Iwanik, Anzelm

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Institute of Mathematics AS CR, v. v. i.

Acta Universitatis Carolinae. Mathematica et PhysicaAnzelm IwanikPeriod structure for pointwise periodic isometries of continuaActa Universitatis Carolinae. Mathematica et Physica, Vol. 29 (1988), No. 2, 19--21Persistent URL: http://dml.cz/dmlcz/701939Terms of use:© Univerzita Karlova v Praze, 1988Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.cz1988 ACTA UNIVERSITATIS CAROLINAE—-MATHEMAT1CA ET PHYSICA VOL. 29. NO . 2 Period Structure for Pointwise Periodic Isometries of Continua ANZELM IWANIK*) Wroclaw, Poland Received 1 March, 1988 Let (X, d) be a compact metric space. An isometry T of X is said to be pointwise periodic if for every xeX there exists n e N such that Tnx = x. We denote by P(x) the minimal (or essential) period of x. We let S(X, T) = {P(x):xeX} . As in [1], we say that a subset S of r\J is finitely based if it contains finitely many numbers nl9..., nk such that each element of S is a multiple of one of the n/s. This means S has finitely many minimal elements when partially ordered by the divisibility relation m | n. It was shown in [ l ] S(X, T) must be finitely based and, conversely, any finitely based subset S of N is equal to an S(X, T) for some pointwise periodic isometry T on a compact space X. (in fact the more general compact equicontinuous systems are dealt with in [ l ] . In the metrizable case equicontinuous system becomes isometric provided a suitable equivalent metric is chosen.) In the present note we consider pointwise periodic isometries of continua, i.e. connected compact metric spaces. If X is not only connected but also locally Euclidean then S(X, T) is known to be finite [2]. As we shall see, for pointwise periodic isometries of general continua the set of periods need not be finite. On the other hand, not all finitely based sets can occur. We say that a subset S of M is connected if for any m,ne S there exists a path n0,..., nk in 5 such that nQ = m, nk = n, and either n^^ \ nj or n} \ n^Y holds for each j = 1, . . . , k. This clearly means that S forms a connected subgraph of the (graph of) partially ordered set (N, |). Theorem. If 0 #= S c N is finitely based and connected then there exists a con-tinuum X and a pointwise periodic isometry T such that S = S(X, T). Conversely, if Tis a pointwise periodic isometry of a continuum X then S(X, T) is finitely based and connected. *) Institute of Mathematics, Technical University, 50-370 Wroclaw, Poland. 19 Proof. To prove the first part of the theorem we first observe that for any pair m, n of natural numbers such that m | n there exists a continuum X with a pointwise periodic isometry T such that S(X, T) = {m,n}. To this end consider a fc-fold covering h: Xx -> X2 of the unit circle by itself where fc = njm. Let Tl9 T2 be rational rotations of the circles with periods w, m, respectively and such that hTx = T2h. Now we can connect X1 with X2 in order to obtain a continuum X and a pointwise periodic isometry T extending Tt and T2. This can be done as follows. First define Y = Xt x [0,1) and extend Tx in a natural way by letting T^x, t) = (Txx, f). Now compactify Yby adjoining X2 = X2 x {1} so that lim (x, t) = h(x) as f -> 1. The system (K, T), where T = 2\ u T2, is clearly equicontinuous, hence isometric for an appropriate choice of the metric. Evidently S(X, T) = {m, n). The above construction easily extends to any finite chain n09...,nk such that nj-i I nJ 0 = !»•••> ^)- More generally, if 0 4= S0 c f\J is finite and connected then there exists a continuum X0 with a pointwise periodic isometry Tsuch that S(X0, T) = = S0. Finally, if S 4= 0 is finitely based and connected, then there exists S0 as above such that each element of S is a multiple of a number from S0. It now suffices to enlarge (K0, T) by adding new multiple cycles in the same way as it was done in the beginning of the proof. If the number of added cycles is infinite we take care of compactness by gradually decreasing to zero the distance between X0 and the newly added cycles (this can be accomplished in the Hilbert cube). The pointwise periodic equicontinuous system (X, T) obtained in this manner becomes a continuum with S(X, T) = S. Now we prove the second part of the theorem (note that this part is valid for a general non-metrizable case). For each xeX denote by <x> the set of all y e X such that the numbers P(x), P(y) can be joined by a path in the partially ordered set S(X, T). It is easy to see that the sets <x>, xeX form a partition of X. Besides, if y e <x> then [y] c <x> where [y] = {zeX:P(y)\P(z)} as in [ l ] . Since [y] is an open set ([1], Lemma 1), all the sets <x> are open. By connectedness, X = <x0> for a single x0, which implies that S(X, T) is connected in (N, I). Theorem 1 in [1] says that S(X, T) is finitely based, which concludes the proof of the theorem. We would like to remark that an intrinsic characterization of the sets of periods for arbitrary isometries of compact metric spaces (with the presence of aperiodic points) seems to remain an open problem. In fact the period structure for arbitrary isometry can be more complicated than incorrectly stated in Theorem 3 of [1]. To see this we fix an infinite set of odd primes and arrange it in two disjoint arrays (Pij), (qij). Now we construct a compact isometric system (K, T) with S(X, T) = = {2hpn ... Piflij'- i,j, ke N}. This can be done by modifying the construction in 20 [1], Proposition 2. For each I G N we obtain a system (Xh Tt) with S(Xh Tt) = = W&u>0ii0*2&j2>---> °°} w h e r e («ii» «i2» -••) = (2, . . . , 2 , p f l , p i 2 , . . . ) , the initial 2 being repeated i times, and b ^ = qtj. Now (X, T) is the disjoint union of the systems (Xi9 Tt) plus the compact group of dyadic integers with translation by its topological generator (1,0,0,. . .) . On the other hand, S(X,T)r\N cannot be represented as a finite union of a finitely based set and sets of type oo. References [1] ІWANIK A., JANOS L., KOWALSKI Z., Periods in equicontinuous topological dynamical systems, in: Nonlinear Analysis, World Scientific Publishers, Singapore, to appear. [2] MONTGOMERY D., Pointwise periodic homeomorphisms, Amer. J. Math. 59 (1937), 118—120. 21 

Period structure for pointwise periodic isometries of continua

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Period structure for pointwise periodic isometries of continua

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Institute of Mathematics AS CR, v. v. i.