6.
7.
8.
9.
i0.
Yu. M. Berezanskii and Yu. S. Samoilenko, Preprint No. 79.16, Inst. Mat., Kiev (1979).
Yu. L. Daletskii, in: Dokl. Akad. Nauk SSSR, 227, No. 4, 787-794 (1978).
N. N. Frolov, "Self-adjointness of elliptic operators with an infinite number of variables," Funkts. Anal. Prilozhen., 14, No. i, 85-86 (1980).
L. Gross, "Logarithmic Sobolev inequalities," Am. J. Math., 97, No. 4, 1061-1083 (1975).
S. Albeverio and R. Hoegh-Krohn, "Energy forms, Hamiltonians, and distorted Brownian
paths," J. Math. Phys., 18, No. 5, 907-917 (1977).
THE M A X I M U M - E N T R O P Y
OF THE RIEMANN
M. Yu.
MEASURE
OF A RATIONAL
ENDOMORPHISM
SPHERE
Lyubich
UDC 517.53
Let f(z) be a rational function of a complex variable regarded as an analytic endomorphism of the Riemann sphere S 2.
In the present note an existence and uniqueness theorem is
established for the maximum-entropy measure ~ of the endomorphism f. We prove that the roots
of the equation fmz = ~ (z) are asymptotically equally distributed with respect to the measure
Z, where ~ is an arbitrary rational function, apart, possibly, from two exceptional constants.
In particular, the full inverse images f-nc (where c is not an exceptional constant), as well
as the periodic points of the endomorphism f, are asymptotically equidistributed according to
the measure ~.
We shall construct a maximum-entropy measure by investigating a special operator in the
space of continuous functions.
Let A be a bounded operator in the complex Banach space ~ .
We consider its subspaces: ~ u, the closure of the linear hull of the eigenvectors of the operator A corresponding to unitary eigenvalues (J~] = i); 9 0 : {(~ ~ JI[A~'~[I~ 0 ( m ~ ) }
.
Definition.
The operator
~ 9 is strongly
A is called
almost
periodic
of any vector
if the orbit {A ~ (P~m=1
~
precompact.
Theorem on the Decomposition of a Unitary Discrete Spectrum (see [i]).
If A is an almost
periodic operator in the Banach space 9 , then we have the direct decomposition ~ : ~ 0 + 9u.
COROLLARY.
Let the almost-periodic operator A have no unitary eigenvalues other than i,
and let the subspace of invariant vectors be one-dimensional (= Lin{h}).
Then there exists
an A*-invariant functional V ~ 9" , ~(h) = 1 such that for any vector 9 ~ ~ll A~--V(~p) hiI-~
O(m + o~).
uous
We apply this
functions:
result
to the following
operator:
Aq~ (z) = - ~
(n= deg/, ~
C($2), z ~ $2), where
of multiplicity.
We shall use
as defined, e.g., in [2].
By
function identically equal to
A: C(S 2) -~ C(S 2) in the space of contin-
q) ({)
the roots of the equation fE = z are counted with their numbers
the concepts of an "irregular point" and an "exceptional point ;~
F we denote the compact of irregular points.
By 1 we denote the
i. We ]put JI~[!K= sup [~(z) i
z~K
THEOREM
i.
There exists
an A*-invariant
probability
measure#
~ on the sphere
S 2 such
that
A m,T -- ( I (p dpO l K --) O
*By "measure"
Vol.
we shall always mean a complex
( m ~ oo )
Borel regular measure.
Tashkent State University.
Translated from Funktsional'nyi Analiz i Ego Prilozheniya,
16, No. 4, pp. 78-79, October-December, 1982.
Original article submitted July 24, 1981.
0016-2663/82/1604-0309507.50
© 1983 Plenum Publishing
Corporation
309
for any compact
tion.
K C S 2 containing
no exceptional
We denote by 6~ a unit mass concentrated
We consider the following measures:
points
as the point ~.
I
n
of the function
Let ~(~)
f, and any ~
C(S~) .
be a rational
func-
1
/m~=~(~)
,~/m~=~( D
where the roots of the equation /m~=~(~) are counted, respectively, with and without their
degrees of multiplicity, and ~ . ~ is the number of roots counted, ignoring their degrees of
multiplicity.
THEOREM 2.
For all rational functions 9, apart, possibly, from two exceptional constants,
the measures ~m,~ and ~ . ~
converge weakly to a certain probability measure ~ independent of
If T ~ c ,
where the constant c is not exceptional, the convergence F m . ~
(m~)
follows
from Theorem 1 with K = {c}.
This result was obtained in [3] for a polynomial f by methods
of potential theory, which apparently do not work in the general ease of a rational function.
COROLLARY.
The periodic points of a rational
tributed according to the measure ~.
endomorphism
are asymptotically
Proposition.
a) The carrier of the measure ~ is the set F of irregular
F @ S 2, the measure ~ and the Lebesgue measure are mutually singular.
THEOREM 3.
a) The measure
c) The entropy h~(f) = log n.
~ is f-invariant,
b) The dynamical
system
points,
4.
Rational
endomorphisms
of the sphere
are asymptotically
Remark.
A rational endomorphism (and even its restriction
speaking, not h-expansive (for the definition, see [7]).
THEOREM
measure.
5.
A rational
The following
lemmas
endomorphism
of the Riemann
are used to prove
Theorem
b)
If
(f, y) is exact.
It was proved in [4] and [5] that the topologic entropy h(f) = in n.
Thus,
m a x i m u m - e n t r o p y measure of the endomorphism f. We remark that the mere existence
mum-entropy measure of a rational endomorphism can be deduced from fairly general
tions.
Namely, asymptotically h-expansive endomorphisms were defined in [6], and
the existence of a maximum-entropy measure was established.
THEOREM
equidis-
~ is the
of a maxiconsiderafor these
h-expansive.
to the set F) is, generally
sphere has a unique maximum-entropy
5:
LEM~iA 1 (see [8]).
Let f: X * X be a continuous endomorphism of a metric compact X, and
let ~ be an f-invariant ergodic probability measure.
Let Y C X and D(Y) > 0.
Then h~(f)
hf(Y), where hf(Y) is the topologic entropy of f with respect to Y, as defined in [7].
Let E = {i, ..., n}, 0 ~ < ~ i. We consider the set G m ( × ) C E m
m in which the element 1 appears at least ~m times.
LEMMA 2.
Let × >
Vn.
Then there exist K > 0 and 0 < ~ < n such thatlGm(×)t ~
We shall say that a system of sets distinguishes
some set of the system.
LEMMA 3.
rem i.
There
Let the measure
exists
- -
of all sequences
v be mutually
such a natural
the measures
KO m
.
~ and v if ~(Z) @ v(Z)
singular with the measure
r and such a covering
of length
D constructed
for
in Theo-
D = { D i } ~ ~ of the sphere by closed
O
sets that D + = D ~ ; 2)D~ N D ~ =
~ (~j); 3 ) ( ~ + w ) ( U 0 D 0 = 0: 4) the intersection
of D i and f-~z
is composed of not more than one point (i ~ i ~ n r, z ~ $2); 5) the covering D distinguishes
from v.
Added in Print.
The results in this note were presented at the Fifteenth Voronezh Mathematical Winter School (January, 1981); the theses are deposited with the All-Union Scientific
and Technical Information Institute (VINITI) (No. 5691, pp. 65-66).
310
LITERATURE
1.
2.
CITED
K. D e l e e u w and I. Glicksberg, " A p p l i c a t i o n s of a l m o s t - p e r i o d i c c o m p a c t i f i e a t i o n s , "
Acta
Math., 105, 63-97 (1961).
P. Montel, "Legons sur les Familles Normales de Fonctions A n a l y t i q u e s et leurs Applications (Lectures on Normal Families of A n a l y t i c Functions and their A p p l i c a t i o n s ) , "
Gauthier-Villars,
Paris (1927).
H. Brolin, Arkiv Math., 6, No. 2, 103-144 (1965).
M. Gromov, "Entropy of h o l o m o r p h i c m a p s , " Preprint, U n i v e r s i t y of C a l i f o r n i a (1980).
M. Yu. Lyubich, Funkts. Anal., 15, No. 4, 83-84 (1981).
M. M i s i u r e w i c z , Bull. Acad. Pol. Sci., 21, 903-910 (1973).
R. Bowen, Trans. Am. Math. Soc., 164, 323-331 (1972).
R. Bowen, Trans. Am. Math. Soc., 184, 125-136 (1973).
INVARIANT
SOLUTION
G.
ORDERINGS
IN SIMPLE
TO E. B. V I N B E R G ' S
LIE GROUPS.
THE
PROBLEM
I. O l ' s h a n s k i i
UDC 519.46
Let ~ be a real simple n o n c o m p a c t Lie
K simply c o n n e c t e d Lie groups c o r r e s p o n d i n g
is a b o u n d e d s y m m e t r i c domain.
It is known
convex G - i n v a r i a n t cones in g distinct from
a minimal cone, Cma x and Cmi n in Con; they
Con has been d e s c r i b e d in [2] and [3], [4]
algebra; g = ~: ~ the Cartan decomposition; G and
to the algebras
g and t. We assume thai D = G/K
[i] that in this case the set Con of all closed
{0} and g is not empty.
There is a maximal and
are unique up to a m u l t i p l i c a t i o n by --i. The set
(also see Sec. 2).
Let C ~ Con and let P = P(C) be the closed semigroup in G t o p o l o g i c a l l y g e n e r a t e d by the
set exp C.
It defines an invariant p a r t i a l o r d e r i n g in G for w h i c h {gLg ~ e} coincides w i t h
P.
The ordering is n o n t r i v i a l if P # G.
E. B. V i n b e r g [i] proved that P ¢ G and P ; p-1 =
{e} for C = Cmi n and raised the question w h e t h e r it was true for all C ~ Con.
S. Paneitz [3]
proved that P(C) # G for all C ~ Con if D is a classical domain of tubular type; the definition can be found in [5].
THEOREM.
(i) P ( 6 ' ) ~ G ~ C
~ 6"o (the cone Co ~ Con is defined in Sec. 3), w i t h Co = Cma x
for tubular D and Co # Cma x for n o n t u b u l a r D.
(ii) If P = P(C) # G, then P ~ p-l = {e} and
the "tangent cone" C(P) (cf. [i]) coincides w i t h C.
If g # sp(n,
variant orderings
R), su (2, i), EIII, then
in G follows from (ii).
Co # Cmi n and the existence
i.
Notation. ~ C ~
is the o n e - d i m e n s i o n a l center;
Weyl group for (~<,~c); A + c ~ R e , set of n o n c o m p a c t p o s i t i v e
p a i r w i s e o r t h o g o n a l roots, where r = rank D; E k and E_k,
roots i~k(k = i, ..., r) normed in such a way that H k =
i(E k -- E_k) ~ D ,
~k(Hk) = 2; Z, element of i~ defined by
Ho = Z -- HI -- ... -- Hr); and <,>, Killing form.
of a c o n t i n u u m
~ - f, Cartan subalgebra; ~Re = ~; W,
roots; { ~ ,
..., ~r } C A t family of
root vectors c o r r e s p o n d i n g to the
[Ek, E_k] ~ R e ;
Xk = Ek + E-k ~ P ,
the c o n d i t i o n that ~(Z) = 2 ~ ~ A ÷ ;
2.
Information from [i], [2].
Each C ~ Con lies in ~2~ and is u n i q u e l y
the cone c = iO~ ~Re (we w r i t e C +-+ c); here c contains either Z (then we write
--Z.
Let
I f C*={X ~ g ] - - , X
Cmin C bRe b e
the
If c --~l~{. is a closed
3.
Y:,~O V Y ~ C } , c * = { X ~ S R e I < X , Y , ~ O V Y ~ c } ,
cone
spanned
convex
The D e f i n i t i o n
by
cone,
of the
Cone
A+ a n d
then
Co.
Cma x = C ~ i n ;
of in-
then
then
C ++
Co +-+ Co, where
c implies
Cmi n +-+ Cmi n a n d
(c +-> C for some c ~ Con +) ~ ( W c =
d e t e r m i n e d by
C ~ Con + ) or
C, C m m ~ C ~ e , , ~ a
c~ is the cone
C* +-* c * .
Cma x +-+ Cma x .
spanned
x) .
by Cmi n and
14Ho.
If D is of tubular
type,
Ho # O, Ho ~ %mx, Ho ~ cmin ,
and
then Ho = 0 and Co = Cma x.
If D is not
of tubular
type,
then
Co # Cma x .
T r a n s l a t e d from F u n k t s l•o n a l i nyl- Analiz i Ego Prilozheniya, Vol.
October-December,
1982.
Original article submitted June 4, 1981.
0016-2663/82/1604-0311507.50
© 1983 P l e n u m
Publishing
16
No.
4, pp.
Corporation
80-81,
311