Some Systems with Unique EquilibriumStates
by
RUFUS BOWEN*
Department of Mathematics
University of California
Berkeley, California 94720
We shall be dealing with a h o m e o m o r p h i s m f : X --~ X of a compact metric
space and a continuous ~o: X ~ R. Let My(X) denote the set of all f-invariant
Borel probability measures on X. tz ~ My(X) is called an equilibrium state (for
f and ~o) if
h~,(f) +tz(~o) =
sup
veMs(X)
(h~(f) + v(q))),
where h , ( f ) is the entropy of/z. We want conditions o n f a n d ~owhich guarantee
a unique equilibrium state.
f i s called expansive if there is an ~ > 0 such that for any two points x ¢ y
in X there is an n e Z so that d ( f " ( x ) , f " ( y ) ) > E. fsatisfies specification if for
each 8 > 0 there is an integer p(8) for which the following is true: if I~,. •., I,
are intervals o f integers contained in [a, b] with d(ll, I j) >_p(8) for i ¢ j and
x l , . . . , x , eX, then there is a point x e X with fb-"+PO)(X)= X and
d(fk(x), fk(xi)) < 8 for k e Ii. This condition allows us to construct a lot of
periodic points.
F o r ~oe C(X) and n _ 1 let
( S.cp) (x) = cp(x)+ q)(f(x)) + " " + q)(f"- 1(x)).
Let V ( f ) be the set o f ~ e C(X) for which an E > 0 and a K exist for which the
following is true: d(fk(x),fk(y)) < ~ for all 0 < k < n =~ IS.~(x)-S.~(y)l <_ K,
T H E O R E M . Let f: X --->X be an expansive homeomorphism of a compact
metric space satisfying specification. Then each ~ E V ( f ) has a unique equilibrium
state t%.
Remark. Let 8 be any expansive constant for f. Then, if ¢p E V ( f ) , I olf =
sup {Is.~(x)-S.~(y)l: n ___ 1 and d(fk(x),fk(y)) < 8Vk e [0, n)} is finite (if ,, K
are as in the definition of ~o e V ( f ) and d(x, y) < , follows from d(fi(x), fJ(y))
_< 8 for all IJl -< N (there is such an N by expansiveness), then I~olf -< K + 2N1¥[I ).
* Partially supported by NSF grant GP-14519.
193
MATHEMATICALSYSTEMS THEORY,WoE 8, NO. 3.
© 1975 by Springer-Verlag N e w Y o r k Inc.
194
R u F u s BOWEN
V ( f ) is a linear subspace of C(X) and becomes a Banach space under the norm
Jll~olrl = II~oll+ I~oly. We now give a condition on ~ that guarantees q~ e V ( f ) and
occurs in examples.
P R O P O S I T I O N . For E > 0 and m > 0 define Varm (% f ~) = sup {]9(x)cp(y)[: d(fJ(x),fJ(y)) < ~for all ]j] < m}. Then q~ v(U) /f Z~=o Var= ( % f E)
<
00.
Proof Suppose d(fk(x), fk(y)) < ~ for all 0 < k < n. Then for m k =
min {k, n - k - l }
one has d ( f J ( f k x ) , f J ( f k y ) ) < , for all IJl -< mk. So
I~ofk(x)--~fk(y)l ~
Var,,~ (~0,f ~)
and
IS.q~(x)-S,,q~(y)l ___2 ~ Vars,,(~,f,
,)
m=0
since m k = m for at most two k's.
Example 1. One-dimensional lattice systems [13], [14]. Here X = I-[z T where
T is a finite set with the discrete topology a n d f i s the shift map, i.e., for t =
(t~)i~z one defines f(t)~ = t~+ ~. X is the space of all possible configurations of an
infinite one-dimensional lattice where T is the set of possibilities at a single
point. Often T = {0, 1 } and denotes whether a position is occupied or not.
Furthermore we are given a chemical potential c: T ~ R and pair-potentials
qb: Z+× T× T - + R where O(n, t, t') is the potential energy due to a state t'
being n places to the right of a state t. For t ~ X define U(t) = C(to) + ~joo=1
qb(j, to, tj). Let LI@[I~= max,,,,~r [qb(j, t, t')[. Then Ue C(X) provided Y'~=~
I[@jlt < oo.
Let d be a metric on X. Pick E > 0 so that d(t, s) < ~ implies t o = S O. If
d(fk(t), fk(s)) < ~ for 0 _< k < n, then tk = Sk for 0 _< k < n and
[ U ( t ) - U(s)[ _< ~
j=l
I@(j, to, t j ) - @ ( j , t o, si) [ < 2 ~ II~jll.
j=n
Hence
V a r , ( U , f , E) < 2 ~
n=l
~ II@jll < 2 x~ jll*jll.
n=l j=n
j=l
Thus U~ V ( f ) i f ~ = t
jl[~j[I < oo.
This condition ~]~=lJil~j]! < oo is a standard one [13] and our theorem is
already known for a one-dimensional lattice system satisfying it. Dobrushin
([9], [10]) proves that in this case there is a unique invariant Gibbs state and it
is known that Gibbs states and equilibrium states are the same for lattice
systems ([8, T h e o r e m 3], [21, Theorem A.1], [22, Theorem 1]). Ruelle [15] has
generalized the notion of specification to Z"-actions and Dobrushin ([9], [10])
has theorems on uniqueness of equilibrium state for the case of v-dimensional
lattice systems; at present our approach doesn't seem to work for v > 2.
Lanford [23] contains an exposition of equilibrium states in statistical mechanics
which covers the results mentioned above.
Some Systems with Unique Equilibrium States
195
Example 2. Axiom A diffeomorphisms [17]. Let g: M ~ M be a diffeomorphism of a compact manifold satisfying Smale's Axiom A and f~s c M a
basic set for g [17, p. 777]. Then g(f~s) = f~s and g[f~, is expansive, gifts may not
satisfy specification but one can express f2, as a disjoint union of compact sets
~s = X1 u . . . U X, so that g ( X i ) = X i + 1 (Xn+ 1 = X1) and g"lX i satisfies
specification [2]. If ~oe V(g[f~s), then ~b = ~0+q~ o g + . . . +~o o g , - I e V(g"IX~). If
tz~ is the unique equilibrium state for g"lX~ with ~blXi, the one can show that
tz = n- l(/z t + . . . +tz,) e Mg(X) and is a unique equilibrium state for gifts and ~0.
We claim that if F: M ~ R is differentiable (or of positive HOlder exponent),
then ~0 = Flf2 s ~ V(glf~s). First, from the fact that g is Axiom A, there are
, > 0 and ~ > 1 so that if x, y e f~s and d(fk(x), fk(y)) < , for all Ikl <-- n, then
d(x, y) < 0~-". As F is differentiable, it has a Lipschitz constant L. Then Var,
(Flf~ s, gifts, ,) < Lc,-" and ~ L x Var, (F[f~s, gl~qs, ") < oo.
For the case of an Anosov diffeomorphism Sinai [16] constructed the
measure we constructed (in a somewhat different way) but did not give in our
theorem.
Example 3. Intrinsic ergodicity. ~ = 0 obviously is in V(f). As v(0) = 0,
tz is an equilibrium state for ~0 = 0 if and only if hu(f) >_by(f) for all tz e MI(X),
i.e., tz is an invariant measure maximizing the entropy. A homeomorphism
having a unique such t~ is called intrinsically ergodie [18]. By our theorem any
expansive homeomorphism satisfying specification is intrinsically ergodic.
The first theorem of this kind is due to W. Parry [12]: a subshift of finite
type is intrinsically ergodic. A subshift of finite type may not satisfy specification
but one can reduce back to the case satisfying specification as in Example 2
(in fact all subshifts of finite type occur as basic sets [26]). Intrinsic ergodicity
for basic sets was first proved in [3] using symbolic dynamics. For them tL was
first constructed in [2]. The t~ in our present more general case was constructed
by Ruelle [13]. The estimates we prove in the present paper are modifications of
ones in [4], [5], [6].
Another class of intrinsically ergodic systems are the sofic systems of B. Weiss
[19]; irreducible sofic systems satisfy specification the way we stated it. Finally
we mention that H. Brascamp [25] and W. Krieger [24] proved our theorem for
the case f a subshift of finite type and ~o locally constant.
We now proceed with the proof of the theorem. Throughout, f and q> are
fixed and satisfy the hypotheses of the theorem.
Definition. E c X is an (n, ,)-separated set (with respect to f ) if for x, y
distinct points in E, there is an integer k such that 0 _< k < n and d(fk(x),
fk(y)) > ¢. Define Z,(% ,) = sup ~2~x~r exp (S,q>(x)): E is (n, ,)-separated}.
(Recall S,~o(x) = q~(x)+ qJ(fx) + . . . + q~(f"- ix)).
Ruelle [15] has proved that (under conditions more general than ours)
P(9~) = lira 1 log Z,(cp, ,)
n~oo
/'/
exists and does not depend o n , for E small enough to be an expansive constant.
For q~ = 0, one is looking at the topological entropy.
196
RUFUS BOWEN
F o r v ~ M y ( X ) and A = {A1,-.., Am} a finite v-measurable partition of X
one defines H,(A) = - ~ = 1 v(Ai) log v(A~). F o r n > 1 let
A; = {A~on f - ~ A ~ c~...c~f-"+~A~._,: A~k ~A}.
The following limit exists (see [20]): H , ( f , A) = lim,_.~ n - x H,(A~). Finally
one defines the entropy h~(f) = sup A h,(f, A) where A varies over all finite
v-measurable partitions of X.
Let P~(~)= h,(f)+~(~). It is known [15, Theorem 5.1] that e(~o)= sup
{P~(~): v ~ Ms(X)}. Furthermore one can construct a /~ with P,(~o) = P(~o).
Let Per, = {x ~ X: f " ( x ) = x} and z ~ r ( ~ ) = ~ ,
exp (S,~o(x)). Define
/%,,, = [Z,,~r (~o)1- ~ y '
exp (S.~(x))/zx
XePer.
where t~ is the probability measure with /z~({x}) = 1. One sees that /%,,
M f ( X ) . N o w M y ( X ) with the weak topology is a compact metrizable space [11];
hence one can choose n~ ~ ~ so that /~ = l i m ~
/ % , , , ~ M f ( X ) exists.
Throughout this section the sequence {n~ } will be fixed a n d / z defined as this
limit. Ruelle [15, Theorems 3.2 and 5.1] proved that P~(~o) = P(~o). We want to
show that if P,(~o) = P(~o), then v =/~. (Note that this will imply that if/~' =
l i m ~ / % , . ~ t h e n / , ' = / z ; from this it follows that one has t* = lim._,~ t%,..)
L E M M A 1. For any sufficiently small positive E and 3 there is a constant
Ca, ` so that Z.(% 3) < Ca, ` Z,(% , ) f o r all n > O.
Proof. We assume c small enough to that 2E is an expansive constant. Then
there is an N so that d(fkx, f k y ) < 2~ for all Ikl <- N =~ d(x, y) < 3. Choose
> 0 so small that if d(x, y) < ~, then d ( f k x , f k y ) < 3 for all Ikl -< N. Let
F be a maximal (n, ~)-separated set and E and (n, 8)-separated set. F r o m the
maximality of F it follows that for each x ~ E there is an a(x) ~ F so that
d ( f i x , fia(x)) < • for all i e [0, n) (otherwise F u {x} is (n, ~)-separated). F o r
a ~ F let E , = {x ~ E: a(x) = a}. I f x, y ~ E., then d ( f i x , f i y ) < 2~ for all
i ~ [0, n) and so d ( f i x , f i y ) < 3 for all i E [N, n - N ) . As {x, y} is (n, 3)-separated,
one must have either d(x, y) > ~ or d ( f " ( x ) , f " ( y ) ) > ~. Let M be the m a x i m u m
number of points one can choose from the compact metric space X × X so that
every two points are distance ~ apart. {(x, f " ( x ) ) : x ~ E~ } is such a set of points;
so card E~ < M. F o r x ~ E., [S,~o(x)- S.cp(a)l < K. Hence
exp (S,~(x)) <_ ~ (card E,)e k exp (S,~o(a)) <_ MekZ,(% ~).
x~E
aEF
We get our result with Ca,, = M e k.
L E M M A 2. For ~ small there are positive constants E, and D, so that
k
k
lq E,z., (% ,) _< z.,+... +.k(~, ") -< j]-I
&z.,(% ~)
j=l
=l
whenever nl," " ", nk > 1.
Proof. Let E be (n 1 + . . . + n k, E)-separated and, for each j, Fj a maximal
(n j, ½~)-separated set. F o r x e E pick g(x) = (gl(x)," " ", g~(x)) e F 1 × . . . × Fk so
Some Systems with Unique Equilibrium States
197
that d ( f "~+"" +"J-~ + i(x), figs(x)) <_ ½E for all 0 < i < nj (use the maximality of
Fs). The m a p g: E - + F, x . . . x Fk is one-to-one since E is (nx + " " +nk, E)separated. N o w
k
IS.,+ ... +.alP(x) -
X S,/P(gj(x))I
j=l
k
j=l
IS.j~(f "'+''" +"~-'(x))- S.s~(g j(x)) [ < k K
Hence
k
~2 exp (S,,,+... +,kq~(x)) < 1-I e r z , j(% ½")"
x~E
j= l
Letting D r = erC,/2.,, Z,,+... +.k(qo, E) < 1--[~=, D,Z,s(% ~).
Let E s be an (nj, 3e)-separated set for e a c h j . Let I s = [as, a j + n s - I] where
aj = n l + " " " + n s - x + (J-1)p(E) and p(E) is as in the definition of specification.
Then for each z = (Zl,"' ", z , ) e El x . . . x E, one can find an x = x(z) such
that d(ff~+i(x),fi(zj)) < ~ for 0 _< i < n s. (Apply specification to x s = f - " S z / ) .
Using the triangle inequality one sees that E = {x(z): z e E, x . . . x E,} is an
(m, ,)-separated set where m = ni + " • +na + ( k - Up(e) and
k
S,,,~(x(z)) >_ -kp(,)[I ~i[-kK + j ~,
S, fi~(zj).
=l
Hence
k
Z,,,(% e) > exp ( - ( p @ ) l l ~ l ] + g ) k )
~ Z~j(~, 3 G
J
By the first part of this l e m m a
Zm(%') < D,k Z,,+... +,k(~v, e)Zp(,)(qo, ,)k-~
Thus
z,,~+... +,,~(~, ~) >
--
exp
(-(p(.) Jl~oll+K)k)
k
!
[l
By L e m m a 1, Z.j(% ,) < C,, 3,Z.~(% 3E). Let
G=
exp (-p(¢)I[~o]l+K))
D, max {1, Zp(o( % ¢)}C~,3,
L E M M A 3, For small ~ and all n
1 ee . < Z,(%~)_<
D,
1 ee .
E~
where P = P(~).
Proof. Recall that P = l i m , ~ n -1 log Z,(% e) for all small ~. Suppose
Z.(% c) = ~ > ee"/E,. Then Zk,(% ~) > (E,~) k by L e m m a 2 and P = l i m k ~
(1/kn) log ZR,(% E) > n- 1 log E,~ > P, a contradiction. The other inequality is
provide d similarly.
198
RUFUSBOWEN
L E M M A 4. There are positive dl and d 2 so that dae TM <_ Z ~ r (9) <- d2ee~
for sufficiently large n.
Proof. For E small, Per. is (n, ~)-separated. For if x, y ~ Per. and d(fk(x),
i f ( y ) ) <_ E holds for k ~ [0, n), then it holds for all k ~ Z and x = y by expansiveness. Hence Z ~ ~ (9) <-- Z.(% ~) and L e m m a 3 gives us d2.
Consider now n _> p(~) and an (n-p(~), 3E)-separated set E. By specification,
for each z ~ E there is an x(z) ~ Per. with d(fkz, fkx(z)) <_ ~ for k ~ [0, n -p(~)).
Then x(z) # x(z') for z # z' and S,9(x(z)) >_ S._m)9(z)-K-p(~)1191[. Hence
ZnTM (99) > exp (-K-p(,)[[911) Z . - m ) ( % 3¢) > exp (-- K--p(E) [19[[)eV..
-
-
D , e e~'(')
L E M M A 5. For each small E there is an A, > 0 so that, for every y ~ X
and n > 1,
tz{x c X: d(fkx, fky) <_ E V k E [0, n}) _> A, exp ( S . 9 ( y ) - n P ) .
Proof. Let V be the set above. Let E,, be an (m, 3E)-separated set and r =
n + m + 2p(E). By specification, for each z ~ E,. we can find an x(z) ~ Per, so that
d(fkx(z), fky) < E for 0 < k < n and d(fJ+"+m)x(z), fJz) < ~ for 0 < j < m.
Notice that x(z) ¢ x(z') for z # z' (in E,,), x(z) ~ V n Per~ and
ISrg(x(z))- s.9(y)- smg(z)l -< 2p(,) El9 IJ+ 2/(.
Hence
/%,,(V) = [ Z , ~ ( 9 ) ] - I
~
exp(S,9(x))
xEVnPr
>_ [Z~ ~r (9)] -1 exp (S.9(y)-2p(E)]19]l-2K)
~
exp (S~9(z)).
z EZra
Using Lemmas 3 and 4 and Z,.(% ,) = supz° ' ~'.z~E,. exp (Stag(z)) we get I%,~(V)
> A, exp (S.9(y) - n P ) where A, = [d2 D, exp (2p(.) 11911+ 2p(0P + 2K)]-~. This
holds for all r ___ n+2p(E). Let r = n k and k - > oo: since t~.k-+ t~ weakly and
V is closed,
t~(V) > lim sup t % , . k ( V ) > A , e x p
(S,,9(y)-nP).
tz is called partially mixing if there is a constant c > 0 so that lim inf,,.~
t~(P n f -"Q) > ctz(P)tz(Q) for all t~-measurable sets P and Q. One sees that such
a tz is ergodic.
L E M M A 6. tz is partially mixing.
Proof. Fix some small E > 0. Let A, B be compact and U, V be the 8neighborhoods of A, B for some ~ > 0. There is an N(3) so that d(x, y) <
whenever d(fJ(x), f~(y)) < E for all IJ[ < N(3). Consider now n > 2N(3) and
any other positive integers s and t. Let Es be an (s, 3e)-separated set and Et a
(t, 3a)-separated set.
Some Systems with Unique Equilibrium States
199
Define Ig = [ag, b j] for j = 1, 2, 3, 4 by
b2 = a2+s
b 3 = a 3+n
b4 = a4+t.
a2 = b l + p ( ~ )
a3 = b2 + P ( 0
a4 = b3 + P ( 0
For each z = (zl, z2, z3, z4) in
f -[~l (Per, c3 A) x Es
x f -Ill (Per.
n B) x E,
use specification to find a point x = x(x) so thatfb'-°l+P(')x = x and d(f"J+kx,
fkzi) < E for 0 < k < b j - a j . By the definition of N(8) one sees that this
implies x • U and fs+2P(')x • V. Because the various coordinate sets for z =
(Zl, z2, zz, z4) are 3E-separated, one gets x(z) # x(z') for z # z'. Let m = b 4 - al
+p(e) = t + s + 2n + 4p(E). Then
Smq~(x(z)) > S.~(zl) + Ss~o(z:) + S~(za) + $6o(z4)- 4 K - 4p(~) ]l ~oil.
Hence
exp (Smq~(x(z)) > b ~ exp (S.rp(z,)) ~ exp (S.~o(z3)) Z~(% 3~)Z,(% 3e),
Z
Zl
~3
where b = exp ( - 4 K - 4 p ( ~ ) I [ ~o[I). A s f " ( z l ) = zl, S.go(zt) = S~(ft"/zlZl). So
~] exp (S.~o(z,)) =
z1
Z
exp (S.q~(z)) = / % , . ( A ) Z ~ r (~).
z E Pern N A
Similarly ~z3 exp (S.~(z,,)) = ~ , . ( B ) Z ~ r (~). Putting the inequalities together
with L e m m a s 3 and 4:
Id,tp,m(U n f
-~- 2p(.) V)
[groper (~°)]-
1
2 exp (S,.q~(x(z)))
g
> b(d 2 le-P.) (t%,.(A)dle TM) (t%,.(B)dle TM) (D;.le Ps) (D;.'e P')
bdf
>- d2D], exp ( - 4 p ( c ) P ) t%,.(A) t%,.(B).
Let c = bdz~d~e-4p(')PDL 2. N o w fix n > N(3) and s. Letting t ~
(m = nk)
oo we get
~(U r ~ f - s - 2 P ( ' W ) > lira sup/%,.~(U c~f-~-2P(~)V) >__cI%,.(A)I%,.(B).
k ~
Letting s --> oo we get
lira inf/~(U n f - ' 7 )
> cI%,.(A)I%,n(B).
r'-* o0
Letting n -+ oc we get
lira inf/~(U c ~ f -~ V) >_ c lim sup/%,.~(A)/%,.~(B).
r--* oo
k-'~ oo
200
RuFus BOWEN
This now holds for any compact A and B and any neighborhoods U D A and
V ~ B. By a standard limit argument we get that t~ is partially mixing.
L E M M A 7. Suppose 0 < a~,. . ., a m < 1, s = a~ + . . . + % < 1 and bt,. " ,
b,, are real. Then
(log i=l
~ eb'--log S ) .
i=1
~ ai(bz-l°gai)<-s
Proof. This is standard and can be proved by calculus. The maximum of the
left side is attained at aj = sebS/Y'4 e b'.
LEMMA 8. I f v e My(X) and P~(q~) = P(go), then v = ix.
Proof. The proof closely parallels [6], which was in turn a modification of
ideas of Adler and Weiss [1]. We first suppose that v is singular with respect
to/z. Then there is a (t~+v)-measurable set B ~ X with f ( B ) = B, tz(B) = O,
and v(B) = 1. For each n > 1 we define a partition A, as follows. First fix a
small ¢ > 0. Let E~ be a maximal (n, 2E)-separated set and define
V,(x, ,) = {z e X: d ( f k x , f k y ) <_ ~ Vk e [0, n)}.
Then V,(x, ~) n V,(y, e) = ~ for x, y e E,, x ¢ y. E,, (n, 2E)-spans X and so
X c U,~E,, V,,(x, 2E). Pick Borel sets Ax such that V,(x, E) c A , c V,(x, 2E) and
A, = {A,: x s E,} is a partition of X. Now diam f-t"/21 A, --> 0 as n ~ oo by
expansiveness. Hence there are sets C,, each a union of elements of A,,, so that
(ix+v) (f-r"/2IC,,AB) -+ 0 as n -+ 0 (see Lemma 2 of [6] for instance). As B is
f-invariant, (t~ + v) (C, AB) --> O.
As 2e is an expansive constant for f, the iterates of A,, under f " generates the
a-algebra of Borel sets and so h~(f") = h~(f", A.) < H~(A"). Thus h~(f) = n - 1
h~(f") < n - l H~(A,). Adding v(q~) = n-~ v(S, qO to both sides:
P = hv(f) + v(cp) _< _1 [Hv(A.) + v(Snq~)],
n
.I, _
E [ - (ax) log
+
A x E An
Now S.cp < K+S.rp(x) on Ax; so v(S.rP.XAx) < Kv(Ax)+ S.~o(x)v(A,,) and
nP <_ K +
•
v(Ax) ( S . 9 ( x ) - l o g v ( A x ) )
A x ~ Cn
+
v(A~,) (S,q~(x)- log v(A~,)).
~
AxnCn
= ;~
Now apply Lemma 7 to each sum"
r i P - K < v(C.)
log ~
exp S.cp(x)
A x = Ca
+v(C~) log
~
AxnCn
exp S.~o(x)+2K*
=
Some Systems with Unique Equilibrium States
201
where K* = max, ~ro. i1 ( - t log t). Rearranging terms,
-2K*-K
<_ v(C.) log
exp ( S . r f ( x ) - n P )
~
A x ~ Cn
+v(C~) log
~
AxnCn
exp (S.~o(x)-nP)
= ,~
< v(C,) log A~lt~(C,)+v(C,~) log A~I[~(C~,).
(We use L e m m a 5.) As n--> ~ , v(C,)--->1 and p,(C,) --->0; hence v(C,) log
Ad~( C.) -+ - ~ and v( C,~) log Ad~( C,~) -+ O. This contradicts the above inequality.
In general, if v' e M r ( X ) , then v' = , v + ( l - ~)/~' where ~ e [0, 1], v e M r ( X )
is singular with respect to t~ and /~'e M I ( X ) is absolutely continuous with
respect to tL. As /~' and v are supported on disjoint sets P~,(~) = ,P~(w)+
(1-,)Pu,(c;). Suppose now that P~,(~) = P. Since P~(~) < P and Pu,(cp) _< P,
we must have P~(~) = P or ~ = 0. We just showed we can't have P~(~) = P.
Thus v' = /~' is absolutely continuous. As ~ and v' are invariant, the RadonN i k o d y m derivative dv'/dl~ is invariant. As/~ is ergodic (Lemma 5), this function
must be constant and v' = ~.
Flows. The concepts of expansiveness [7] and specification ([4] but modified
to include the slightly stronger condition used in the p r o o f of L e m m a 1.1 of [5])
can be defined for continuous flows ~F = (~bt: X - + X } . A n equilibrium state
for q~ is a t~ e ~ , M~,(x) maximizing hu(~) +t~(~). ~ e V(~F) provided there are
K and E so that
whenever d(¢z(x ), et(Y)) < • for all t e [0, T]. Then one can show that ~ e V(~F)
has a unique equilibrium state if ~F satisfies expensiveness and specification.
An example of this is a differentiable function on a basic sect for an Axiom A
flow. This theorem for flows has been proved by E. Franco.
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(Received 20 November 1972)