOXl ••• Xk = YOYl ••• Yk.

Therefore B{3 E S({O,1}N).

2. Coincidence of the Gibbs Measures

Let

(~M,

a) be a mixing one-sided subshift of finite type and let 4>

E S(~M).

Then it is known that there is one and only one equilibrium state J.Lt/> of 4>, called the Gibbs measure of 4>. We can easily extend the result when 4>

E S(~M)

+

8C(~M).

The following theorem is called Ruelle's operator theorem.

THEOREM

2.1. Let

(~M,a)

be a mixing one-sided subshift of finite type, and let 4>

E S(~M)

+

8C(~M).

Then there are a positive real number A, a continuous function f which

is

positive everywhere on

~M,

and a positive Borel measure J.L such that (a) Lt/>f

=

Af, L;J.L = AJ.L, and fE M fdJ.L = ^1,

(b) for

all

9

E C(~M), IIA-nL~g

- (fEMgdJL)fll

~

0 as n

~ ^00,

and (c) the Gibbs measure of 4>

is

given by f dJL.

Moreover, the A--eigenspaces of Lt/> and L; are 1-dimensiona!.

Proof. For the special case when 4>

E S(~M),

see [4]. The general case follows from the special case, because for all 4>,

{3 E C(~M)

we have

Lt/>+8!3

=

Li,1

0

Lt/>

0

LI,{3,

and 4> and 4> + 8P have the same set of equilibrium states.

0

From here on, any triple (A, f, dJ.L) which satisfies the conditions (a) and (b) of the above theorem will be called a Ruelle triple of 4>(

E S(~M)

+

8C(~M».

LEMMA

2.2. Let

(~M,

a) be a mixing one-sided subshift of finite type, and let (A, f, dJL) be a Ruelle triple of 4>

E S(~M)

+

8C(~M)'

Then for

all

nonempty elopen subset U

of~M,

we have J.L(U) > O.

Proof. Let U be an arbitrary elopen subset of

~M

and let hu denote the characteristic function of U. Since

(~M,

a) is mixing, there is a positive integer N such that

L:hu(x)

=

L ^{exp(4)(Y) + 4>(ay) + ... + 4>(a}

^{N- 1}

^{y»hu(Y) > 0}

yEu-N(x)

for

all

x

E ~M.

Since

~M

is compact, the continuous function L1Jhu has a positive minimum, so that

o

THEOREM

2.3. Let

(~M,U)

be a

mixing

one-sided subshift of finite type,

</>,'Ij;E S(~M) +ac(~M);

and assume that (1,!, dp,) and (1,

g,

dv) are Ruelle triples of

</>

and

'Ij;,

respectively. Then the following are equivalent:

(a)

</>= 'Ij;.

(b)

p,=

cv for some positive constant c.

(c) Le/>

⁼

L'l/J.

Proof The implication (a)=? (b) follows from Ruelle's operator theo- rem, and (c)=?(a) follows from Lemma 1.2. To prove (b)=?(c), suppose that

p, =

cv for some positive constant c. Then Lemma 1.1 implies that

{ hLe/>h'dp,

= {

Le/>((h

⁰

u)h')dp,

J'EM J'EM

= {

(h

⁰

u)h'dp,

J'EM

= { L'l/J((h

⁰

u)h')dp,

J'EM

= {

hL'l/Jh'dp, (h,h'

E

C(~M))' J'EM

Therefore we obtain Le/>

=

L'l/J by Lemma 2.2. o

Under the assumptions of Theorem 2.3, it is not true in general that

! =

9

implies

</>

=

'Ij;,

as we see in the following example.

EXAMPLE

2.4. Let ho and hI be the characteristic functions of Uo =

{x E {O,l}N: Xo = O} and U I = {x E {O,l}N: Xo = 1} respectively, and let a, b

E

lR satisfy ea + e

^b=

1. Define

</>

and

'Ij;

by

</>

= aho + bhl,

'Ij;

= bho + ahl.

Then </>,'l/J

E

ll({O,l}N). Let JL and v denote the Bernoulli measures

on {O,l}N defined by

JL(U

_o)

=

^ea,

^JL(U

1)

=

^eb,

^v(U

o)

=

^eb,

^V(Ul) =

^ea.

Then it is easy to see that (1,1, JL) and (1,1, v) are Ruelle triples of

</>

and 'l/J respectively.

Now we prove the main theorem of this section in a simple and direct manner. In [1] and [4], one can find other proofs of the theorem in weaker forms.

THEOREM

2.5. Let

(~M,

0") be a

mixing

one-sided subsbift oEfinite type, and let 4>,'l/J

E S(E_{M )}

+

8C(E_{M ).}

Then

</>

and 'l/J have the same Gibbs measure if and only if there are a constant K and a continuous function (3 :

~^M ~

lR such that

</>=

'l/J + K + 8{3.

Moreover,

if

4>, 'l/J

E

ll(E M) ^{then (3}

^E

ll(EM).

Proof Let (A,f,dJL) and (AI,g,dv) denote Ruelle triples of 4> and 'l/J respectively. Then, by Ruelle's operator theorem, fdJL is the Gibbs measure of 4> and gdv is that of 'l/J.

Suppose that 4> = 'l/J + K + 8{3 for some constant K and some continuous {3. Then it is easy to see that (AIe

^K ,

e-

f3g,

e

^f3

dv) is a Ruelle triple of 4>. Therefore we have

fdJL

=

e-

^f3

ge

^f3

dv

=

gdv, by the uniqueness of the Gibbs measure.

Conversely, suppose that fdJL

=

gdv. Then we will show that

4> = 'l/J + (log A - log AI) + 8(log 9 - log f).

First of all, we may assume that A

=

A'

=

1 since otherwise we can replace 4> and 'l/J by 4> -log A and 'l/J -log A' respectively. Since (1,

g,

dv) is a Ruelle triple of'l/J, it follows that

f 9

(1, gg, ^jdv)

is a Ruelle triple of'l/J + 8(logg - log/). Now the assumption that fdJL

=

gdv implies that (g/ /)dv

=

dv. Therefore by Theorem 2.3 we see that

</> =

'l/J + 8(log

9 -

log /). Finally, the last statement follows

from Theorem 1.3. 0

EXAMPLE

2.6. Let ho and hI be the characteristic functions defined

in

Example 2.4. For

i,j E

{0,1} let U

ij

= {x

E

{0,1}N :

XOXI

=

ij}

and let h

_ij

be the characteristic function of U

ij .

Let a and b be arbitrary real numbers and define 4> and 'l/J by

'l/J = ^ahoo + bhol + ah

lO

+ bhu . Then it is easy to see that

4> - 'l/J = 8/3,

where /3 = -bho - ahl. Therefore the Gibbs measures of 4> and 'l/J are the same.

3. Conditions for a Function to be a Coboundary

In this section, we discuss conditions for a function 4> to be a cobound-

ary,

Le., 4>

=

8/3 for some function /3.

Let

(~M,

q) be a mixing one-sided subshift of finite type, and let

4>

E C(~M)' If

there is a function /3 such that 4>

=

8/3 then we have

p-I

L ^4>(qi

^{X ) =}

^0,

i=O

for all

p

= 1,2, ... and for all x

E _~M

satisfying

^qPX

= x. The following theorem shows that the converse is true provided that 4>

E S(~M)

and /3

E_C(~M)'

THEOREM

3.1. Let

(~M,

q) be a mixing one-sided subshift offinite type, and let 4>

E _S(~M)'

Then there is a function /3

E _C(~M)

such

that 4> = ^8/3

^if

^{and only}

^if

p-I

L ^{4>(qi x )}

⁼

^0,

i=O

for all p = 1,2, ... and for all x

E

:EM satisfying

qPX

= x. Moreover,

if

4>

E

1l(:EM) then /3

E

1l(:EM) also.

Proof

First of all, recall that we have already proved the last state- ment in Theorem 1.3. Then we need only to prove the 'if' part. So assume that </> satisfies

p-I

L </>(ix)

=

0,

i=O

for all

p =

1,2, ... and for all x

E _~M

satisfying uPx

=

x.

Since

(~M,U)

is mixing, there is a point

Z = ZOZIZ2 •••

in

~M

such that the forward orbit r

=

{z, uz,u

²

z, ... } of

Z

is dense in

_~M.

Then define /30 : r

^----7

lR by

i-I

/30(u

ⁱ

z)

= -

L</>(ulz)

l=O

(j

=

0,1,2, ... ).

It is obvious that </>(x) = /3o(x) - {3o(ux) for all x

E

r. Now a similar argument as in the proof of Theorem 1.3 shows that if x,

YE

r satisfies

XOXI ••• Xk = YOYI ••• Yk

then we have

00

l/3o(x) - /30(Y) I ~ L varl</>.

l=k+1

Therefore /30 is uniformly continuous on r, so that there is a continuous function /3 on

~M

such that /3lr = /30. Now it is clear that <p = ^{B/3. 0}

REMARK.

Using the same method as in the above proof, we can show that for

<P E S(~M)

and K

E

lR there is a function /3

^E C(~M)

such that <p

=

K + B/3 if and only if

p-l

L</>(uix)

=

pK,

i=O

for all

p =

1,2, ... and for all x

E ~M

satisfying uPx

=

x.

Finally, we consider coboundary relations on functions defined on

two-sided subshifts of finite type. Let

(~M,U)

be a mixing one-sided

subshift of finite type. Then the natural extension of (:E

M,

u) is the mixing two-sided subshift of finite type (i:

^{M ,}

a) defined by

and

In this case, the natural factoring

7r :

i:

M ^-+

EM, defined by

is a continuous open surjective mapping and satisfies

7r0

a

=

u

0 7r.

For each u-invariant Borel probability measure

J.L

on EM there is a unique a-invariant Borel probability measure [J, on i:

M

such that

(J

E C(~M».

It is easy to see that

J.L^I--t

[J, is a one-to-one correspondence between the set of all u-invariant Borel probability measures on

~M

and the set of all a-invariant Borel probability measures on i:

M .

Moreover, for each <P

E C(~M),

a u-invariant measure

J.L

is an equilibrium state of <P if and only if jl is an equilibrium state of <P

0 7r.

For <P

E

C(i: _M), and k = 0,1,2, ... , define the k-,--th variation of <P by

vark<P = sup{I4>(x) - 4>(y) I : x,y

E

i:

M , Xi

=

^Yi

for lil S k}.

Then define S(i:M) ^and ll(i:M) in the obvious way. From the discus- sions given in [1, Chapter 1], it follows that for all 4>

E

S(i:M)+&c(i:M) there is one and only one equilibrium state of <P, which is also called the Gibbs measure of 4>. Moreover, Theorem 2.5 holds for (i:

M ,

a) also.

The Gibbs equivalence relation

^rv

on i:

M

is defined as follows: for

X,

Y

E

i:

M , X ^rv y

if and only if there is a positive integer n such that

Xi

=

^Yi

^whenever lil ~ n. For

^X E

i::

M

let [x] denote the equivalence class of x, Le., [x] = {y

E

i::

M :

x ,...., y}.

Let 4>

E

S(i::

M ).

If there are a constant K and a function P

^E

C(i::

M )

such that 4> = K + Gp, then it is easy to see that

00 00

L (4)(o-lx) - 4>(o-ly)) = L (4)(o--ly) - 4>(o--lx)) ,

1=0 1=1

whenever x, y

E

i::

_M

and x ,...., y. Note that since 4>

E

S(i::

_{M ) ,}

the infinite sums in the above equation converge absolutely. The following theorem shows that the converse is also true.

THEOREM

3.2. Let (i::

^M,

0-) be a mixing two-sided subshift offinite type and 4>

E

S(i::

M ).

Then there are a constant K and a function

P

E

C(E

M )

such that 4>

=

K + Gp if and only if

00 00

L (4)(o-lx) - 4>(o-ly)) = L (4)(o--ly) - 4>(o--lx)) ,

1=0 1=1

whenever x, y

E

i::

M

and x ,...., y. Moreover, if 4>

E

1l(t

M )

then P

E

1l(t

M )

also.

Proof. Again, we need only to prove the 'if' part. We first prove the theorem in the special case that (tM,o-) has a fixed point. So assume that there is a point z

E

t

M

such that O-Z = z, and that 4> satisfies the condition stated in the theorem. Since (i::

^M,

0-) is mixing, it follows that [z] is dense in t

M •

Let K = ^-4>(z), and define a function f30 : [z] -lR by

00

f30(x) = L ^{(4)(o-lx) + K)}

1=0

(x

E

[z]).

Since z is a fixed point of 0-, Po is a well-defined function on [z]. Then it is clear that

4>(x) = ^K + Po(x) - Po(o-x) _ (x

E

[z]).

Moreover, the assumption implies

00 00

f3o(x)

=

L (</>(ulx) + K)

= -

L (</>(u-lx) + K)

l=O l=l

(x

E

[z)).

Now assume that k is a positive integer, x, y

E

[z], and that Xi

=

Yi for lil ::; k. Then there is a point W

E

[z] such that Wi

=

Xi for i 2: -k and Wi

=

Yi for i ::;

k

so that

lf3o(x) - {3o(y) I ::; lf3o(x) - f3o(w)/ + lf3o(w) - f30(Y) I

00 00

::; L I</>(ulx) - </>(a-lw) I + L I</>(u-ly) - </>(u-lW) I

l=O l=l

00 00

::; L varl</> + L varl</>.

l=k l=k+l

Since [z] is dense in EM, this inequality implies our assertion.

In the general case, there is a positive integer

p

and a point z

E

EM

such that uPz

=

z. Then (EM,uP) has a fixed point z, and therefore there is a constant C and a function f3

E

EM such that

</>(x) + </>(ux) + ... + </>(u

^p- 1

x) = ^C + f3(x) - f3(uPx) Moreover, if

</>

is Holder then f3 is Holder. Let

(x

E

EM).

'lj;(x) = </>(x) - f3(x) + f3(ux) (x

E

EM).

Then we must show that

'lj;

is a constant function. From the definition of'lj;, we have

so that

'lj;(x) + 'lj;(ux) + ... + 'lj;(a-

^{p- 1}

x) = ^{C (x}

^E

^EM),

'lj;(x)

=

'lj;(uPx)

Since (EM,a-) is mixing, there exists a point x

E

EM whose forward orbit {x,u

^p

x,u

^2p

x, ... } is dense in EM. Therefore we conclude that

'lj;

is a constant function. 0

References

[1] R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomor- phisms, Springer, Heidelberg, 1975.

[2] D. Ruelle, Thermodynamic Formalism, Addison-Wesley, Reading, 1978.

[3] K. Schmidt, The cohomology of higher-dimensional shifts of finite type, Pacific J. of Math. 170 (1995), 237-269.

[4] P. Waiters, Ruelle's operator theorem and g-measures, Trans. Amer. Math.

Soc. 214 (1975), 375-387.

Young-One Kim

Department of Mathematics Sejong University

Seoul 143-747, Korea

E-mail: [email protected] Jungseob Lee

Department of Mathematics Ajou University

Suwon 442-749, Korea

E-mail: [email protected]