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Topology Proceedings Volume 24, Summer 999, 597 609 TOPOLOGICAL ENTROPY OF A CLASS OF TRANSITIVE MAPS OF A TREE Yong Su and Xiangdong Ye Abstract Let T be a tree, x T and Val T (x) be the valence of x. For a continuous map f of T, h(f) is the entropy of f. We study the class of transitive maps f of T with f (x) ={x}. We prove that h(f) > Val T log 3, and show that there exist a sequence (x) of transitive maps f n of T with fn (x) ={x} and h(f n ) Val T log 3, n. (x). Introduction Topological entropy of a transitive map of a tree has been studied by many authors [ABLM], [AKLS], [BC], [KH], [W] and [Y]. Particularly, in [ABLM] the authors obtain a lower bound of the topology entropies of the transitive maps of a given tree with a given point such that the preimage of this point under the maps is itself. In this paper we aim at proving the lower bound mentioned before can NOT be reached, but it is the infimum. Let f be a continuous map of a topological space X into itself. We say f is transitive if for each pair of non-empty open subsets (U, V )ofx there exists n N such that f n (U) V. Bya tree we mean a connected compact one-dimensional polyhedron, which does not contain any subset homeomorphic to a circle and which contains a subset homeomorphic to [0, ]. A subtree Project supported by NNSF 962503 and 973 project. Mathematics Subject Classification: Primary 34C35, 4H20; Secondary: 58F03. Key words: Entropy, transitivity, tree map, atlas, PL-map.
598 Yong Su and Xiangdong Ye of a tree T is a subset of T, which is a tree itself. If T is a tree and x T, then the number of connected components of T \{x} is called the valence of x in T and will be denoted by Val T (x). A point of T of valence is called an end of T. The set of the ends of T and the number of ends of T, will be denoted by E (T ) and End (T ) respectively. The set of the vertices of T is consisted of all the points of valences different from 2 and finitely many points of valence 2, which depends on our need. The set of the vertices of T will be denoted by V (T ). The closure of each connected component of T \ V (T ) is called an edge of T. The set and the number of the edges of T, will be denoted by Edge (T ) and EdgeN (T ) respectively. Note that a point x Int (T )=T \ E(T ) will be said in the interior of T or an interior point of T. Let T be a tree and A T. We will use [A] and cl(a) to denote the smallest connected and closed subset containing A and the closure of A respectively. If A = {a, b}, then we use [a, b] to denote [A]. We define (a, b) =[a, b] \{a, b} and we similarly define (a, b] and [a, b). We will use F (T,x) to denote the set of transitive maps of a tree T such that the preimage of x T under this map is x itself. The following theorem is the Proposition 4.2 of [ABLM]. Theorem.. Let f : T T be a transitive map of a tree T and x T with {x} = f (x). Then f has 3-horseshoe if x is an end of T, and f k (k =Val T (T )) has 3-horseshoe if x is in the interior of T. We have the following immediate consequence. Corollary.2. Let T be a tree, and x T. Then h(f) log 3 for each f F (T,x). Val T (x)
TOPOLOGICAL ENTROPY OF A CLASS OF TRANSITIVE... 599 2. Entropy Estimate In this section we shall prove that the topology entropy h(f) can t reach the lower bound Val T log 3, for each f F (T,x), (x) where T is a tree and x T. That is, we will prove Theorem 2.. Let T be a tree, x be a point of T and f be a transitive map of T such that f (x) ={x}. Then the topological entropy h(f) > Val T log 3. (x) To prove this theorem, we need the following lemmas Lemma 2.2. Let T be a tree and f : T T be transitive. Then exactly one of the following alternatives holds. () f n is transitive for each n N. (2) There is n 0 > such that there are an interior fixed point y of f and subtrees T,...,T n0 of T with n 0 i=t i = T, T i T j = {y} for all i j and f(t i )=T i+(mod n0 ) for i n 0. Moreover, f n 0 Ti is transitive for each i n 0. The proof of Lemma 2.2 can be found in [AKLS]. And the following corollary is obvious. Corollary 2.3. Let T be a tree, and E(T ) contain a fixed point of f, then f s is transitive for each s. Lemma 2.4. Let T be a tree and f be a map of T such that f s is transitive for each s N. Suppose H is a subtree of T with H End (T )=, and J is an open subtree of T. Then there is n N such that H f n (J). The proof of Lemma 2.4 is similar to that of Proposition 44 of [BC]. Lemma 2.5. Let T be a tree, x E(T ) and f F (f,x), then h(f) >log3.
600 Yong Su and Xiangdong Ye Proof. We identify each edge containing x with the unit interval [0, ] (with x = 0), and use the usual ordering < to describe the relative position of points of this edge. Points not on this edge (if any) will be unimportant. Set y 0 = min{f(y) :y T \[0, ],f(y) [0, ]}, y = min{y [0, ] : f(y) =} and y 2 = min{y 0,y }. We need from transitivity only that if y [0, ] then none of the sets [0,y] and T \[0,y) is f-invariant (plus we use the assumption f (0) = {0}), and we will mean that when we say by transitivity. Let A l (resp. A r ) be the set of those points of [0,y 2 ] that lead movement to the left (resp. right). That is, y A l if y [0,y 2 ], f(y) y and f(z) f(y) for all z [y,]; similarly y A r if y [0,y 2 ], f(y) y and f(z) f(y) for all z [0,y]. Clearly, A l and A r are closed. By transitivity, A l A r = {0}. Also by transitivity and by the definitions of A l and A r, one can easily show the following properties: () A l (0,y] for each y y 2, (2) A r (0,y] for each y y 2, (3) (f(y),y) A l for each y A l, and (4) (y,f(y)) A r for each y A r such that f(y) y 2. Since A l and A r are closed and by properties () and (2) above, there are points w<twith w A r, f(w) y 2, t A l and no points of A l A r in (w, t). By properties (3) and (4) above, there are points v (f(t),t) A l and u (w, f(w)) A r. Hence, we get f([v,w]) [f(v),f(w)] [v,u], f([w, t]) [f(t),f(w)] [v,u], and f([t, u]) [f(t),f(u)] [v,u]. Notice that f(w) > u,f(t) < v. Therefore, there are three pairwise disjoint intervals I,I 2 and I 3 [v,u] such that f(i i )= [v,u],i=, 2, 3. Thus, there exists an open interval J [v,u] \ (I I 2 I 3 ). By Lemma 2.4, there exists k N such that
TOPOLOGICAL ENTROPY OF A CLASS OF TRANSITIVE... 60 f k (J) [v,u]. It s obvious that there are 3 k pairwise disjoint closed intervals J i I I 2 I 3,i=,...,3 k such that f k (J i )= [v,u]. Hence the intervals among {J} {J i } 3k i= are pairwise disjoint, in other words, f k has (3 k + )-horseshoe. This implies kh(f) =h(f k ) log(3 k +)>k log 3. That is, h(f) > log 3. This ends the proof of Lemma 2.5. Proof of Theorem 2.: Set h = Val T (x). By the assumption f (x) ={x} and the transitivity of f, we have f(t i )= T i+(mod h), where {T i } h i=0 is the set of the closures of the components of T \{x}. Moreover, T i is f h -invariant and f h Ti is transitive, 0 i h. By Lemma 2.3, we have that hh(f) = h(f h ) h(f h Ti ) > log 3. Consequently, h(f) > log 3 = h Val T log 3. (x) 3. The Lower Bound is Infimum In this section we shall prove that the lower bound obtained in Theorem 2. is infimum. That is, Theorem 3.. Let T be a tree, x T, and ɛ>0. Then there exists f F (T,x) such that h(f) < Val T log 3 + ɛ. Consequently, (x) Val T log 3 = inf{h(f) f F (T,x)}. (x) To prove Theorem 3., we need the following lemmas. Lemma 3.2. [KH P.23] Let T be a tree, f be a continuous map of T and d be a metric on T such that there exists a positive real number L with d(f(a),f(b)) Ld(a, b) for each a, b T. Then h(f) max{0, log L}. For each edge e of a tree T, e is homeomorphic to some interval I e under some homeomorphism ϕ e : e I e. D T is called an atlas of T,ifD T is a map which maps each edge e of T to some homeomorphism ϕ e from e to I e =[0,t e ](t e > 0).
602 Yong Su and Xiangdong Ye Let T be a tree and S be a subset of T. S is called measurable, if for each edge e, ϕ e (S e) is Lebesgue measurable as a subset of I e. For each edge e, we can define a measure m e such that m e (A) =µ Ie (D T (e)(a)) = µ Ie (ϕ e (A)) for each measurable subset A of e, where µ Ie is the Lebesgue measure of I e. For a measurable subset S T, let m DT (S) = m e (S e). e Edge(T ) Clearly, m DT is a measure of T completely determined by D T. For any two points a and b of T, let d DT (a, b) =m DT ([a, b]). Clearly, d DT is a metric of T which is completely determine by D T. And it is compatible to the original topology on T (inherited from R 2 ). Definition 3.3. Let T and T be two trees and f be a map from T to T. D T and D T are atlases on T and T respectively. We call f a PL-map(or, piecewise linear map) if for each e Edge (T ), J = f(e) is an interval of T, and f e is a non-degenerate linear function from e to J, i.e. there exists c>0 such that d DT (f(a),f(b)) = cd DT (a, b),and we call c the scale of f on e. Here is a property of a PL-map: Lemma 3.4. Let T and T be two trees, and f be a PL-map from T to T. Then there is L > 0, such that m DT (f(s)) L m DT (S) for each subtree S of T. Proof. Let S be a subtree of T. By the definition of m DT, m DT (S) = e Edge (T ) m DT (S e). Then we have m DT (f(s)) = m DT (f( e Edge (T ) (S e))) = m DT ( e Edge(T ) f(s e)) c e m DT (S e),
TOPOLOGICAL ENTROPY OF A CLASS OF TRANSITIVE... 603 for each edge e of T, where c e is the scale of f on e. It is easy to see that there exists e 0 Edge (T ) such that m DT (S e 0 ) EdgeN (T ) m D T (S). Therefore, m DT (f(s)) c e0 m DT (S e 0 ) min{c e e Edge (T )} m DT (S) EdgeN (T ) = L m DT (S), by letting L = min{c e e Edge (T )}. EdgeN (T ) This ends the proof of Lemma 3.4. We call L an L -constant of f, and write L (f) =L. We have some properties showing relations between atlas, measure, metric and PL-map. Note that if D T is an atlas of T, and c, c > 0, then cd T will denote the atlas with (cd T )(e) =cd T (e) for each e Edge(T ). Remark 3.5. (i) m cdt = cm DT, d cdt = cd DT, and (ii) if f is a PL-map from T (related to D T )tot (related to D T ), then f is also a PL-map from T (related to cd T )tot (related to c D T ). Moreover, () L cdt,c D T (f) = c L c D T,D T (f), L cd T,c D (f) = T c c L D T,D (f), where L T D T,D T (f), L D T,D T (f) denote the Lipschitz constant and L -constant (related to D T and D T ) respectively. (2) if c = c, then L cdt,c D T (f) =L DT,D T (f) and L cd T,c D T (f) = L D T,D T (f).
604 Yong Su and Xiangdong Ye Fig.. Proof of Theorem 3.: Let h =Val T (x). Then T \{x} has exactly h connected components, whose closures will be denoted by T i, i =0,..., h. We choose an interior point y i in the edge of T i, which contains x. And we add y i as a new vertice of T and T i. Then T i can be represented as P i T i, where P i =[x, y i ] which is an edge of T i. For any given ɛ>0, which is small enough, we want to construct a continuous map f F (T,x) which has the topological entropy less than log 3 + ɛ. Val T (x)
TOPOLOGICAL ENTROPY OF A CLASS OF TRANSITIVE... 605 To do this, first we choose a homeomorphism ϕ Pi : P i [0, ] with ϕ Pi (x) = 0 for each 0 i h, and then we construct maps f i from T i to T i+(mod h) (see Figure ) with the following properties: (a) For 0 i h 2, let f i map P i onto P i+ with f i = ϕ P i+ ϕ Pi satisfying that () f i T i : T i T i+ is a surjective PL-map, and (2) there is a constant L i 3 h...( ), such that d DTi+ (f i (x),f i (y)) L i d DTi (x, y) for any x, y T i,0 i h 2. By Lemma 3.4, there exists L i > 0 such that m DTi+ (f i (S)) L i m DTi (S) for any subtree S of T i. (b) For convenience, we identify P i (0 i h ) with [0, ], i.e. for each a, b [0, ] denote the point (D Ti (P i )) (a) simply by a, and denote [(D Ti (P i )) (a), (D Ti (P i )) (b)] P i simply by [a, b] [0, ]. Note that, this will be also used in the rest part of this paper. Let f h map T h onto T 0 satisfying () f h (u l 0)=f h (u r 0)=,f h (v j )=v j,j, f h (u j )=v j + ɛ,j, and f 2 j h (w j )=v j + ɛ,j 0, 2 j where v j =, u 2 j+ j = v j +, w 3 2 j+ j = v j + 2, j 0, 3 2 j+ u l 0 = u 0 δ, and u r 0 = u 0 + δ (δ will be determined later). (2) Moreover, f h is linear on intervals {[v 0,u l 0], [u r 0,w 0 ], [w 0,v ]} {[v j,u j ], [u j,w j ], [w j,v j ]} j= respectively (see Figure 2). For each j, let u l j <ur j be those of [v j,v j ] such that f(u l i)=f(u r i )=v i. And for each j 0, let w l j <w r j be those of [v j,v j ] such that f(w l j)=f(w r j)=v j.
606 Yong Su and Xiangdong Ye Fig. 2. (3) For each j 0, let I j =[v j,u l j ],J j = [u l j,ur j ], K j = [u r j,wl j ],L j =[wj l,wr j ], and M j =[wj,v r j ]. Then we can choose a PL-map f h J0 from J 0 onto T 0 such that L DJ0,D T (f h J0 ) 3 h for some 0 atlases D J0 (we take J 0 as a new edge of T ) and D T 0... (**). This can be done by using Definition 3.3 and Remark 3.5 (ii). (4) It is easy to see that we can obtain a PL-map f h T h onto φ T 0 [ δ, ] (by choosing δ ), such that L(f n T 0 ) 3 h... (***).
TOPOLOGICAL ENTROPY OF A CLASS OF TRANSITIVE... 607 Let f = h i=0 f i. Then f is a continuous map of T with f (x) ={x}. Therefore, to prove Theorem 3. it s enough to show that we can choose ɛ, δ, and δ such that f is a transitive map of T and its entropy is very close to Val T (x) log 3. Let F = f h T0 = f h f h 2... f 0. By (*),(**) and (***), we know that the Lipschitz constant L(F ) 3+6ɛ 6δ, where 3+6ɛ lim ɛ 0 +,δ 0 + 6δ =3. By Lemma 3., h(f ) log( 3+6ɛ ). Hence, 6δ h(f) = h h(f h T 0 ) = h(f ) Val T (x) Val T (x) log 3(ɛ 0+,δ 0 + ). By Lemma 3.2 there exists L i > 0 such that m DT (f i (S)) L i m DT (S), 0 i h, for any subtree S of T i, 0 i h. Thus, there exists L J 0 > 0 such that m(f(s)) L J 0 m(s) for each subtree S of T 0. Let L = h i=0 L i. Now, we need to choose δ > 0 and n N such that () δ (3 + 6ɛ) n < 6, (2) L 3 n >, and (3) L J 0 L 3 n >. This can be done by letting (i) n > (ii) δ < 2 max( L, L L J ) 0 log(3+6ɛ) (3+6ɛ) n, and In order to preserve the constants L i,l i,l J 0 and L J 0,we need to change the atlas D T of T on the subforest ( 0 j h T j) f h (J 0 ) and decrease δ (by Remark 3.5 (ii)).
608 Yong Su and Xiangdong Ye Now, it is easy to see that (I) For each closed subtree T of T 0 \{x}, and each subtree S of T 0 containing some point from V = {v i,u l i,u r i,w l i,w r i } i=0 {}, there exists n N such that F (S) T. (II) We claim that, for each subtree S of T 0, there exists N N {0} such that F N (S) V. Assume the contrary, i.e., F (S) U \ V, for each n N {0}, where U {I i,j i,k i,m i,l i } i=0 {T 0}. Then () If U {J i,l i } i= {L 0} then m DT0 (F (S)) 3 m 2 D T0 (S), (2) if U = J 0, then F n (S) M 0 (by (****)) and m DT0 (F n (S)) L J 0 3 n m DT0 (S). (3) if U {I i,k i,m i } i=0 {I 0,K 0 }, then m DT0 (F (S)) 3m DT0 (S), (4) if U = T 0, m D T0 (F n (S)) L 3 n m DT0 (S). For m DT0 (S) > 0, L >, we have lim sup m DT0 (F n (S)) = +, n + a contradiction. This proves the claim. By (I) and (II) we get that F is transitive. It follows that f is transitive, too. References [ABLM] L. Alseda, S. Baldwin, J. Llibre and M. Misiurewicz, Entropy of transitive tree maps, Topology, 36(997), 59-532. [AKLS] L. Alseda, S.F. Kolyada, J. Llibre and L. Snoha, Entropy for transitive maps, Preprint, Barcelona, 995, Trans. Amer. Math. Soc., to appear.
TOPOLOGICAL ENTROPY OF A CLASS OF TRANSITIVE... 609 [BC] L. Block and W.A. Coppel, Dynamics in one dimension, Lec. Notes in Math., 53, Springer-Verlag, 992. [KH] A. Katok and B. Hasselblatt, An introduction to the Modern Theory of Dynamical Systems, Cambridge Univ. Press. U.K., 995, 23-24 [W] P. Walters, An introduction to ergodic theory, Springer- Verlag, New York, 982. [Y] X. Ye, Topological entropy of transitive maps of a tree, Ergod. Th. & Dynam. Sys., 20 (2000), 289-34. Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, P.R.China E-mail address: yexd@ustc.edu.cn