Symbolic Representations of Iterated Maps November 4, 2000 Xin-Chu Fu, Weiping Lu 2, Peter Ashwin and Jinqiao Duan 3. School of Mathematical Sciences,

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2 Symbolic Representations of Iterated Maps November 4, 2000 Xin-Chu Fu, Weiping Lu 2, Peter Ashwin and Jinqiao Duan 3. School of Mathematical Sciences, Laver Building University of Exeter, Exeter EX4 4QJ, UK 2. Department of Physics, Heriot-Watt University Riccarton, Edinburgh EH4 4AS, UK 3. Department of Applied Mathematics Illinois Institute of Technology, Chicago, IL 6066, USA Abstract This paper presents a general and systematic discussion of various symbolic representations of iterated maps through subshifts. We give a unied model for all continuous maps on a metric space, by representing a map through a general subshift over usually an uncountable alphabet. It is shown that at most the second order representation is enough for a continuous map. In particular, it is shown that the dynamics of one-dimensional continuous maps to a great extent can be transformed to the study of subshift structure of a general symbolic dynamics system. By introducing distillations, partial representations of some general continuous maps are obtained. Finally, partitions and representations of a class of discontinuous maps, piecewise continuous maps are discussed, and as examples, a representation of the Gauss map via a full shift over a countable alphabet and representations of interval exchange transformations as subshifts of innite type are given. Key words: Continuous Map, Discontinuous Map, Representation, Subshift, Symbolic Dynamics Introduction It has long been known that interesting behaviour can occur when iterating continuous maps. Such maps dene discrete dynamical systems, which have been used as simplied prototypical models for some engineering and biological processes; see, e.g. [20, 5, ]. Through the Poincare section maps, discrete dynamical systems have also been used to study continuous dynamical systems, e.g. [4, 23]. Considerable progress has been made in the last two decades in the understanding of dynamical behaviour of nonlinear continuous maps. These systems, while mostly studied individually because of their distinct nature of nonlinear interactions, show 2

3 similar dynamical features, especially when the parameters of the systems are close to some critical values where abrupt change in behavior takes place. The universality of such behavior has been a key subject in the study of nonlinear dynamics. A mathematical framework has, however, yet to be established under which a class of dynamical systems, such as one-dimensional continuous maps, can be described by a unied model. Such a framework will improve our understanding of general properties of dynamical systems and may be useful in our eort to classify dynamical systems. Shifts and subshifts dened on a space of abstract symbols are special discrete dynamical systems which are called symbolic dynamics systems. Symbolic dynamics is a powerful tool to study more general discrete dynamical systems, because the latter often contain invariant subsets on which the dynamics is similar or even equivalent to a shift or subshift. Moreover, there are a number of denitions of chaos, namely () the Li-Yorke denition; (2) Devaney's denition; (3) topological mixing; (4) Smale's horseshoe; (5) transversal homoclinic points; and (6) symbolic dynamics. The symbolic dynamics denition is especially important of these denitions, as it unies aspects of many of the denitions. More precisely, it implies the rst three denitions, is topologically conjugate to the fourth one and occurs as a subsystem of the fth. Furthermore (see for example Ford [4]) symbolic dynamics is very important for analysis of applications of nonlinear dynamics in physical sciences. For a dynamical system, we can study it either directly or via other systems which are better understood. Symbolic representations are methods to study dynamical systems through shifts and subshifts. In the study of hyperbolic dynamics of homeomorphisms, symbolic dynamics is one of the most fundamental models. Since the discovery of the Markov partitions of the two dimensional torus by Berg [8] and the related work by Adler and Weiss [3], symbolic representations of hyperbolic systems through the Markov partitions have been studied extensively (e.g., see [2]). The equivalence between Smale's horseshoe and the symbolic dynamical system ((2); ) (see [22]) implies that the former has a symbolic representation through the full shift ((2); ). A similar symbolic representation has also been revealed by Wiggins (see [23]) on higher dimensional versions of the Smale's horseshoe. Earlier works in [24] and [9] established that, under certain conditions, the restrictions of a general continuous self-map f : X! X to some horseshoe-like invariant subsets are topologically conjugate to j (N ) or j (Z + ) (see [9]), where ((Z + ); ) is the symbolic dynamics system with a countable alphabet. These results actually demonstrated the partial symbolic representation of a class of maps as a full shift over a countable alphabet. Motivated by the above work, we study in this paper symbolic representations of continuous self-maps by using a more general and systematic approach. We present a unied model for all continuous maps on a metric space, by representing a map as a general subshift ((X); ). We show that the subshifts of ((X); ) may be used as such a unied model for all continuous self-maps on a metric space X. And it is shown that at most the second order representation is enough for a continuous map. In particular, when X is a closed interval, we show that the dynamics of one-dimensional continuous maps to a great extent can be transformed to the study 3

4 of subshifts of a symbolic dynamical system. We also discuss quasi-representations, and by introducing distillations, partial representations of some general continuous maps are obtained. Finally, partitions and representations of a class of discontinuous maps, piecewise continuous maps, are discussed, and as examples, a regular representation of the Gauss map via a full shift over a countable alphabet and regular representations of interval exchange transformations as subshifts of innite type are given. 2 Some Denitions, Notation and Lemmas Before turning to the next section to discuss representation theorems, we recall some denitions, introduce some notation, and provide some lemmas. Let (X; d) be a metric space and denote by (X) the space X Z + which consists of functions from the nonnegative integers Z + to the metric space X. x 2 (X) may thus be denoted by x = (x 0 ; x ; ; x i ; ), x i 2 X; i 0. Further, let (X) be endowed with the product topology, so (X) is metrizable. The metric on (X) can be chosen to be (x; y) = +X i=0 2 i d(x i ; y i ) + d(x i ; y i ) ; x = (x 0; x ; ); y = (y 0 ; y ; ) 2 (X): The shift map : (X)! (X) is dened by ((x)) i = x i+ ; i = 0; ; : Since ((x); (y)) 2(x; y); is continuous. ((X); ) is a general symbolic dynamics system (see [23, 0, ]). We call X the symbol space or alphabet, and (X) the symbol sequence space. When X is chosen as f0; ; ; N? g and the metric d on f0; ; ; N? g is the discrete metric: 0; m = n d(m; n) = ; m 6= n ; then ((X); ) becomes the usual symbolic dynamics system ((N); ) as in [4]. Let (X) be closed, and invariant for, i.e., (), then (; ) forms a subsystem of ((X); ). We call (; ) a subshift of the full shift ((X); ), denoted by (; ) ((X); ). We denote by C(X) the set of all continuous self-maps on X and M(X) the set of all self-maps on X. We also call the iteration system of an f 2 M(X) a dynamical system, denoted by (X; f). For two dynamical systems f : X! X and f 2 : X 2! X 2, where X i are metric spaces and f i 2 M(X i ); i = ; 2; if there exists a homeomorphism h : X! X 2 such that hf = f 2 h, then we call f is topologically conjugate to f 2, denoted by f f 2. If h is continuous and surjective, but not necessarily invertible, and if hf = f 2 h, then we call f is topologically semi-conjugate to f 2, and also call f 2 a factor of f (and f an extension of f 2 ). We call f and f 2 are weakly conjugate if each is a factor of the other. We give the following denitions for various symbolic representations: Definition 2. If for an f 2 M(X), there exists a subshift (; ) of certain a symbolic dynamical system and a surjective map h :! X, and a countable subset D, such that h is one-to-one and continuous on n D, nite-to-one on D, and 4

5 f h = h, then we call the subshift (; ) a regular representation of the dynamical system (X; f). If D =?, we call the subshift (; ) a faithful representation of the dynamical system (X; f). If h is a topological conjugacy, we call (; ) a conjugate representation of (X; f). If j and f are weakly conjugate, we call (; ) a weakly conjugate representation of (X; f). If h is a topological semi-conjugacy, i.e., (X; f) is a factor of (; ), we call (; ) a semi-conjugate representation of (X; f). If (; ) is a factor of (X; f), we call (; ) a quasi-representation of (X; f). In all the above representations, we call a correspondence between symbol sequences in and points in X (either from to X or from X to ) a coding. From the denition above, among the six representations, conjugate representation is the strongest one, since it implies all the others. A weakly conjugate representation is always a semi-conjugate representation and a quasi-representation. A faithful representation is also a semi-conjugate representation. Quasi-representation is the weakest one, but it still tells us some information. For example, if (; ) is a quasi-representation of (X; f), then h top (f) h top (j ), while the latter is usually easier to calculate (h top () denotes the topological entropy). In the denition of a regular representation, we allow h to be possibly discountinuous on D so that we can apply this denition to symbolic representations of some discontinuous maps. Note that weak conjugacy is strictly weaker than conjugacy ([9]). For example, the Fibonacci subshift (consists of all sequences of 0's and 's with 's separated by 0's) and the even subshift (consists of all sequences of 0's and 's with 's separated by an even number of 0's) are weakly conjugate, but they are not conjugate since one is a subshift of nite type and the other is a subshift of innite type. Throughout this paper, for a 2 I = [0; ], we denote by r m (a) the m-adic fraction r m (a) = +X i=0 a i m i+ ; and when a 6= 0, we require that there are innite number of non-zero a i 's, i 0, and denote a i by r i m(a). Unless indicated otherwise, we will always make this choice when we write an m-adic fraction expansion of a real number. For an integer sequence x 2 (N); x = (x 0 ; x ; ; x i ; ), we also denote in form by r m (x) the m-adic fraction For a; b 2 I; let r m (x) = +X i=0 d 0 (a; b) = ja? bj; d (a; b) = x i ; where m N: i+ m +X i=0 2 i jri 2(a)? r i 2(b)j; and denote by I k the metric space (I; d k ); k = 0; : 5

6 Dene the metric k on (I k ) as k (x; y) = +X i=0 d k (x i ; y i ) ; k = 0; : 2 i + d k (x i ; y i ) The lemmas below are necessary for the proof of the theorems in the following sections. Lemma 2. Suppose x = (x 0 ; x ; ; x i ; ); y = (y 0 ; y ; ; y i ; ) 2 (X), then (x; y) < 2 2N + =) d(x i ; y i ) < 2 N ; 8 i N; For > 0; d(x i ; y i ) < ; 8 i N =) (x; y) < N : Proof. Otherwise, if there exists i 0 N; such that d(x i0 ; y i0 ) =2 N ; then (x; y) 2 N 2 i N + 2 N 2 2 N This is a contradiction. So we have the rst inequalities. We can check directly the second inequality. 2N + : Lemma 2.2 If A I is compact in I, then A is also compact in I 0 ; and if (I) is compact in (I ), then is also compact in (I 0 ). Proof. 8a; b 2 I, d 0 (a; b) = ja? bj = j +X i=0 2?(i+) (r i 2(a)? r i 2(b))j 2 d (a; b): So any limit points of A in the metric d are also limit points of A in the metric d 0. Hence compactness of A in I implies compactness of A in I 0. Similarly, we can check the compactness of in (I 0 ). We say a subshift (; ) satises the condition (2.) if 8 x = (x 0 ; x ; ); y = (y 0 ; y ; ) 2 ; x = y if and only if x 0 = y 0 : (2.) Lemma 2.3 For any (; ) ((I ); ), there exists ( ; ) ((I ); ) such that j j, and ( ; ) satises the condition (2.). Proof. 8 2 ; = ( 0 ; ; ; i ; ): Rewrite i by Let i = r 2 ( i ) = +X j=0 ij ; i = 0; ; : j+ 2 r() = 0: k0 k?; ;k? 0k (binary fraction), R() = (r(); r(()); ; r( k ()); ): 6

7 From R() = R() we have r() = r(), thus ij = ij ; i; j = 0; ; : As a result, =, that is, R is one-to-one. Whenever ; 2 with (; ) < =2 2N +, from Lemma 2. we have d ( i ; i ) < =2 N ; i N: So ij = ij ; i; j N. Denote we have r() = r() = +X k=0 +X k=0 2?(k+) (r()) k ; 2?(k+) (r()) k ; (r()) k = (r()) k ; k (N + )(N + 2); 2 (r(())) k = (r(())) k ; k N(N + ); 2 (r( l ())) k = (r( l ())) k ; l N; k (N +? l)(n + 2? l); 2 Take N = 2M: Denote N l = (N +? l)(n + 2? l). Then for l M, we have 2 therefore d (r( l ()); r( l ())) = (R(); R()) +X +X k=0 k=n l + 2 k j(r(l ())) k? (r( l ())) k j MX l=0 2 = k 2 N l 2? 2 (M +)(M +2) < 2 M ; 2 l 2?M + 2?M + 2 M < 3 2 M ; which implies that R :! R() is continuous. Take = R(). Let ~ ; ~ 2, then 9 ; 2 ; such that Let N = (2k + )(2k + 2), when 2 from Lemma 2. we have ~ = (r(); r(()); ; r( k ()); ); ~ = (r(); r(()); ; r( k ()); ) ( ; ~ ~) < ; 22N + d (r( i ()); r( i ())) < 2 N ; i N: We get ij = ij ; i k; j 2k, and therefore d ( i ; i ) 2 2k ; i k. Again from Lemma 2. we have (R? ( ~ ); R? (~)) < 2 2k? + 2 k : 7

8 Therefore R? :! is also continuous, hence R is a homeomorphism. And 8 2 ; R() = (r(()); r( 2 ()); ; r( k ()); ) = R(); so j j, while the subshift ( ; ) satises the condition (2.). Lemma 2.4 The full shift ((I ); ) is a faithful representation of ((I 0 ); ). And 8 (; ) ((I ); ) with compact, there exists a subshift ( 0 ; ) ((I 0 ); ), such that j j 0. Proof. 8a; b 2 [0; ], d 0 (a; b) 2 d (a; b): So 8; 2 (I ), 0 (; ) (; ); hence the identity mapping i : (I )! (I 0 ) is continuous. Moreover, the following diagram commutes: (I )?! (I ) i # # i (I 0 )?! (I 0 ) So ((I ); ) is a faithful representation of ((I 0 ); ). 8 (; ) ((I ); ) with compact, i :! 0 = i() = (I 0 ) is a homeomorphism, therefore (; ) ((I 0 ); ) and j j 0. The result in Lemma 2.4 can be generalized to: Proposition 2.5 If (; ) is a compact subshift of a faithful representation of (X; f) under coding h, then (; ) is a conjugate representation of (h(); f), a compact subsystem of (X; f). Remark 2. 8f 2 M(I); fj I0 is a factor of fj I. For the identity mapping i : I! I 0, from the proof of Lemma 2.4, we have d 0 (i(x); i(y)) = d 0 (x; y) 2 d (x; y); 8x; y 2 I : So i : I! I 0 is continuous. 8f 2 M(I), it is obvious that the following diagram commutes: I i # I 0 f?! I # i f?! I 0 So fj I0 is a factor of fj I. 8

9 3 Conjugate Representations As discussed in [2], representing a general dynamical system by a symbolic one involves a fundamental complication: it is dicult and sometimes impossible to nd a coding of a continuous one-to-one correspondence between orbits and symbolic sequences, and especially when we desire the shift system to be one of nite type and with a nite (or at most countable [0, 9, 8]) alphabet. In this section, we will discuss conjugate representations of general continuous maps through general subshifts, which usually involve an uncountable alphabet. For a metric space (X; d), there exists a huge number of continuous self-maps on X. The dynamics on X is therefore diverse. The following theorem shows that the general subshifts of ((X); ) can be used as a unied model for all continuous self-maps on the space X. Theorem 3. Let (X; d) be a metric space, 8f 2 C(X); 9(; ) ((X); ), such that (; ) is a conjugate representation of (X; f). Proof. Dene a mapping h : X! (X) as h(x) = (x; f(x); f 2 (x); ; f n (x); ); then h is a one-to-one mapping. Since f is continuous, for any positive integer N and any sequence fx n g X with lim n!+ x n = x 0 2 X, we have lim d(f k (x n ); f k (x 0 )) = 0; k = 0; ; ; N: n!+ From 0 (h(x n ); h(x 0 )) = NX i=0 +X i=0 2 i d(f i (x n ); f i (x 0 )) + d(f i (x n ); f i (x 0 )) 2 i d(f i (x n ); f i (x 0 )) + d(f i (x n ); f i (x 0 )) + + X i=n + 2 i ; we have 0 lim (h(x n); h(x 0 )) ; 8N ; n!+ 2N that is, h : X! (X) is continuous. Suppose we have a sequence f (n) = ( (n) 0 ; (n) ; ; (n) ); g h(x) with lim n!+ (n) = = ( 0 ; ; ; i ; ) 2 h(x); i then From lim n!+ ((n) ; ) = 0: 0 d((n) 0 ; 0 ) + d( (n) 0 ; 0 ) ((n) ; )! 0 as n! +; 9

10 we obtain lim n!+ d((n) 0 ; 0 ) = 0: So h? : h(x)! X is also continuous, and therefore h is a homeomorphism from X to h(x). Moreover, hf(x) = (f(x); f 2 (x); ) = h(x); 8x 2 X; i.e., hf = h: Take = h(x), then is invariant for the shift map, and is a closed subset of (X). So (; ) ((X); ), and f is topologically conjugate to j. Remark 3. The theorem above shows how to symbolize (X; f) to (; ). Although the phase space of the latter may be more complex than that of the former, the map action of the latter is denitely simpler than that of the former. If we can understand the construction of the space by some means, then the dynamics of f on X can be known accordingly. We note that the subshift (; ) in the proof of Theorem 3. satises the condition (2.), i.e., 8 x = (x 0 ; x ; ); y = (y 0 ; y ; ) 2 ; x = y if and only if x 0 = y 0 This is a restriction under which Theorem 3. is reversible, as stated below. Theorem 3.2 Suppose (X; d) is a compact metric space, (; ) ((X); ), and (; ) satises condition (2.); then there exists a compact subset X 0 X, and a continuous self-map f on X 0, such that (; ) is a conjugate representation of (X 0 ; f). Proof. Let (; ) be a subshift of ((X); ) satisfying condition (2.). Dene ' :! X as '((x 0 ; x ; )) = x 0 ; 8 (x 0 ; x ; ) 2 : Then ' is a one-to-one mapping. Denote '() by X 0. 8(x 0 ; x ; ); (y 0 ; y ; ) 2 ; d('((x 0 ; x ; )); '((y 0 ; y ; ))) = d(x 0 ; y 0 ) < ( + d(x 0 ; y 0 ))((x 0 ; x ; ); (y 0 ; y ; )); 8" > 0; let = "=( + "), then 0 < < : Whenever ((x 0 ; x ; ); (y 0 ; y ; )) <, d(x 0 ; y 0 ) + d(x 0 ; y 0 ) d(x 0 ; y 0 ) < < ((x 0 ; x ; ); (y 0 ; y ; )) < ;? ; d('((x 0 ; x ; )); '((y 0 ; y ; ))) < ( + ) = ";? so ' :! X 0 is continuous. 0

11 Since X is compact, (X) is also compact and so is. As a result, ' :! X 0 is a homeomorphism and hence X 0 X is compact. Take f : X 0! X 0 as f = ''?, then f is continuous, and fj X0 j. Naturally, one would like to ask if Theorem 3.2 still holds for subshifts (; ) which don't satisfy the condition (2.)? The answer is yes if one can construct another subshift ( ; ) such that (; ) ( ; ), and ( ; ) satises the condition (2.). This can be done at least for one-dimensional self-maps, as shown in the following example. Example Let X = [0; ]; d(x; y) = jx? yj; = (2) = f(x 0 ; x ; ; x i ; ) : x i = 0; ; i 0g. Take = R() = fr(x) : x 2 g, where R :! (X) is dened as R(x) = (r 0 (x); r 0 ((x)); ; r 0 ( k (x)); ); r 0 (x) = 0:x 0 x x k (decimal fraction); 8x = (x 0 ; x ; ; x k ; ) 2 ; then it can be veried (see Remark 5.3 in Section 5 for detail) that R :! is a topological conjugacy. So j j, while ( ; ) satises the condition (2.). In the following theorem, we reach a more general conclusion for the case of X = [0; ]. That is, when X = [0; ], Theorem 3.2 holds for all subshifts (; ) ((I 0 ); ) with compact in ((I ); ), regardless of the condition (2.). Theorem (; ) ((I 0 ); ) with compact in (I ), there exists an f 2 C(I 0 ) and I 0 with compact and invariant for f, such that (; ) is a conjugate representation of (; f). Proof. Since is compact in (I ), (; ) is also a subshift of ((I ); ). From Lemma 2.3, there exists a subshift ( ; ) of ((I ); ) with also compact, such that j j, and ( ; ) satises the condition (2.). From Lemma 2.4, ( ; ) is also a subshift of ((I 0 ); ) satisfying the condition (2.) Similar to the proof of Theorem 3.2, it can be shown that there exists a compact subset I 0, and a continuous self-map f on such that ( ; ) is a conjugate representation of (; f), a subsystem of (I 0 ; f). Therefore (; ) is a conjugate representation of (; f). Putting Theorem 3. and Theorem 3.3 together, we have the following representation theorem for one-dimensional continuous self-maps. Theorem 3.4 8f 2 C(I 0 ); 9(; ) ((I 0 ); ), such that (; ) is a conjugate representation of (I 0 ; f); conversely, 8(; ) ((I 0 ); ) with compact in (I ); 9f 2 C(I 0 ) and I 0, where is compact and invariant for f, such that (; ) is a conjugate representation of (; f).

12 4 The Second Order Representations Theorem 3. indicates that 8f 2 C(X); (X; f) can be embedded in ((X); ), denoted by (X; f),! ((X); ). This embedding relation means that the system (X; f) may be represented by a subshift of ((X); ). Naturally, one may further consider the representation of the system ((X); ) itself. To do so, one can regard (X) as a symbol space and denote by 2 (X) the symbol sequence space ((X)), i.e., 2 (X) = fx = (x 0 ; x ; ; x i ; ) : x i 2 (X); i 0g: We specify a metric (2) on 2 (X) by (2) (x; y) = +X k=0 (x k ; y k ) 2 k + (x k ; y k ) ; where is the metric on (X). We denote by 2 the shift map on 2 (X), and if no confusion caused, we also denote by the shift map on 2 (X). And so we get a general symbolic dynamics system ( 2 (X); ). Similarly, we can dene general symbolic dynamics system ( k (X); k ) (or denote by ( k (X); ) if no confusion caused) for k 3; and call it the k-th order symbolic representation of dynamics on X. For higher order symbolic sequence spaces, we have the following general result. Theorem 4. Suppose (X; d) is a metric space. Then 2 (X) is homeomorphic to (X). In general, for k 2; k (X) is homeomorphic to k? (X). Proof. For ~x; ~y 2 2 (X), denote and ~x = (~x 0 ; ~x ; ; ~x i ; ); ~y = (~y 0 ; ~y ; ; ~y i ; ); ~x i = (x i0 ; x i ; ; x ij ; ); ~y i = (y i0 ; y i ; ; y ij ; ); where x ij ; y ij 2 X. Dene h : 2 (X)! (X) as: h(~x) = (x 00 x 0 x 0 x i0 x i?; x ;i? x 0i ); then h is one-to-one and onto. When (2) (~x; ~y) < from Lemma 2., we have 2 2(2N +)+ ; (~x i ; ~y i ) < ; i 2N + ; 22N + therefore d(x ij ; y ij ) < 2?N ; i 2N + ; j N, so we get (h(~x); h(~y)) < 2 + ; N? 2 N 2

13 where N = (N + )(N + 2) + N +. 2 On the other hand, let x = (x 0 x x k ); y = (y 0 y y k ) 2 (X), when (x; y) < 2?(2N +), where we suppose N = (2k + )(2k + 2), we have d(x 2 i; y i ) < 2?N ; i N. Let h? (x) = ~x = (~x 0 ; ~x ; ; ~x i ; ); h? (y) = ~y = (~y 0 ; ~y ; ; ~y i ; ); ~x i = (x i0 ; x i ; ; x ij ; ); ~y i = (y i0 ; y i ; ; y ij ; ); then d(x ij ; y ij ) < 2?N ; i + j 2k +, therefore d(x ij ; y ij ) < 2?N ; i k; j k. This implies (~x i ; ~y i ) < 2?N + 2?k ; i k, therefore (2) (~x; ~y) < 2 2?N + 2?k + 2?k, i.e., we have (2) (h? (x); h? (y)) < 2 + N?2 2 + k? 2 : k Hence both h and h? are continuous, and therefore 2 (X) is homeomorphic to (X). Similarly, we can check that for k 2; k (X) is homeomorphic to k? (X). On the other hand, dene ' : (X)! 2 (X) as '(x) = (x; (x); ; k (x); ); then if X is compact, ' can be veried to be a topological conjugacy from (X) to '((X)) 2 (X). So ((X); ),! ( 2 (X); ). In general, for a compact metric space X, we have the following embedding sequence: (X; f),! ((X); ),! ( 2 (X); ),!,! ( k (X); ),! (4.) That's why we discuss the higher order representations. And the topic is also motivated by Nasu ([2]) and is helpful to study maps in symbolic dynamics discussed in [2], as well. As we discussed earlier that (X; f),! ((X); ) shows a transformation between dynamics on X and subshifts structure in ((X); ); ( k (X); ),! ( k+ (X); ) indicates that the subshifts structure in ( k (X); ) can be transformed to the subshifts structure in ( k+ (X); ). In other words, the symbolic dynamics system ( k (X); ), and its subshifts, can be further represented by the one order higher symbolic dynamics system ( k+ (X); ). If the embedding sequence (4.) is nite, then the above transformation or representations will not go on without limit. This means that the piling up of symbol sequence spaces k (X) will not cause an unlimited increase of the complexity of the corresponding shift systems. In particular, if we have ((X); ) ( 2 (X); ), then ((X); ) is an ultimate (or nal) representation. We may ask under what conditions does ((X); ) ( 2 (X); ) hold? We have the following theorems. Theorem 4.2 ( 2 (X); ) is topologically conjugate to ((X); ) if and only if (X) is homeomorphic to X. In general, for k ; ( k+ (X); ) ( k (X); ) if and only if k (X) is homeomorphic to k? (X). 3

14 Proof. Suppose : (X)! X is a homeomorphism. Dene ' : 2 (X)! (X) as: '() = (( 0 ); ( ); ; ( k ); ); = ( 0 ; ; ; k ; ) 2 2 (X); then ' is a homeomorphism, and ' 2 = '. Therefore ( 2 (X); ) ((X); ). Conversely, let ' : 2 (X)! (X) is a topological conjugacy. 8 x 2 (X), denote ~x = (x; x; ; x; ), then ~x is a xed point of 2 in 2 (X). Since 2, '(~x) is also a xed point of in (X). So 9 x 2 X, such that '(~x) = (x ; ; x ; ). Dene : (X)! X as: (x) = x. Then is a homeomorphism. Similarly, we can prove the general case of the Theorem. From Theorems 4. and 4.2, the following corollary is immediate: Corollary 4.3 Suppose (X; d) is a metric space, then we always have ( k (X); ) ( 2 (X); ); 8 k 3: So the embedding sequence (4.) is nite, and the third or higher order representations are therefore not necessary. The following results show that we further have ((X); ) ( 2 (X); ) when X = I. Theorem 4.4 ((I ); ) is topologically conjugate to ( 2 (I ); ), i.e., ((I ); ) ( 2 (I ); ). Proof. 8(x 0 ; x ; ; x k ; ) 2 2 ([0; ]), where x i 2 ([0; ]); x i = (a (i) 0 ; a (i) ; ; a (i) ; ); a(i) 2 [0; ]; i 0; k 0: k k Rewrite a (i) k by a (i) k X + = r 2 (a (i) ) = k 2?(j+) a (i) kj j=0 ; i; k 0; then dene : ([0; ])! [0; ] as: (x i ) = +X l=0 lx j=0 k+j=l k0 2?( 2 l(l+)+j+) a (i) kj ; i 0; So (x i ) is also a binary fraction. Further, : ([0; ])! [0; ] is a one-to-one and onto mapping. Choose d as the metric on [0; ], and correspondingly as the metric on ([0; ]). Then 8x i ; y i 2 (I ), x i = (a (i) 0 ; a (i) ; ; a (i) k ; ); y i = (b (i) 0 ; b (i) ; ; b (i) ; ); k 4

15 rewrite a (i) k and b (i) k as X + a (i) k = r 2 (a (i) ) = k 2?(j+) a (i) ; kj j=0 X + b (i) k = r 2 (b (i) ) = k 2?(j+) b (i) : kj j=0 Whenever we have therefore So we have (x i ; y i ) < ; 22N + d (a (i) ; k b(i) ) < k 2 ; k N; N a (i) = kj b(i) kj ; k N; j N? : d ((x i ); (y i )) 2? 2 N (N +) ; namely, : (I )! I is continuous. Similarly, we can check that? : I! (I ) is also continuous. Hence ([0; ]) is homeomorphic to [0; ]. From Theorem 4.2 we have The Theorem is proved. ((I ); ) ( 2 (I ); ): We may ask if we also have ((I 0 ); ) ( 2 (I 0 ); )? We guess this is not true but only have: Proposition 4.5 ((I ); ) is a faithful representation for both ((I 0 ); ) and ( 2 (I 0 ); ). Proof. From Lemma 2.4, ((I ); ) is a faithful representation of ((I 0 ); ). Since d 0 (; ) d 2 (; ); 8; 2 [0; ], we have 0 (a; b) (a; b); 8a; b 2 ([0; ]); (2) 0 (x; y) (2) (x; y); 8x; y 2 2 ([0; ]): Similar to the proof of Lemma 2.4, we can prove that ( 2 (I ); ) is topologically semiconjugate to ( 2 (I 0 ); ) under the identity mapping i : 2 (I )! 2 (I 0 ). Therefore ( 2 (I ); ) is a faithful representation of ( 2 (I 0 ); ). So ((I ); ) is a faithful representation for both ((I 0 ); ) and ( 2 (I 0 ); ). Note that from Theorem 4.2, ( 2 (N); ) ((N); ): 5

16 We suspect that ( 2 (I 0 ); ) ((I 0 ); ): And we also suspect in most circumstances ( 2 (X); ) ((X); ): So it is very necessary to study the second order representations. ( 2 (N); ) ((N); ) also means that shift maps with a nite or countable alphabet is not sucient for the study of symbolic representations of dynamical systems, and so it is necessary to study symbolic dynamics with an uncountable alphabet. 5 Quasi-Representations While we have shown the existence of the topological conjugacy between a onedimensional map and a subshift of a symbolic dynamics system, in practice it is often dicult to construct such conjugacy for applications. Instead, it may be easier to nd a topological semi-conjugacy, which sometimes is sucient for the problems under investigation. In this section we discuss quasi-representations of one-dimensional dynamical systems. The symbolic dynamics system ((I); ) is an extension of the usual symbolic dynamics system ((N); ). Equip (N) with a metric as follows: (x; y) = +X i=0 jx i? y i j 2 i + jx i? y i j : Take from I a convergent sequence fa k g with the limit a 2 I, where a i 6= a j ; 8 i 6= j. Let S = fa ; a 2 ; ; ag; = f(x 0 ; x ; ) 2 (I ) : x i 2 S ; i 0g; then is compact in (I ), and ( ; ) ((I 0 ); ). From Theorem 3.3, 9f 2 C(I 0 ) and I 0 ; is compact and invariant for f, such that fj j. 8(; ) ((N); ), we have h top (j ) h top (j (N ) ) = log N 6= h top (j ) = +; So j 6 j, and fj 6 j. Therefore the following result is obvious: Theorem 5. There exists an f 2 C(I 0 ), such that 8(; ) ((N); ); fj j, where I 0 is a closed invariant subset for f. 6 Theorem 5. indicates once again that the scope of application for ((N); ) is quite limited. So it is necessary to study its extensions, such as ((I 0 ); ), etc. Nevertheless, the system ((N); ) still has its special signicance. For example, ((N); ) can be used eectively to characterize the dynamics of multimodal onedimensional maps (see [5] and the references therein). The signicance of ((N); ) can also be shown by the following theorem. 6

17 Theorem 5.2 8f 2 C(I 0 ); 9(; ) ((N); ); N 2; such that (; ) is a quasirepresentation of (I 0 ; f). Proof. First, we prove ((I 0 ); ) is topologically semi-conjugate to ((N); ). Let [0; ] = N? [ k=0 A k ; where A 0 = [0; N ]; A k = ( k N ; k + ]; k = ; 2; ; N? : N Dene : [0; ]! f0; ; ; N? g as (a) = k; a 2 A k : And dene ' : (I 0 )! (N) as: '((x 0 ; x ; ; x k ; )) = ((x 0 ); (x ); ; (x k ); ): Then If k 6= l, we have therefore If k = l, it is obvious that j(x i )? (y i )j = jk? lj; x i 2 A k ; y i 2 A l : N jx i? y i j ; j(x i )? (y i )j N? ; j(x i )? (y i )j + j(x i )? (y i )j < N 2 jx i? y i j + jx i? y i j : j(x i )? (y i )j + j(x i )? (y i )j N jx 2 i? y i j + jx i? y i j : So ('(x); '(y)) N 2 0 (x; y), thus ' is continuous. ' is also surjective. And the following diagram commutes: (I 0 ) ' # (N)?! (I 0 ) # '?! (N) So we have proved that ((I 0 ); ) is topologically semi-conjugate to ((N); ). 8f 2 C(I 0 ), there exists ( ; ) ((I 0 ); ) such that fj I0 j. Denote the topological conjugacy by. Then we can dene : I 0! '( ) as = ', I 0?! '?! '( ) f # # # I 0?! '?! '( ) So is a continuous and onto map, and f = : Since ( ) ; ('( )) '( ). So '( ) is invariant for : is closed, (I 0 ) is compact, so '( ) is closed. So ('( ); ) ((N); ). Take = '( ), then fj I0 is topologically semi-conjugate to j : From the above discussions, we have the following remarks. 7

18 Remark 5. 9( ; ) ((I 0 ); ), such that 8(; ) ((N); ); j 6 j. Remark 5.2 ((N); ) is a factor of ((M); ) when M > N. And ((N); ) is a factor of ((I 0 ); ). Remark 5.3 ((N); ) can be embedded in ((I 0 ); ), i.e., 9(; ) ((I 0 ); ), such that j (N ) j, where N 2: In fact, the subshift may be chosen as: = R N ((N)), where R N (x) = (r N (x); r N ((x)); ; r N ( k (x)); ); r N (x) = +X i=0 Then if R N (x) = R N (y), we have 8k 0; x i N i+ ; 8x = (x 0; x ; ) 2 (N): r N ( k (x)) = r N ( k (y)); r N ( k+ (x)) = r N ( k+ (y)); These imply x k = y k ; 8k 0: So x = y, and R N : (N)! is a one-to-one mapping. When x; y 2 (N) and (x; y) < =2 M +, we have x i = y i ; i = 0; ; ; M: So jr N ( k (x))? r N ( k (y))j +X i=m?k+ jx k+i? y k+i j < ; 0 k M: N i+ N M?k Therefore Since N 2; =N =2, 0 (R N (x); R N (y)) MX k=0 2 k N M?k X k=m + 2 k : 0 (R N (x); R N (y)) < M M : So R N is continuous, and therefore is a homeomorphism. It is obvious that R N j (N ) = j R N : So j (N ) j. So we have the result in Remark Partitions and Representations In Section 5 quasi-representations of one-dimensional dynamical systems are discussed. From the proof of Theorem 5.2, when the whole interval (phase space) is divided into some smaller pieces, and code each piece with a symbol, then we get a quasi-representation of the original system. This shows a direct connection between partitions and representations. Partitions are natural ways to associate a symbolic sequence with an orbit by tracking its history. As pointed out in [2], in order to get a useful symbolism, one needs to construct a partition with special properties. For 8

19 example, when a partition is Markov, the sysytem can be represented by a subshift of nite type. Some hyperbolic systems, such as Anosov systems, axiom A systems, psuedo-anosov systems, and hyperbolic automorphisms on n-tori, n 2, admit Markov partitions. However, for non-hyperbolic systems, there may be no Markov partitions. Therefore other than Markov partitions, some more general partitions for these systems need to be used, so that if a certain kind of partition exists for a non-hyperbolic system, the system then can be represented by a subshift of innite type. In this section we generalize some concepts and main results discussed in [2]. The discussion below are for dynamical systems (X; '), where X is a compact metric space with metric d( ; ) and ' is a homeomorphism of X onto itself. Definition 6. A nite or countable family of subsets R = fr i ; i 2 Ig is called a topological partition for a dynamical system (X; ') if: () each R i is open; (2) R i \ R j =?; i 6= j; (3) X = S i2i R i; (4) 8i 2 I; cardfk 2 I; 'R i \ R k 6=?g < +. Remark 6. Other authors have taken the sets R i to be closed sets with the property that each is the closure of its interior. The variation introduced by Adler here is slightly more general, just enough to make some notation and certain arguments simpler. In fact, there is a example of partition whose elements are not the interiors of their closures ([2]). Remark 6.2 When card I < +, then Denition 6. coincides with the denition of topological partitions in [2]. Given two topological partitions R = fr i ; i 2 I g and S = fs j ; j 2 I 2 g; we dene their common topological renement R _ S as R _ S = fr i \ S j ; i 2 I ; j 2 I 2 g: Lemma 6. For dynamical system (X; ') with topological partition R, the set ' n R dened by ' n R = f' n R i ; i 2 Ig is also a topological partition; and 8m n; W n k=m 'k R is again a topological partition. A topological partition is called a generator for a dynamical system (X; ') if _ n lim D ' k R n!?n! = 0: Where D(R) denotes the diameter of a partition R, D(R) = max R i 2R D(R i) 9

20 where D(R i ) sup x;y2ri d(x; y): We say that a topological partition R for a dynamical system (X; ') satises the n-fold intersection property for a positive integer n 3 if R sk \ '? R sk+ 6=?; k n? ) n\ k= '?k R sk 6=?: Furthermore, we call a topological partition Markov if it satises the n-fold intersection property for all n 3: A homeomorphism ' is said to be expansive if there exists a real number c > 0 such that if d(' n (x); ' n (y)) < c for all n 2 Z, then x = y. Suppose a dynamical system (X; ') has a Markov generator R = fr i ; i 2 Ig. We dene an associated subshift of nite type ( R ; ) over a nite or countable alphabet by R = fs = (s n ) n2z : R sn? \ '? R sn 6=?; s n 2 I; n 2 Zg: Similar to Theorems 6.5 and 6.3 in [2], we have the following result. Theorem 6.2 If the dynamical system (X; ') is expansive and has a Markov generator R = fr i ; i 2 Ig, then the map : R! X dened by (s) = \ n=0 ' n R s?n \ ' n? R s?n+ \ : : : \ '?n R sn gives a regular representation ( R ; ) of (X; '). Moreover, a subshift of nite type is a semi-conjugate representation of an expansive dynamical system (X; ') if and only if (X; ') has a Markov partition. 7 Partial Representations It would be better, under certain conditions, to symbolize a dynamical system by a conjugate representation and using a nite or countable alphabet. This section will show that if there exists a distillation, then we can achieve this target, although we have to abandon the quest of representing all points in the phase space, instead, only represent an invariant subset, that is, we obtain a partial representation. In contrast with partitions, if there exist pairwise disjoint non-empty closed or compact subsets A 0 ; A ; ; A N? of the phase space X (here the union of all A i 's need not to cover X), satisfying certain conditions, then the restriction of the system to a invariant subset of X can be represented by the full shift on N symbols. When the number of such closed (compact) subsets need to be countably innite, then a subsystem can be represented by the full shift with a countable alphabet ([9]). The family of such closed (compact) subsets with certain conditions is called a distillation. More precisely, we give the following denitions. 20

21 Definition 7. If for an f 2 M(X), there exists an invariant subset X and a subshift (; ) of a certain symbolic dynamical system such that (; ) is a symbolic representation of (; f), then we call the subshift (; ) a partial representation of (X; f). A partial representation is also helpful to our understanding of the original system, especially when is a maximal invariant subset or a global attractor of (X; f). In these cases, apart from some transient states in X n, all signicant dynamical behaviour will asymptotically take place in. Definition 7.2 Let X be a topological space, f 2 M(X). We call a nite family of subsets A = fa 0 ; A ; ; A N? g a quasi-distillation of order N for the system (X; f) if: () each A i is compact and non-empty; (2) A i \ A j = S?; 8i 6= j; (3) f(a i ) A j2a(i) j; 80 i N?. Where a(i) f0; ; ; N? g, and each a(i) is non-empty and is maximal in the sense that 8j =2 a(i); f(a i ) \ A j =?. We call T A a distillation of order N for (X; f) if it satises a further condition: + (4) card f?s (A s=0 is ) ; 8 (i 0 i i s ) 2 (N). Where card() denotes the cardinal number of a set. Definition 7.3 Let X and f be as above. We call a countable family of subsets A = fa 0 ; A ; ; A k ; g a quasi-distillation of order innity for the system (X; f) if: () each A i is closed and non-empty; (2) there are open subsets O i X; i 2 Z +, such that 8 i 2 Z + ; A i O i and O i \ ( [ i6=j A j ) =?; S (3) f(a i ) A j2a(i) j; 8i 2 Z +. Where a(i) Z +, and each a(i) is non-empty and is maximal in the same sense as in Denition 7.2. We call A a distillation of order innity for a self-map on a metric space (X; d) if it satises a further T condition: n (4) lim n!+ D( f?s (A s=0 is )) = 0; 8 (i 0 i i s ) 2 (Z + ). Where D(S) denotes the diameter of a subset S, i.e., D(S) = sup a;b2s d(a; b). At rst we give a general lemma. Lemma 7. For a nite or countable family of S subsets fa i ; i 2 Ig of a topological space X and a map f : X! X, if f(a j ) i2i A i; 8j 2 I, then 8l ; 8i k 2 I; k = 0; ; ; f l ( l\ s=0 f?s (A is )) = A il : 2

22 Proof. From we have So when l =, we have f(a j ) [ i2i A i ; 8j 2 I; f(a i ) \ A j = A j ; 8i; j 2 I: f(a i0 \ f? (A i )) f(a i0 ) \ A i = A i : 8x 2 A i, since f(a i0 ) A i ; 9y 2 A i0 such that f(y) = x. Therefore y 2 f? (A i ), and hence y 2 A i0 \ f? (A i ), and x = f(y) 2 f(a i0 \ f? (A i )). That is, So we get A i f(a i0 \ f? (A i )): f(a i0 \ f? (A i )) = A i : By the similar arguement the Lemma can be proven inductively. Some known results about distillations and symbolic representations are Smale's horseshoe theorem ([22]), higher dimensional versions of horseshoes ([23]) and some generalizations to horseshoe-like invariant sets ([9, 24]), and etc. Below we give some more general results about distillations and representations. The following theorem gives results on partial representations over a nite alphabet. Theorem 7.2 Suppose X is a Hausdor space and f 2 C(X), and (X; f) has a quasi-distillation of order N. Then there exists a subshift of nite type (; ) ((N); ) such that (; ) is a partial quasi-representation of (X; f). If (X; f) has a distillation of order N, then there exists a subshift of nite type (; ) ((N); ) such that (; ) is a partial conjugate representation of (X; f). Proof. Let A = fx 2 (N) : x = (x 0 x x k ); a xk x k+ is the transition matrix, = ; 8 k 0g, where A From Lemma 7., 8l, ; A = (a ij ) NN ; a ij = f l ( l\ s=0 j 2 a(i) 0; otherwise : f?s (A is )) = A il ; 8(i 0 i ) 2 A : So T + s=0 f?s (A is ) 6=?, since X is a Hausdor space. Let = + \ s=0 [ N? f?s ( i=0 A i ) = 22 [ (i 0 i )2 A + \ s=0 f?s (A is );

23 then is an invariant compact subset for f. Dene a coding ' :! A as: '(x) = (i 0 i i k ); 8 x 2 f?s (A is ): + \ s=0 Then ' is surjective. 8x 2, suppose x 2 A i0 ; f s (x) 2 A is ; s = 0; ; ; K. Then '(x) 2 K\ s=0 'f?s (A is ): From the continuity of f s,there exists a neighborhood V s (x) of x such that f s (V s (x) \ ) A is ; s = 0; ; ; K: Let V (x) = T K s=0 V s(x), then V (x) \ T K T f?s (A s=0 is ). So 8y 2 V (x) \, we have K '(y) 2 'f?s (A s=0 is ). Thus the rst K entries of '(y) and '(x) agree, therefore ('(x); '(y)) 2 K : So ' is continuous. The commutativity 'fj = j A ' is obvious. So ( A ; ) is a partial quasi-representation of (X; f). When (X; f) has a distillation of order N, then ' is also one-to-one. Note that X is a Hausdor space, ' is therefore a homeomorphism, and hence ( A ; ) is a partial conjugate representation of (X; f). The following two theorems give results on partial representations over a countable alphabet. Theorem 7.3 Let X be a sequentially compact T space and f 2 C(X). If (X; f) has a quasi-distillation of order innity, then there exists a countable state Markov subshift (; ) ((Z + ); ) such that (; ) is a partial quasi-representation of (X; f). Proof. 8(s 0 ; s ; ) 2 A, where A is dened similarly as in the proof of Theorem 7.2, but here A is a matrix of innite order. From + \ s=0 f?s (A is ) = l\ + \ l=0 s=0 f?s (A is ); T + s=0 f?s (A is ) is the intersection of a decreasing sequence of nonempty closed subsets in a sequentially compact T space, so it is nonempty. Dene a coding ' :! A as: '(x) = (i 0 i i k ); 8 x 2 := + \ s=0 + [ f?s ( A i ) = i=0 [ (i 0 i )2 A + \ s=0 f?s (A is ); 23

24 then ' is onto. Let N > 0 satisfy " > P + n=n. Suppose x 2 A 2 n i0 and f s (x) 2 f s (x) 2 A is, then x 2 f?s (A is ) and '(x) 2 'f?s (A is ); s = 0; ; ; N. From the continuity of f s, there exists a neighborhood T V s (x) of x such that f s (V s (x)) N O is ; s = 0; ; ; N: Let V (x) = V s=0 s(x), then f s (V (x)) O is, therefore f s (V (x)\) A T is ; s = 0; ; ; N; T T N and V (x)\ f?s (A s=0 is ). So y 2 V (x)\ N implies '(y) 2 ' f?s N (A s=0 is ) 'f?s (A s=0 is ). Thus the rst N entries of '(y) and '(x) agree, hence ('(x); '(y)) < ". This proves the continuity of '. Therefore ( A ; ) is a partial quasi-representation of (X; f). Theorem 7.4 Let (X; d) be a complete metric space and f 2 C(X). If (X; f) has a distillation of order innity, then there exists a countable state Markov subshift (; ) ((Z + ); ) such that (; ) is a partial conjugate representation of (X; f). Proof. As being shown in the proofs of Theorems 7.2 and 7.3, T + s=0 f?s (A is ) is a nonempty set for all (i 0 i i k ) 2 A. From the condition (4) in Denition 7.3, T + s=0 f?s (A is ) is a one-point set. Dene a coding ' :! A as: + \ '( f?s (A is )) = (i 0 i i s ); s=0 T + where = + T ( s=0 i=0 i) = + (i 0 i )2 A (A s=0 is ): Then as shown in the proof of Theorem 7.3, ' is continuous. For x = (x 0 x ) 2 A ; 8 " > 0; 9 N > 0; whenever n N, we have n\ D( s=0 f?s (A as )) < ": 8 y 2 A ; when (x; y) < =2 N, the rst N + entries of x and y agree. Thus d('? (x); '? (y)) D( N\ s=0 f?s (A xs )) < "; hence '? is continuous. This shows that ' is a homeomorphism. Therefore ( A ; ) is a partial conjugate representation of (X; f). 8 Representations for Discontinuous Maps This section will discuss some specic examples and show that it is possible to use symbolic dynamics as a tool for further studies of dynamics of a class of discontinuous maps: piecewise continuous maps. Let X S R n, and P = fp 0 ; P ; ; P N? g be a nite family of subsets of X, N? satisfying P i=0 i = X, and P i \ P j =? for i 6= j. A piecewise continuous map is a map f : X! X whose restriction to each P i ; 0 i N?, is continuous, and P is minimal in the sense that f is not continuous on P i [ P j for i 6= j. 24

25 A partition P = fp 0 ; ; P N? g associated to a piecewise continuous map provides a natural coding ' : X! (N) by '(x) = (i 0 i i k ) where f k (x) 2 P ik. Let G = f(x; '(x)); x 2 Xg be the graph of ' : X! (N) endowed with the metric d G ((x; '(x)); (y; '(y))) = maxfd(x; y); ('(x); '(y))g; where d is the metric on X. Then the extension map f G : G! G; f G (x; '(x)) = (f(x); '(f(x))), is continuous ([2]). This result may be useful since sometimes it is a bridge between discontinuous and continuous maps. Here we give a denition of partitions for piecewise continuous maps. Definition 8. For X R n ; f 2 M(X). We call a nite family of subsets P = fp 0 ; P ; ; P N? g a partition S of (X; f) if: () each P i is open and convex; (2) N? P i \ P j =? for i 6= j; (3) X = P i=0 i; (4) 8 0 i N? ; fj Pi is continuous and can be extended to a continuous map on P i. We call a partition minimal if f is not continuous on P i [ P j for i 6= j. A cell, denoted by C(x), is a set of points in X encoded by the same symbolic sequence '(x), i.e., C(x) = fy 2 X : '(y) = '(x)g, where ' is called the coding associated with the partition P, ' : X! (N) is dened by '(x) = (i 0 i i k ) where f k (x) 2 P ik. Example 8. Symbolic representation for the Gauss map. Let (I; g) be the iterated system of the Gauss map ([]) g : x! x? (mod ) for 0 < x, and g(0) = 0. Let a sequence s = (s n ) n2z+ 2 (N) correspond to the continued fraction expansion [s 0 s s 2 ]. This denes the map from (N) to I = [0; ]: (s 0 ; s ; ) = [s 0 s s 2 ]: It can be veried that (i) g =, (ii) is continuous, (iii) is onto, (iv) there is a bound on the number of pre-images (in this case two), and (v) there is a unique pre-image of \most" numbers in I (here those with innite continued fraction expansions). So ((N); ) is a regular representation of (I; g). The map is not a homeomorphism, but we do have a satisfactory representation of the dynamical system by a one-sided full shift over a countable alphabet in the sense of Adler [2], i.e., orbits are preserved; every point has at least one symbolic representative; 25

26 there is a nite upper limit to the number of representatives of any point; and every symbolic sequence represents some point. There is an alternate denition of in terms of a countable partition of I: (s 0 ; s ; ) = R = fr i ; i 2 Ng; R i = ( i + ; i ): \ n=0 R s0 \ g? (R s ) \ \ g?n (R sn ): Note that in a suitable torus topology the Gauss map becomes continuous everywhere except at x = 0. Below we give another example where the map is \more discontinuous" but is still easy to symbolize. Example 8.2 Symbolic representations for interval exchange transformations. An interval exchange transformation f : I! I; I = [0; ], over a partition P = fi 0 ; I ; ; I N? g is a one-dimensional piecewise isometry, so we can follow the arguements in [6, 7, 3] to discuss symbolic representation for f. By naturally coding orbits of f using the partition P; ' : I! (N), '(x) = ( i? i 0 i ); where f k (x) 2 I ik ; k 2 Z, we obtain a subset (N) of bi-innite symbolic sequences, = f'(x) 2 (N); x 2 Ig. is closed and shift invariant, so (; ) ((N); ). Note that if a subinterval S I is invariant under f m, then f m j S is either the identity or the reection in the midpoint of S (in this case f 2m j S is the identity). So the disk/polygon packing discussed in [6, 7, 3] now reduce to rigid interval ([7]) packing, i.e., the set J = I n \ n2zf n (^I); ^I = N? [ k=0 inti k of points that never hit or approach the discontinuity can be decomposed into nite ([7]:Lemma 4.5.4) cells, or rigid intervals J k, J = K[ k= J k ; K 2(N? ); all points in J have periodic codings, and f is periodic on each J k. As discussed in [7], when f is generic (i.e., J =?), we can dene a map h :! I, h(( i? i 0 i )) = \ k2zf?k (I ik ): h satises: () fh = h, 26

27 (2) h is continuous, (3) h is onto, (4) there is a bound on the number of pre-images (in this case 2 N ), and (5) there is a unique pre-image of \most" points in I (here those no image or pre-image of them are discontinuity points). So (; ) is a regular representation of (I; f). Although the map h is not a homeomorphism, we do have a satisfactory representation of the dynamical system by a two-sided subshift over a nite alphabet with the properties listed near the end of Example 8.. If (; ) is topologically transitive, then (; ) is a subshift of innite type, because otherwise the periodic points of j would be dense in. 9 Final Remarks When we consider the symbolic representation of an iterated map, one may hope to nd one of the following: (A) a full representation (i.e., all points in the phase space are symbolically represented); (B) a conjugate representation; (C) a representation via a subshift of nite type over a nite or countable alphabet. Unfortunately, it is very rare to get a symbolic representation satisfying all of (A), (B) and (C). Usually, if we'd like to get a symbolic representation satisfying (A) and (B), we may need to pay the price of giving up (C), i.e., we need to use an uncountable alphabet, as discussed in Sections 3 and 4; if we'd like to obtain a representation with properties (A) and (C), we may have to give up (B), as discussed in Sections 5 and 6; if we'd like to nd a representation with (B) and (C), we may have to give up (A), i.e., we only get a partial representation, as discussed in Section 7. When we consider a symbolic representation of a discontinuous map, we at least have to give up (B), as discussed in Section 8 using special cases of piecewise continuous maps. Symbolic representations provide a powerful method to investigate general discrete dynamical systems through shifts or subshifts. This is similar to the situation for the study of nite groups in algebra, where all nite groups can be represented by subgroups of symmetric groups S n, which are better understood. We can view S n as `symbolic' groups. There is a large body of theory that describes the dynamics of continuous smooth dynamical systems. However, if there are discontinuities in the system such as those caused by collisions (impacting) or switching there is still comparatively little in 27

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