Kneading Sequences for Unimodal Expanding Maps of the Interval

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1 Acta Mathematica Sinica, New Series 1998, Oct., Vol.14, No.4, pp Kneading Sequences for Unimodal Expanding Maps of the Interval Zeng Fanping (Institute of Mathematics, Guangxi University, Nanning , China) Abstract Let P and AC be two primary sequences with min{p, AC} RLR,ρ(P)andρ(AC) be the eigenvalues of P and AC, respectively. Let f C 0 (I,I) be a unimodal expanding map with expanding constant λ and m be a nonegative integer. It is proved that f has the kneading sequence K(f) (RC) m P if λ (ρ(p )) 1/2m,andK(f) > (RC) m AC E for any shift maximal sequence E if λ>(ρ(ac)) 1/2m. The value of (ρ(p )) 1/2m or (ρ(ac)) 1/2m is the best possible in the sense that the related conclusion may not be true if it is replaced by any smaller one. Keywords Periodic point, Kneading sequence, Topological entropy 1991MR Subject Classification 58F08, 58F20 Chinese Library Classification O Introduction Let C 0 (I,I) denote the set of continuous maps from the interval I = [0, 1] into itself. Let ϕ C 0 (I,I). ϕ is called a unimodal map relative to some point τ (0, 1) if ϕ [0,τ]is increasing and ϕ [τ,1] is decreasing. The orbit of the point ϕ(τ) is called the kneading orbit of ϕ and its itinerary (for definition see [1]) under ϕ the kneading sequence, written K(ϕ), of ϕ. In the study of symbolic dynamics of the unimodal maps, kneading sequence plays an important role, roughly speaking, all types of other orbits of a unimodal map are determined by its kneading sequence, see [1 5] for more details. In this paper we study the kneading sequences for unimodal expanding maps (see [6, 7] for definition) of the interval. The main goal of this paper is to establish a general relationship between the kneading sequences of unimodal expanding maps and their expanding constants. In Section 2 the concept of eigenvalues of shift maximal sequences is introduced and the primary sequences defined in [1, pp.176] are extended. A complete characterization for kneading sequences of the tent maps is given in Section 3. In Section 4 the general relationship between the kneading sequences of unimodal expanding maps and their expanding constants is established. As a corollary, we obtain a general relationship between the existence of periodic orbits with all possible itineraries in unimodal expanding maps and their expanding constants, which generalizes the main results in [6, 8] and the related results in [9]. Received November 29, 1996, Accepted April 21, 1997 Project supported by the National Natural Science Foundation of China

2 458 Acta Mathematica Sinica, New Series Vol.14 No.4 2 Eigenvalues of Shift Maximal Sequences Shift maximal sequences are important and closely related to the kneading sequences of unimodal maps. Denote by M the set of such sequences. Theorem A (See [1 5]) M = {K(ϕ) :ϕ C 0 (I,I) is a unimodal map}. For a g C 0 (I,I), let T(g) denote the set of orbit types (see [7, 10]) of periodic orbits of gands(θ) the spectral radius of the Markov graph induced by θ for any θ T(g). Denote by h(g) the topological entropy (see [11]) of g. Theorem B (see [7, 10]) h(g) = sup{log s(θ) : θ T(g)}. Let ε(l) =1,ε(R) = 1 andε(c) = 0. For each M = M 0 M 1 M, define M i 1 D M (t) =1+ t i ε(m j ), (1) i=1 j=0 where M denotes the length of M and t is a positive number. Lemma 1 Let PC,Q M and put n = PC. Then D PC Q (t) =D PC (t) D Q (t n ). (2) Proof Put α i = ɛ(p i ),i = 0, 1,,n 2,γ j = ɛ(q j ) and β j = ( 1) N(P ) γ j,j = 0, 1,, Q 1, where N(P ) is the parity of R s in P. Clearly, we have α 0 α 1 α n 2 β k = γ k for any 0 k< Q. From the definition of -product (see [1, 3]) it follows that D PC Q (t) = 1+α 0 t + + α 0 α 1 α n 2 t n 1 + α 0 α 1 α n 2 β 0 t n + +(α 0 α 1 α n 2 ) 2 β 0 β 1 t 2n + ( ) n 1 Q j 1 = 1+ α 0 α 1 α i 1 t i 1+ (α 0 α 1 α n 2 ) j t jn i=1 = D PC (t) D Q (t n ). Lemma 1, therefore, is proven. The D M (t) is closely related to the characteristic polynomial of Collet-Eckmann (see [1, pp.171]) and also to the kneading determinant D(t) (when l = 2) of Milnor-Thurston (see [12]). In fact, by [12, Lemma 4.5] we have j=1 k=0 β k { DM L (t) =D M (t) D L (t M ), if M is finite, D(t) = D M (t), if M is infinite. (3) From [12, Theorem 6.3] and (3) it follows that there exists an s>1 such that D M (1/s) =0 and D M (t) 0for0<t<1/s if M>(RC) (= R,see[1,pp. 174]). Denote by ρ(m) such an s and set ρ(m) =1ifM (RC). Definition 1 For an M M, suchaρ(m) is called the eigenvalue of M. Theorem C (see [12]) Let ϕ be a unimodal map. Then h(ϕ) =logρ(k(ϕ)). Proposition 1 Let D, E M with D E. Thenρ(D) ρ(e). Proof From Theorem A it follows that there exist two unimodal maps ϕ i C 0 (I,I)such that K(ϕ i )=i for i {D, E}. Suppose ϕ E has a periodic orbit with itinerary Q. Then from the condition D E and [1, Theorem II.3.8 and Lemma II.3.4] it follows that ϕ D has a periodic

3 Zeng Fanping Kneading Sequences for Unimodal Expanding Maps of the Interval 459 orbit with itinerary Q. This implies T(ϕ E ) T(ϕ D ). Thus by Theorems B and C we have ρ(d) ρ(e). Proposition 1, therefore, is proven. Set M 0 = {M M : M (RC) } {RC}, M 1 = {BC D : BC M with BC > (RC),D M {C}}, P = M (M 0 M1 ). Definition 2 For an M M, M is called a primary sequence if M P. Theorem 1 Let PC,Q M. Then (i) ρ(pc Q) =(ρ(q)) 1/2m if PC < (RC),where2 m = PC ; (ii) ρ(pc Q) =ρ(pc) if PC > (RC) ; (iii) there exist an AC P with AC > RLR and an m Z + (set of nonegative integers) such that ρ(q) =(ρ(ac)) 1/2m if Q M 1. Proof (i) In this case we have PC =(RC) m C for some m Z +.Then D PC Q (t) =D RC (t) D (RC) (m 1) Q(t 2 )=(1 t) D (RC) (m 1) Q(t 2 ) = (1 t) (1 t 2m 1 ) D Q (t 2m ). Then by Definition 1 we have ρ(pc Q) =(ρ(q)) 1/2m ; (ii) From [1, Theorem II.2.7] it follows that there exist a BC M with BC > RLR and an r Z + such that PC =(RC) r BC. Then by (i) we have ρ(pc)=(ρ(bc)) 1/2r and ρ(pc Q) =(ρ(bc Q)) 1/2r. Now it suffices to show that ρ(bc Q) =ρ(bc). Let p = BC. By (2) we get D BC Q (t) =D BC (t) D Q (t p ). Since BC > RLR,Q RL and p 3, by Proposition 1 we have ρ(bc) ρ(rlr )=ρ(rc RL )=(ρ(rl )) 1/2 = 2 > 3 2 (ρ(rl )) 1/p (ρ(q)) 1/p. Thus by Definition 1 we get ρ(bc Q) =ρ(bc); (iii) It follows from [1, Theorem II.2.7], (i) and (ii). Theorem 1, therefore, is proven. 3 Kneading Sequences of Tent Maps Let 1 < 2andT (x) =min{x, (1 x)} for any x I. Each map T : I I is called a tent map. It is well known that the tent maps are the important maps in dynamical systems and are extensively studied (see [1, 3, 8, etc.]). In this section we give a complete characterization for kneading sequences of the tent maps, which extends the related results in [1, pp ]. From the Bowen s definition of topological entropy (see [11]) it is easy to verify that h(t )=log, for any (1, 2]. (4) Theorem 2 Let 1 <α<β 2. ThenK(T α ) <K(T β ). Proof Suppose to the contrary. Then by Proposition 1 we have ρ(k(t α )) ρ(k(t β )) and thus by Theorem C we have h(t α ) h(t β ). On the other hand, by (4) we have h(t α ) <h(t β ). There is a contradiction. Theorem 2, therefore, is proven. Theorem 3 Let (1, 2],m Z + and m = 1/2m.ThenK(T m )=(RC) m K(T ). Proof Obviously, it suffices to show that the theorem is true for m =1.

4 460 Acta Mathematica Sinica, New Series Vol.14 No.4 [ ] Let J = 1 1+, 1+. Define the map π : J I by π(x) = 1+ 1 Then π is an orientation preserving linear homemorphism. Note that x + [ ], if x 1 T 2 2, 1 2, (x) = x + ( ], if x 1 2, 1 2 1, ( ) and T 2 1+ = T = ( x 1+. It is not difficult to verify that ) 1+. π (T 2 J) =T π. (5) Clearly we have 1 T = 2 2 > 1 2, (6) T 1 2 = 2 2 < 1 2, (7) and T 3 12 = Let J 0 =[ /2, 1 + /2], J 1 =[(1 /2), /2]. Then it is easy to see that T (J 0 )=J 1 and T 2 (J 0 ) J 0. Then we have T 2k+1 ( 12 ) J 1, and 1 T 2k+1 T (8) for any k {1, 2, }. Note that J 0 J and π is orientation-preserving. Thus by (3.3.2) we have T 1 2k > if and only if T k < 1 (9) 2 2 and T 1 2k = if and only if T k 1 = (10) From (6 10) and the definition of -product it follows that K(T )=RC K(T ). Theorem 3, therefore, is proven. Theorem 4 P {RC} = {K(T ):1< 2}. Proof Let A = {P P : P RLR } and B = {K(T ): 2 2}. From Definition 2 and [1, Theorem II.2.7] it follows that P {RC} = {(RC) r P : r Z +,P A}. (11) On the other hand, K(T 2 )=RLR,K(T 2 )=RL, from Theorems 2 and 3 it follows that {K(T α ):1<α 2} = {(RC) r K(T ):r Z + and [ 2, 2]}. (12) By (11) and (12) we know that it suffices to show A = B. The proof of the part A B is similar to that of [1, Lemma III.1.6] and is omitted. Now we prove B A. Suppose to the contrary that there exists a 0 [ 2, 2] such that K(T 0 ) A. Then by Theorem 2 we have K(T 0 ) >RLR and K(T 0 ) M 1. From [1, Theorem II.2.7] it follows

5 Zeng Fanping Kneading Sequences for Unimodal Expanding Maps of the Interval 461 that there exist an AC P with AC > RLR and an E M {C} such that K(T 0 )=AC E. Since A B, there exists a 1 [ 2, 2] such that K(T 1 )=AC. Then K(T 0 ) K(T 1 ), and by Theorem 2 we have 0 1. On the other hand, from (4), Theorem C and the part (ii) of Theorem 1 it follows that log 0 = h(t 0 )=logρ(ac E) =logρ(ac) =h(t 1 )=log 1. Then we have 0 = 1. There is a contradiction. Theorem 4, therefore, is proven. Proposition 2 Let A, B P with A>B.Thenρ(A) >ρ(b). Proof It follows from Theorems 4, 2 and C. 4 Kneading Sequences of Unimodal Expanding Maps In the study of the dynamics of unimodal expanding maps, an interesting and important problem is: For a given M M with M>(RC), what is the smallest value of the expanding constants of the unimodal expanding map f such that K(f) M? In this section we will study and solve this problem. Lemma 2 Let ϕ C 0 (I,I) be an unimodal map, P be an infinite primary sequence and AC be a finite primary sequence with AC RC. Then (i) K(ϕ) P if h(ϕ) log ρ(p ); (ii) K(ϕ) AC L if h(ϕ) log ρ(ac). Proof We prove (ii), the proof of (i) is similar and omitted. Suppose to the contrary. Then we have AC L >K(ϕ). Now we argue in the following two cases Case 1 K(ϕ) P. NotethatAC > AC L >K(ϕ). Then by Proposition 2 we have ρ(ac) >ρ(k(ϕ)) and by Theorem C we have h(ϕ) < log ρ(ac), a contradiction. Case 2 K(ϕ) P. Sinceh(ϕ) log ρ(ac) > 0, we have K(ϕ) > (RC). There exist a BC P {RC} and an E M {C} such that K(ϕ) =BC E. Since AC is a primary sequence and AC RC, wehaveac > RC. Thus by Theorems C, 1 and Proposition 2 we have h(ϕ) = logρ(k(ϕ)) = logρ(bc) < log ρ(ac). This is a contradiction. Lemma 2, therefore, is proven. In what follows we fix the map f C 0 (I,I) to be a unimodal expanding map relative to τ with expanding constant λ. Lemma 3 K(f) {BC L : BC M}. Proof Since f is expanding, it is not difficult to prove that I f (x 1 ) I f (x 2 ) for any x 1,x 2 I with x 1 x 2,whereI f (x) denotes the itinerary of the point x under f (see[3]). Suppose K(f) =BC L for some BC M. Let BC = p. Then the points f(τ) and f p+1 (τ) are distinct but have the same itinerary BC L. There is a contradiction. Lemma 3, therefore, is proven. Lemma 4 h(f) log λ. Proof Fix a positive integer n. LetJ be a subinterval of I on which f n is monotone. Since f is expanding, we have J λ n f n (J) λ n,where J denotes the length of the interval J. Letl(f n ) denote the lap number of f n (see [12]). Then l(f n ) [ I /λ n ]=[λ n ], where [λ n ] denotes the integer part of λ n. Note that lim n ([λ n ]) 1/n = λ. By [12, pp.470] we have 1 h(f) = lim n n log(f n 1 ) lim n n log[λn ]=logλ. Lemma 4, therefore, is proven. Theorem 5 Let P and AC be two primary sequences with min{p, AC} RLR. And let ρ(p ) and ρ(ac) be the eigenvalues of P and AC, respectively, and m Z +.Then

6 462 Acta Mathematica Sinica, New Series Vol.14 No.4 (i) K(f) (RC) m P if λ (ρ(p )) 1/2m ; (ii) K(f) (RC) m AC E for any E M if λ>(ρ(ac)) 1/2m. Proof From Lemmas 2, 3 and 4 it follows that K(f) K(T λ ). (i) Let ρ 0 =(ρ(p )) 1/2m. Then by Theorems 4 and 3 we have K(T ρ0 )=(RC) m P. If λ ρ 0 then, by Theorem 2, K(T λ ) K(T ρ0 ). Thus, K(f) K(T λ ) (RC) m P ; (ii) Let ρ 1 =(ρ(ac)) 1/2m. Then by Theorems 4 and 3 we have K(T ρ1 )=(RC) m AC. Also by Theorem 4 we have K(T λ ) P {RC}. Ifλ>ρ 1 then, by Theorem 2 and Definition 2, we have K(T λ ) >K(T ρ1 ) RL. Thus, for any E M, wehavek(t λ ) > (RC) m AC RL (RC) m AC E. Theorem 5, therefore, is proven. Proposition 3 Let AC, ρ(ac) and m be as in Theorem 5. Then (i) f has a periodic orbit with itinerary (RC) m AC or (RC) m AC L if λ (ρ(ac)) 1/2m ; (ii) f has a periodic orbit with itinerary (RC) m AC BC L for any BC M if λ>(ρ(ac)) 1/2m. Proof It follows from Lemma 3, Theorem 5 and [1, Theorem II.3.8 and Lemma II.3.4]. Remark 1 Let n Z + be odd and n 3. It is easy to see that ρ(rlr n 3 C)(or ρ(rlr n 3 C L )) is the unique positive zero of the polynomial λ n 2λ n 2 1. So the part (i) of Proposition 3 implies the main results in [6, 8]. Let k Z + and k 3. Also it is easy to see that ρ(rl k 2 C)(or ρ(rl k 2 C L )) is the greatest zero of the polynomial α k 2α k So the part (ii) of Proposition 3 implies Theorem 2 in [9]. Remark 2 The values of (ρ(p )) 1/2m and (ρ(ac)) 1/2m in Theorem 5 or in Proposition 3 are both the best possible in the sense that the related conclusion may not be true if each one is replaced by any smaller value. For example, consider the tent maps. If λ<(ρ(p )) 1/2m, then by Theorem 2 we have K(T λ ) < (RC) m P ;ifλ<(ρ(ac)) 1/2m,thenwehaveK(T λ ) < (RC) m AC L. Acknowledgement I am grateful to Professor Mai Jiehua for his direction in preparing this paper. This workis also supported in part by Graduate Foundation of Shantou University. References 1 Collet P, Eckmann J P. Iterated Maps on the Interval as Dynamical Systems. Boston: Birkhauser, Beyer W A, Mauldin R D, Stein P R. Shift-maximal sequences in function itineration: existence, uniquencess and multiplicity. J Math Anal Appl, 1986, 115(6): Mai J H. The integrity of kneading sequences in the family of unimodal maps. (in chinese), Acta Math, 1990, 33(2): Mai J H, Zeng F P. Some families of unimodal maps without the integrity of kneading sequences. (in chinese), Sys Sci & Math Scis, 1995, 15(1): Metroplis N, Stein M L, Stein P R. On finite limit sets for transformations on the interval. J Combin Theory(A), 1975, 15(2): Du B-S. Unimodal expanding maps of the interval. Bull Austral Math Soc, 1988, 38(2): Geller W, Tolosa J. Maximal entropy odd orbit types. Trans Amer Math Soc, 1992, 329(3): Heidel J. The existence of periodic orbits of the tent maps. Phys Lett A, 1990, 143(4): Mai J H, Zeng F P. On unimodal expanding selfmaps of the interval. (in chinese), J Math, 1994, 14(3): Block L, Coppel W A. Dynamics in One Dimension. Lecture Notes in Math, 1513, New York: Springer- Verlag, Walters P. An Introduction to Ergodic Theory. New York: Springer-Verlag, Milnor J, Thurston W. On iterated maps of the interval. Lecture Notes in Math, 1342, New York: Springer- Verlag, 1988