Remark' on the supremum theorem for the sequence entropyLet (

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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIII (1983) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXIII (1983)

Wojciech Hyb (Warszawa)

Remark' on the supremum theorem for the sequence entropy

Let (X , T) be a dynamical system (X — a compact metric space, T : X -+ X — a continuous transformation). Let A = be a non-decreasing sequence of non-negative integers. We denote by M(X, T) the set of all Borel regular normed T-invariant measures. Let hA(T) denote the topological sequence entropy of T with respect to the sequence A and for p eM (X , T) let hAfl(T) denote the sequence entropy of the system (X, ц, T) with respect to the sequence A (for definitions see [2]).

T. N. T. Goodman and E. Eberlein have proved that if the sequence A satisfies certain condition, then for every dynamical system {X, T) the supremum theorem

(1) hA(T )= sup hAtft{T)

цеМ(Х,Т) is valid (see [

1

], [

2

]).

T. N. T. Goodman has constructed a sequence A and a dynamical system (X, T) for which (1) is not valid ([2]). Another example (different from Goodman’s example) has been constructed by W. Szlenk (see [3]).

In the present paper we shall define a class К of sequences such that for AgK there exists a dynamical system (X , T) for which hA(T)

> sup hA fl(T). The idea of the construction of such dynamical system цеМ(Х,Т)

is analogous to the construction in [3].

Let A — { t „ } * b e a non-decreasing sequence of non-negative integers.

We define the sequence {pn}^=°i as follows: Pi = f i, p„+i = tn+l — tn. Let i nk}k=i be a sequence of positive integers such that lim p„ — + oo (in the other case formula (1) is valid, see [2]). Denote rn(A) — Card {k: nk ^ n} and a {A) = lim sup - r„(A). We define К = {A: a (A) > 0}.

n П

Now we prove

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216 W. Ну b

Theorem. I f Л е К , then there exists a dynamical system (X , T) for which the supremum theorem is not valid.

Proof. We denote Sk = tnk, qi — St, qk + i = Sk + l — Sk> к = 1 , 2 , . . . Hence qk ^ p„k and lim qk = + oo. The set Y = f ] Z,, Z, = (0, 1} with the

* f=о

metric g,

q((x )ftÔ, (У,

-Л

о) = (1 + max {к: xt = у,- for 7 < к})

1

is a compact space. The shift Q : Y -+ Y given by 0 ( x;);+=o = (,У.)|+=о, у, = xi + 1, is continuous surjection. We shall construct I as a closed subset of Y such that QX c X. Denote by N the set of all positive integers.

1° For n e N we define vn = (X0, X l t ..., XSl„) as follows:

f

0

for j Ф Sm, 2”

1

< m < 2", [ arbitrary for j = Sm, 2"

1

< m < 2".

There are 22"

1

different sequences v„ and we denote them by vj,, ...,v knn (kn = 22" ). Let V„ be the following sequence:

(k„ = 22" 1).

Now we set x = Vt V2 V3, ... We see that Qmi(x) ^ФQm2{x) for ^Фm2 and we define

X = {Qn(x): n = 0, 1 , 2 , . . . } .

It is easy to see that I is a closed subset of Y and QX с X.

We denote T = Q\x . 2° We define successively

$ = ■

1

° ’ e*=(Si),+.? e y , a'= (

0

,

0

,

0

,...), (

1

, I = k,

Bt = {T *x : k =

0

,

1

,

2

, . . . } , S

2

= {ek: * = 0 , 1 , 2 , . . . } .

We prove that X = B x u B 2 u {a}. Let у = (y,)f=o be a cluster point of the set B l с У. Then there exists a sequence {/к}к+=® such that

g(T lkx, y) — ► 0 and lk+1 > lk. Assume that, for an index m, ym = 1 and y,

к -+ + at

= 0 for l < m. Let j > m be fixed. Thus for /с large enough the m-th and j -th coordinates of T*kx have to be 1 and yj respectively. Let T kx = (Z k) f J o- We define dk = inf {/: for some p ^ 0, Zk = Z*+/ = 1}. It is easy to see that dk+1

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Supremum theorem fo r the sequence entropy 217

^ dk and lim dk = + oo. Hence = 0 and we conclude that y = em for some

к

m ^ 0 or у = a. It is easy to check that eke X , к = 0, 1, . , and a e l . 3° If к, l are non-negative integers, then: T lek = e k + h TkX

+

00

c T

~1

(Tk + 1X) and TkB2 => B2. The set Z = f| ^ is closed and Zr\Bx

, k = 0

= 0 . Then we have Z = B

2

u l a}- If T), then ц(Тк + 1Х)

= ц(Т~

1

(7* +

1

X)) ^ ц {Т кХ) for к ^ 0 and we conclude that ц{ТкХ) = 1 for к ^ 0. Hence /x(Z) = 1. It is easy to see that fi({ek}) = ц({Т~ке0}) — ц({е0}).

This shows that ц(В2) = 0. Hence ju({a}) = 1 and hA fl(T) = 0. We conclude that sup hAfl(T) = 0.

tieM (X .T)

4° Let С, = {(у;)/=°о e X : y

0

= *}» i = 0, 1 ; then C = (C0, is an open

2

" _s

cover of X. Then open cover Cn = \J T mC contains only the sets m=

1

Cei...v = {(yj)j=oe X : ys, i = U •••,

2

"}, where £, е{

0

,

1

}, i =

1

,

2

", are fixed. The sequences v>n, j = 1, kn, are different, therefore H(C„)

^ log kn =

2

"_

1

log

2

. ■

5° If a(T) > 0, then there exists a sequence of positive integers

such that lim — rm {A) = v{A). Let be a sequence of non-negative n m„ "

integers such that 2°n ^ rmn(A) < 2a" for n large enough. We have that

2

"

Я(\/ T~S‘C )^

2 "-1

log

2 1

=

1

(see 4°) and now we get

1

m„

M Л ^ hA(T, C) ^ lim sup — H( V T ^ C )

" j = 1

^ lim supг«(Л)

1

m„

^ lim sup T SjC)

n m„

2

"

2=1

r Ml

2

я"

^ lim sup — ---- • a

+1

log

2

= £ a (A) • log

2

>

0

. m„ 2 "

Thus the theorem is proved.

The author is indebted to Doc. dr hab. K. Krzyzewski for suggesting the problem.

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218 W. Hyb References

[1 ] E. E b e r le in , On topological entropy o f semigroups o f commuting transformations, International Conference on Dynamical Systems in Mathematical Physics, Université de

Rennes, 1975. ^

[2 ] T. N. T. G o o d m a n , T opological sequence entropy, Proc. London Math. Soc. 29 (1974), p. 331-350:

[3 ] W. S zlen k , On w eakly* conditionally com pact dynam ical systems, Studia Math. 66 (1979), p. 25-32.