Embedding of topological dynamical systems into symbolic dynamical systems: A necessary and sufficient condition

Vol. 57 (2006)

REPORTS ON MATHEMATICAL PHYSICS

No. 3

EMBEDDING OF TOPOLOGICAL DYNAMICAL SYSTEMS INTO SYMBOLIC DYNAMICAL SYSTEMS: A NECESSARY AND SUFFICIENT CONDITION YANGENG

WANG*

Department of Mathematics, Northwest University, Xian, Shaanxi, 710069, P. R. China (e-mail: ygwang62@ 163.com)

and GUO WEI Department of Mathematics and Computer Science, University of North Carolina at Pembroke, Pembroke, North Carolina, 28372, U.S.A. (e-mail: [email protected]) (Received February 26, 2006) Embedding of a topological dynamical system into another is a weaker condition than topological conjugacy between the two topological dynamical systems. In 2000, under the assumption of E-expansive, compact and totally disconnected systems, Shigeo Akashi found a sufficient condition for embedding a topological dynamical system (X, d, f ) into a symbolic dynamical system. The purpose of this paper is to present and prove a necessary and sufficient condition that determines exactly which of the topological dynamical systems can be embedded into symbolic dynamical systems and which ones cannot be. Particularly, Shigeo Akashi's result becomes a special case of this new sufficient condition.

Keywords: embedding of topological dynamical systems, E-expansive mapping, symbolic dynamical system.

1.

Basic concepts and definitions

Symbolic dynamical systems have various representative and complicated dynamical properties and characteristics. When determining whether or not a given dynamical system has certain dynamical complexity, it is often compared with a symbolic dynamical system, i.e. whether or not it is topologically conjugate with a symbolic dynamical system [1, 2]. Relations that are weaker than topological conjugacy include semi-topological conjugacy, transitive invariant set [3], and topological embedding. In this paper, the focus is to study necessary and sufficient conditions for embedding topological dynamical systems into symbolic dynamical systems. *Supported in Part by NSF of Shaanxi (98SL06), R R. China. [457]

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Y. WANG and G. WEI

Let N denote the set of all nonnegative integers and let ( E ( p ) , a ) be tile symbolic dynamic system where Z ( p ) = {1, 2 . . . . . p}~V and cr is the shift mapping, i,e. ~r(io, il . . . . . in . . . . ) = ( i l . . . . . in . . . . ) for (i0, il . . . . . in . . . . ) ~ Z(p), see [4, 5]. DEFINITION 1. Let (X, Tx) and (Y, Tr) be two topological dynamical systems. if there exists a subspace Z of X and a homeomorphic mapping h from Y onto Z such that Tx o h = h o Tr, then (Y, Tr) is said to be embeddable into (X, Tx). DEFINITION 2. Let (X, d) be a metric space, f : X ~ X be a continuous mapping and E be a positive number. If for all x , y ~ X , x #: y, there exists an n ~ N such that d(fn(x), fn(y)) > ~, then f is said to be E-expansive, and (X, d, f ) is also said to be E-expansive [6]. Under the assumption of E-expansive, compact and totally disconnected systems, Shigeo Akashi obtained a sufficient condition for topological dynamical systems embeddable into symbolic dynamical systems [6]. Shigeo Akashi's result is stated in Theorem 1 below. THEOREM 1. Let ~ be a positive number, ( X, d) a compact and totally disconnected metric space and T an E-expansive continuous mapping on X with values in X. Then (X, d, T) can be embedded into a certain symbolic dynamical system.

In Section 2 below, the authors will present a condition (Theorem 2) that determines exactly which of the topological dynamical systems can be embedded into symbolic dynamical systems and which of the topological dynamical systems cannot be embedded into symbolic dynamical systems. In other words, this is a necessary and sufficient condition.

2.

Main theorem and its proof

THEOREM 2. Let (X, f ) be a topological dynamical system. Then (X, f ) can Oe embedded into a symbolic dynamical system (E(p), or) /f and only if there exists a family of disjoint clopen subsets of X, {Fi}i=v t where l <_ p, such that the following conditions are satisfied." l

U Fi = X;

(1)

i=l

for all {i~}~ o ~ {1, 2 . . . . . l} N, card{["] f-~(Ffs) } _< 1;

(2)

s=0

k

if r ] f-'(Fis) = {x}, then {~'~ f-~(Fis)}k~o forms a topological base at x. (3) s=O

s=O

Proof Necessity. Let (X, f ) be a topological dynamical system that can be embedded into some symbolic dynamical system (E(p),cr). Then there exists a

EMBEDDING OF T O P O L O G I C A L DYNAMICAL SYSTEMS

subspace Z of E ( p ) and a homeomorphic mapping h : X ~ Define

Bi = {{tS}s=O "~ ~ E(p) I io = i},

459

Z satisfying troh = h o f .

Fi - - h -1 (Bi f~ Z), i = 1, 2 . . . . . p.

(4)

These Bi's in (4) are disjoint clopen (i.e. close and open) subsets of N ( p ) with t.JP=lBi = E ( p ) , and thus Fi's are disjoint clopen subsets of X with UP=l/r/ = X. Some of these Fi's may be empty (but not all of them). Without loss of generality, let Fi ¢ : 0 for 1 < i < l and Fi = ~ for l < i < p. Therefore, the condition (1) holds. To examine the condition (2), from cr o h = h o f , it follows that a n o h = h o fn ~ e {1, 2 . . . . . l} N, and h -1 o or-" = f - " o h -1. Then for all {i ,},=0 OO

Oo

N :-'(r,,)= N :-'° h-l(Bi, s=0

N Z)

s=0 O~

-~. [ ~ h -1 o tr-S ( Bi, ~ Z) s----0

= h-l([")tr-S(Bi, N Z)) s----0

= h - l ( i o , il

. . . . .

in . . . . ).

Hence, card{C3~of-S(Fi,)} < 1 and the equal sign holds if and only if (i0, il . . . . . in . . . . ) ~ Z. The condition (2) thus holds. To examine the condition (3), let x0 ~ X with h(xo) = (io, il . . . . . in . . . . ) ~ Z and let U be an arbitrary neighborhood of x0 in X. Since h is a homemorphic mapping, h(U) is a neighborhood of h(xo) in Z, and in turn there exists a basic open subset ' ~ s E ( p ) : j s = i~ f o r O < s < n } , of E ( p ) in the form of oU[i0, il . . . . . ik] = {{J,}s=0 such that oU[i0, il . . . . . ik]Z c_ h(U). Hence, U = h ~1 o h(U) D_ h-l(ou[10, il . . . . . ik] f~ Z) k

= h-l((A

tr-s (Bi,)) ~ Z)

s-----0 k

= N h-l (tr-" ( Bi,) N Z) s=0 k

--A f-S(h-l(Bis S----0 k s~O

Therefore, the condition (3) holds.

f"l Z))

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Y. WANG and G. WEI

Sufficiency. L e t {F/}li=l be disjoint and nonempty clopen subsets of X satisfying all the three conditions (1), (2) and (3). As (E(l),cr) can be embedded into ( E ( p ) , o-), it is sufficient to prove that (X, f ) can be embedded into (E(I), cr). First, we construct a mapping h : X ~ E(I). From the condition (1), Vx e X, Vs E N, there exists a unique is E {1,2 . . . . . 1} satisfying fS(x) E Fis. So x f -1 (Fis) and thus x E n s=0 ~ f - s (Fi~). Then from the condition (2), it follows that {x} = ns~=of-S(Fis). Define h(x) = {S}s=0i ~ E E(I). Again by the condition (2), h is a one-to-one mapping. Let Z = h(X). To prove h : X --~ Z is a homemorphism, the remaining proof that needs to be done is the continuity of h and h -1. i Vx0 6 X, let h(xo) = {s}s=0 E Z and let oU[i0, il . . . . . in] be a basic open neighborhood of h(xo). Then fS(xo) E Fi,, 0 < s < n. As F/s, 0 < s < n, are clopen subsets, f s , 0 < s < n, are continuous and thus there exists a neighborhood U of x0 such that f s ( u ) c_ Fis, 0 < s < n. Hence, h(U) c_ oU[i0, il . . . . . in] and therefore h is continuous at x0. Because x0 is arbitrary, h is continuous in X. V{is}~0 E Z, let h-l(io, il . . . . . is . . . . ) = xo E X and U be a neighborhood of x0 in ~ r-S'~F X. As {x0} = N s=0J '. is ), there exists a k E N such that x0 ~ N~s=0f-S(Fi, ) C_ U by the condition (3). Hence, there exists a basic neighborhood oU[i0, il . . . . . ik] satisfying h-l(oU[i0, il . . . . . i~] n Z)___ U. Therefore, h 1 is continuous in Z. To prove o - o h : h o f , Yx E X, let {x} = n ~ o f - S ( F i , ) . Then f ( x ) n s=0 ~ f - s (F%1) and h o f ( x ) = (il, i2 . . . . . . . . ). On the other hand, h ( x ) = (i0, il, i2 .... ) and o" o h(x) = (il, i2 . . . . ). So cr o h = h o f . []

3.

Corollaries of the main theorem

In this section, two immediate corollaries of the main theorem (Theorem 2) will be presented. The first corollary is about compact topological dynamical systems and the second corollary is a sufficient condition for expansive topological dynamical systems. COROLLARY 1. Let (X, d, f ) be a compact metric topological dynamical system. If (X, f ) can be embedded into some symbolic dynamical system ( E ( p ) , t7), then (X, d, f ) is an expansive dynamical system.

Proof'. Since (X, f ) can be embedded into a symbolic dynamical system (E(p),o-), there exists a family of disjoint clopen subsets {Fi}l=l,l <_ p in X satisfying the three conditions of the main theorem. As X is compact, each Fi is compact and thus d(Fi, F j ) > O for i ~ : j . Let e = m i n { d ( F i , Fj) : i ~ : j , l <_i,j 0. Yx, y E X , x ~ y, by the condition (1) of Theorem 2, Ys E N, there exists a unique is E {1,2 . . . . . l} such that fS(x) E F/s and thus x ~ ns%of-S(Fis). By the condition (2) of Theorem 2, there exists an n E N satisfying y ~ f-n(Fin ), i.e. f " ( y ) q~ Fin. Hence, d ( f n ( x ) , f n ( y ) ) > e. [] Finally, let us prove that Shigeo Akashi's result (Theorem 1) is a special case of the sufficient condition of the main theorem (Theorem 2).

EMBEDDING OF TOPOLOGICAL DYNAMICAL SYSTEMS

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COROLLARY 2. Let (X, d, f ) be an E-expansive topological dynamical system, where X is compact and totally disconnected. Then (X, f ) satisfies the three conditions of Theorem 2. Hence, (X, f ) can be embedded into a certain symbolic dynamical system.

Proof'. Since (X, d) is compact and totally disconnected, there exists a family {Fi}l=l of nonempty clopen subsets of X satisfying the condition (1) of Theorem 2 for each 1 < i < l . and diam(Fi) < ¥{is}s~_0 6 {1, 2 . . . . . l} N, assume that x, y 6 (~s~=0f-S(F/s). As f is e-expansive, there exists an n 6 N such that d(fn(x), fn(y)) > e. As x , y ~ f-n(Fi,), f n ( x ) , fn(y) E Fin and thus d(fn(x), f"(y)) < diam(Fi,) < 2, which contradicts d ( f n ( x ) , fn(y)) >_ e. Hence, {Fi}I=I satisfies the condition (2) of Theorem 2, i.e. card{fq~0f-S(F/s)} < 1. To verify the condition (3) of Theorem 2, let {x0} = Ns~of-S(Fis) and let U be an open neighborhood of x0 in X. If Vk E N, fqk ~=of - s (Fis) \ U # 0, choose k - s (Fis) \ U for k E N. Since X is a compact metric space, {Xk}k=O ~x~ has Xk ~ Ns=of a convergent subsequence {xkt}~=o and let z = limt--,~ xk,. As X \ U is closed in X, z ~ U and thus z # x . On the other hand, Since { Nk= o f -s (Fis)}k=O is a monotone sequence of closed subsets, Inkt r-s(E. t, 's=Oa ~ ,s ) }k=O is its subsequence, and o~ k -s oo -s z 6 ' 0oo 't=0v(t~kt 's=0Jrr-s (Fis)), so z ~ Nt=0(n~=0f (F/s)) -----As=of (Fis). This contradicts z # x. Therefore, {F/}l= 1 satisfies the condition (3) of Theorem 2. [] REFERENCES [1] S. Smale: Diffeomorphisms with Many Periodic Points, in Differential and Combinatorial Topology, S. Cairnes (ed.), 1st ed., Princeton University Press, New Jersey 1965. [2] S. Smale: Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. [3] Z. S. Zhang: On the shift invariant sets of self-maps, ACTA Mathematica Sinica 27 (1984), 564-576. [4] C. Robinson: Dynamical System: Stabili O, Symbolic Dynamics, and Chaos, 2nd ed., CRC Press, Florada 1989. [5] Z. L. Zhou: Symbolic Dynamical Systems, 1st ed., Shanghai Science and Technology Education Press, Shanghai, 1997. [6] S. Akashi: Embedding of expansive dynamical systems into symbolic dynamical systems, Rep. Math. Phys. 46 (2000), 11-14.