TOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS OF INTERVALS

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1 Chung, Y-.M. Osaka J. Math. 38 (200), 2 TOPOLOGICAL ENTROPY FOR DIFFERENTIABLE MAPS OF INTERVALS YONG MOO CHUNG (Received February 9, 998) Let Á be a compact interval of the real line. For a continuous map : Á Á by Misiurewicz et al. ([, 2, 3]) the following relation between the topological entropy ( ) and the growth rate of the number of periodic points is known: ( ) ( ) lim sup Ò½ log Per( Ò) Ò where Per( Ò) denotes the set of all fixed points of Ò for Ò, and the number of elements of a set. (The equality of the expression ( ) does not hold in general. For instance, the topological entropy of the identity map is zero, nevertheless all of points of the interval are fixed by this map.) For a periodic point Ô of with period Ò we put O + (Ô) = Ô (Ô) Ò (Ô) Then we say that Õ is a homoclinic point of Ô if Õ ¾ O + (Ô) and there are a positive integer Ñ with Ñ (Õ) = Ô and a sequence Õ 0, Õ,, Õ, ¾ Á with Õ 0 = Õ such that (Õ ) = Õ ( ) lim Õ O + (Ô) = 0 ½ where Ü = infü Ý : Ý ¾ for Ü ¾ Á, Á. It is known by Block ([2, 3]) that ( ) is positive if and only if has a homoclinic point of a periodic point. In this paper we shall establish more results (Theorems and 2) for differentiable maps of intervals. To describe them we need some notations. Let : Á Á be a +«map («0). A periodic point Ô of with period Ò is a source if (Ô) = ( Ò ) ¼ (Ô) Ò For Ò, and Æ 0 we define an -invariant set by 99 Mathematics Subject Classification. 28D20, 58F, 58F5

2 2 Y-.M. CHUNG Per( Ò Æ) = Ô ¾ Per( Ò) : (Ô) ¼ ( (Ô)) Æ for all 0 Ò Then we have Per( Ò Æ ) Per( Ò 2 Æ 2 ) if 2 Æ Æ 2 and Ô : source of = ½ Æ0 Ò= Per( Ò Æ) One of our results is the following: Theorem. ( ) = max 0 lim lim Æ0 lim sup Ò½ Ò log Per( Ò Æ) By Theorem it is clear that for a +«map of a compact interval if the topological entropy is positive then the map has infinitely many sources. However, the converse is not true in general. In fact, for any Ö it is easy to constract a Ö diffeomorphism of a compact interval having infinitely many source fixed points. But every diffeomorphism of an interval has zero entropy. It is known that if is a 2 map with non-flat critical points, then any REMARK. periodic point of with sufficiently large period is a source ([0]). In Theorem we do not assume any conditition concerned with critical points. Then the map may have flat critical points. For a source Ô of with period Ò we denote by Ïloc Ù (Ô) the maximal interval  of Á containing Ô such that ( Ò ) ¼ (Ü) ( + (Ô))2 Ò for all Ü ¾  We say that a homoclinic point Õ of Ô is transversal if there are non-negative integers Ñ, Ñ 2 and a point Õ ¼ ¾ Ïloc Ù (Ô) such that Ñ (Õ ¼ ) = Õ Ñ +Ñ 2 (Õ ¼ ) = Ñ 2 (Õ) = Ô and ( Ñ +Ñ 2 ) ¼ (Õ ¼ ) = 0 If has a transversal homoclinic point of a source, then there is a neighborhood U of such that every map belonging to U has a transversal homoclinic point of a source. We denote the set of transversal homoclinic points of a source Ô of by TH(Ô), and its closure by TH(Ô) We call TH(Ô) the transversal homolinic closure of

3 TOPOLOGICAL ENTROPY 3 Ô. It is easy to see that Ô ¾ TH(Ô), and TH(Ô) is -invariant. For Ñ and Æ 0 define À (Ô Ñ Æ) = Õ ¾ Ï Ù loc (Ô) : Ñ (Õ) = Ô ¼ ( (Õ)) Æ for all 0 Ñ Then we have À (Ô Ñ Æ ) À (Ô Ñ Æ 2 ) if Æ Æ 2 and TH(Ô) = Æ0 ½ Ñ Ñ= =0 À (Ô Ñ Æ) Ò O + (Ô) The second result of this paper is the following: Theorem 2. If ( ) 0 then and for a source Ô of we have ( ) = sup( TH(Ô) ) : Ô is a source of ( TH(Ô) ) = max 0 lim Æ0 lim sup ѽ Ñ log À (Ô Ñ Æ) A result corresponding to Theorem 2 is known for surface diffeomorphisms by Mendoza ([]). As an easy corollary of Theorem 2 we have: Corollary 3. The following statements are equivalent: (i) ( ) 0; (ii) has a transversal homoclinic point of a source; (iii) has a homoclinic point of a periodic point.. Proofs of Theorems Let : Á Á be a continuous map. For integers Ð we say that a closed - invariant set ¼ is a ( Ð)-horseshoe of if there are subsets ¼ 0,, ¼ of Á such that ¼ = ¼ 0 ¼ (¼ ) = ¼ + (mod ) and ¼ 0: ¼ 0 ¼ 0 is topologically conjugate to a one-sided full shift in Ð-symbols. If ¼ is a ( Ð)-horseshoe, then it is clear that ( ¼ ) = log Ð and ÐÒ [Per( Ò) ¼] Ð Ò

4 4 Y-.M. CHUNG for all Ò. It was proved by Misiurewicz et al. ([, 2, 3]) that if the topological entropy of is positive then there are sequences, Ð of positive integers with a (, Ð )-horseshoe ¼ of ( ) such that ( ) = lim ( ¼ ) = lim log Ð ½ ½ Then the formula ( ) follows from this fact. In order to prove our results, we need the notion of hyperbolic horseshoe and ideas of the theory of hyperbolic measures ([4, 5]). Katok ([9]) has proved that if a +«diffeomorphism of a manifold has a hyperbolic measure then its metric entropy is approximated by the entropy of a hyperbolic horseshoe. The author has shown in [5] that the result of Katok is also valid for +«(non-invertible) maps. Let : Á Á be a differentiable map. For integers Ð, numbers and Æ 0 we say that ¼ is a ( Ð Æ)-hyperbolic horseshoe of if ¼ is a ( Ð)- horseshoe and ( ) ¼ (Ü) ¼ (Ü) Æ (Ü ¾ ¼) The following lemma plays an important role for the proofs of Theorems and 2. Lemma 4. Let : Á Á be a +«map. If ( ) 0, then for a number 0 with 0 exp( ) there exist sequences Ð of positive integers and Æ 0 ( ) such that for there is a ( Ð 0 Æ )-hyperbolic horseshoe ¼ of so that ( ) = lim ( ¼ ) = lim log Ð ½ ½ This is corresponding to the result obtained by Katok for surface diffeomorphisms ([9]). For the proof we use the result stated in [5]. Proof of Lemma 4. For a number 0 with 0 exp( ) we take a sequence of positive numbers ( ) such that exp( ) 3 0 and 0 as ½. By the variational principle for the topological entropy ([6, 7, 8]), we have an -invariant ergodic Borel probability measure on Á such that ( ) 0 where denotes the metric entropy of with respect to. If denotes the Lyapunov exponent of, that is, = log ¼ (Ü) (Ü)

5 TOPOLOGICAL ENTROPY 5 then by the Ruelle inequality ([7]) we have 0 and so is a hyperbolic measure of. Then by Theorem C (and its proof) of [5], we can construct sequences of integers, Ð with ( ) log Ð, numbers and closed sets Á ( ) such that: () ( ) = ; (2) ( ) ¼ (Ü) exp ( ) for all Ü ¾ and ; (3) : is topologically conjugate to a one-sided full shift in Ð - symbols. For we set ¼ = Then ¼ is -invariant. Moreover we put Æ = min ¼ (Ü) : Ü ¾ ¼ 0 ¼ (Ü) = max ¼ (Ý) : Ü Ý ¾ ¼ ¾ [ ½) and take an integer Ò large enough so that exp Ò Then we have ( Ò ) ¼ (Ü) Ò 0 (Ü ¾ ¼ ) This follows from the fact that for 0 and Ü ¾ ( Ò ) ¼ (Ü) = ( Ò ) ¼ ( (Ü)) ( ) ¼ (Ü) ( ) ¼ ( Ò (Ü)) exp( Ò )( ) + exp Ò ( 2 ) exp Ò (( ) 3 ) 0 Ò It is easy to see that Ò : is topologically conjugate to a one-sided full Ò shift in Ð Ò -symbols. Thus ¼ is a ( Ò Ð 0 Æ )-hyperbolic horseshoe, and from which ( ¼ ) = Ò log Ð Ò

6 6 Y-.M. CHUNG = log Ð Since 0 as ½, we have Lemma 4 was proved. ( ) 2 ( ) = lim ½ ( ¼ ) Proof of Theorem. For and Æ 0 we want to find 0 = 0 ( Æ) 0 such that Per( Ò Æ) is an (Ò 0 )-separated set of for all Ò. Take = (Æ) 0 so small that if Ü Ý ¾ Á satisfy Ü Ý then ¼ (Ü) ¼ (Ý) Æ 2 We put Á Æ = Ü ¾ Á : ¼ (Ü) Æ 2 Obviously, Á Æ is closed. For Ò and Ü ¾ Á, Ü Per( Ò Æ) implies that Ü ¾ Á Æ Since a function Ü log ¼ (Ü) is bounded and varies continuously on Á Æ, there is 2 = 2 ( Æ) 0 such that if Ü Ý ¾ Á Æ satisfy Ü Ý 2 then log ¼ (Ü) log ¼ (Ý) log 2 We put 0 = min 2. Then it is checked that Per( Ò Æ) is an (Ò 0 )-separated set of for Ò. Indeed, if a pair Ô Ô ¼ ¾ Per( Ò Æ) with Ô Ô ¼ satisfies max (Ô) (Ô ¼ ) : 0 Ò 0 then we see that for Ü ¾ [Ô Ô ¼ ] and 0 Ò, (Ü) (Ô) 0 (Ü) ¾ Á Æ On the other hand, by the mean value theorem there is a point ¾ [Ô Ô ¼ ] such that Ò (Ô) Ò (Ô ¼ ) = ( Ò ) ¼ () Ô Ô ¼

7 TOPOLOGICAL ENTROPY 7 Since (), (Ô) ¾ Á Æ and () (Ô) 0 for 0 Ò, we have Ò log ( Ò ) ¼ () log ( Ò ) ¼ (Ô) =0 Ò log 2 log ¼ ( ()) log ¼ ( (Ô)) and so ( Ò ) ¼ () ( Ò ) ¼ (Ô) Ò exp 2 log = Ò2 Since Ô Ô ¼ ¾ Per( Ò Æ), we have Ô Ô ¼ = Ò (Ô) Ò (Ô ¼ ) = ( Ò ) ¼ () Ô Ô ¼ = ( Ò ) ¼ () ( Ò ) ¼ (Ô) ( Ò ) ¼ (Ô) Ô Ô ¼ Ò2 Ò Ô Ô ¼ = Ò2 Ô Ô ¼ and so Ô = Ô ¼ because of. Thus Per( Ò Æ) is an (Ò 0 )-separated set of, and then (.) lim sup Ò½ Ò log Per( Ò Æ) lim sup Ò½ Ò log ( Ò 0) lim lim sup log ( Ò ) 0 Ò½ Ò = ( ) for and Æ 0, where ( Ò ) denotes the maximal cardinality of (Ò )- separated sets for. Therefore we have the conclusion of Theorem when ( ) = 0. Thus it remains to give the proof for the case when ( ) 0. Fix 0 exp( ). Take sequences, Ð, Æ and ¼ ( ) as in Lemma 4. Since for all Ò, we have Ð Ò [Per( Ò 0 Æ ) ¼ ] Ð Ò lim lim sup Æ0 Ò½ Ò log Per( Ò 0 Æ) lim Ò½ = log Ð log [Per( Ò 0 Æ ) ¼ ] Ò

8 8 Y-.M. CHUNG If ½, then (.2) lim lim sup Æ0 Ò½ Ò log Per( Ò 0 Æ) ( ) Combining (.) and (.2) we have Theorem was proved. ( ) lim lim sup Æ0 Ò½ Ò log Per( Ò 0 Æ) lim lim Æ0 ( ) lim sup Ò½ log Per( Ò Æ) Ò REMARK. In fact, from the proof of Theorem it follows that if 0 exp( ). ( ) = lim lim sup Æ0 Ò½ Ò log Per( Ò 0 Æ) Proof of Theorem 2. Proof of the first statement. Under the assumption of Theorem 2 we fix a number 0 with 0 exp( ). By Lemma 4, for there are, Ð and Æ 0 with a ( Ð 0 Æ )-hyperbolic horseshoe ¼ = ¼ 0 ¼ such that ( ¼ ) = log Ð ( ) as ½. For define a product space = ½ Ñ= Ð with the product topology and a shift : by (( Ñ ) Ñ ) = ( Ñ+ ) Ñ (( Ñ ) Ñ ¾ ) From the definition of hyperbolic horseshoe, there is a homeomorphism ³ : ¼ 0 such that ³ Æ = ( ¼ 0) Æ ³. Then Ô = ³ ( ) is a source of. For Ñ and Ñ ¾ Ð with = for some Ñ, ³ ( Ñ ) is a transversal homoclinic point of Ô. Thus, TH(Ô ) ¼, from which

9 TOPOLOGICAL ENTROPY 9 ( ) = lim ( ¼ ) ½ lim ½ ( Ì À (Ô ) ) sup( Ì À (Ô) ) : Ô is a source of ( ) The first statement was proved. Proof of the second statement. Let Ô be a source of. Without loss of generality we may assume that Ô is a fixed point, i.e., (Ô) = Ô. To show that for Æ 0 ( TH(Ô) ) lim sup ѽ log À (Ô Ñ Æ) Ñ take 0 = 0 (Æ) 0 so small that if Ü Ý ¾ Á satisfy Ü Ý 0 then ¼ (Ü) ¼ (Ý) Æ 2 Then, for Ñ and a pair Õ Õ ¼ ¾ À (Ô Ñ Æ) satisfying max (Õ) (Õ ¼ ) : 0 Ñ 0 we can find a sequence 0,, Ñ ¾ Á such that and (Õ) 0 + (Õ) + (Õ ¼ ) = ¼ ( ) (Õ) (Õ ¼ ) (0 Ñ ) Since Ñ (Õ) = Ñ (Õ ¼ ) = Ô, we have 0 = Ñ (Õ) Ñ (Õ ¼ ) = ¼ ( Ñ ) Ñ (Õ) Ñ (Õ ¼ ) = = Ñ =0 Ñ =0 ¼ ( ) Õ Õ ¼ ¼ ( (Õ)) Æ Õ Õ ¼ 2 Æ Ñ Õ Õ ¼ 2 and so Õ = Õ ¼. Thus À (Ô Ñ Æ) is an (Ñ 0 )-separated set of TH(Ô), from which it follows that

10 0 Y-.M. CHUNG (.3) lim sup ѽ Ñ log À (Ô Ñ Æ) lim sup ѽ Ñ log ( TH(Ô) Ñ 0) ( TH(Ô) ) If ( TH(Ô) ) = 0, then nothing to prove for the second statement. Thus we must check the conclusion for the case when ( TH(Ô) ) 0. To do so fix a number 0 with 0 min(ô) exp ( TH(Ô) ). By the same way as in the proof of Lemma 4, we can take sequences of integers, Ð, numbers Æ 0 with ( Ð 0 Æ )- hyperbolic horseshoes ¼ = ¼ 0 ¼ containing Ô ( ) such that ( ¼ ) = ( ) log Ð ( TH(Ô) ) as ½ Then there is a homeomorphism ³ : ¼ 0 such that ³ Æ = ( ¼ 0)Ƴ, where : is the shift defined as in the proof of the first statement. Without loss of generality we may assume that ³ ( ) = Ô. By taking an integer Ò large enough we have where ³ ([ ] Ò ) Ï Ù loc (Ô) [ ] Ò = ( Ñ ) Ñ ¾ : Ñ = for all Ñ Ò Since Ò times Þ Ð ß ³ ( Ñ Ò ) ¾ À (Ô Ñ Æ ) holds for all Ñ Ò + and,, Ñ Ò ¾ Ð, we have À (Ô Ñ Æ ) Ð Ñ Ò Thus, lim lim sup log À (Ô Ñ Æ) lim sup log À (Ô Ñ Æ ) Æ0 ѽ Ñ Ñ½ Ñ lim ѽ = log Ð Ñ Ò log Ð Ñ for. If ½, then we have (.4) lim lim sup Æ0 ѽ Ñ log À (Ô Ñ Æ) ( TH(Ô) )

11 TOPOLOGICAL ENTROPY Combining (.3) and (.4), ( TH(Ô) ) lim lim sup log À (Ô Ñ Æ) Æ0 ѽ Ñ ( TH(Ô) ) The second statement was proved. This completes the proof of Theorem Circle Maps In the same way as above, it can be checked that our results (Theorems and 2) are also valid for +«maps («0) of the circle Ë. However, the existence of a homoclinic point does not imply that the topological entropy is positive. In fact, we know an example of a ½ map : Ë Ë such that has a homoclinic point of a source fixed point, nevertheless () = 0 ([6]). It is known that the topological entropy of a continuous circle map is positive if and only if the map has a nonwandering homocinic point of a periodic point ([4]). Since any transversal homoclinic point of a source is nonwandering, we have: Corollary 5. For a +«map : Ë Ë («0) the following statements are equivalent: (i) ( ) 0; (ii) has a transversal homoclinic point of a source; (iii) has a nonwandering homoclinic point of a periodic point. Added in proof. After this manuscript was completed the author learned from A. Katok that he and A. Mezhirov had obtained a result that overlaps with Theorem for maps with finitely many critical points ([8]). References [] L. Alsedà, J. Llibre and M. Misiurewicz: Combinatorial Dynamics and Entropy in Dimension One, Advanced Series in Nonlinear Dynamics 5, World Scientific, Singapore, 993. [2] L.S. Block: Homoclinic points of mappings of the interval, Proc. Amer. Math. Soc. 72 (978), [3] L.S. Block and W.A. Coppel: Dynamics in One Dimension, Lect. Notes in Math. 53, Springer-Verlag, Berlin-Heidelberg, 992. [4] L. Block, E. Coven, I. Mulvey and Z. Nitecki: Homoclinic and nonwandering points for maps of the circle, Ergod. Th. and Dynam. Sys. 3 (983), [5] Y.M. Chung: Shadowing property of non-invertible maps with hyperbolic measures, Tokyo J. Math. 22 (999), [6] E.I. Dinaburg: On the relations among various entropy characteristics of dynamical systems, Math. USSR Izvestia, 5 (97),

12 2 Y-.M. CHUNG [7] T.N.T. Goodman: Relating topological entropy and measure entropy, Bull. London Math. Soc. 3 (97), [8] L.W. Goodwyn: Topological entropy bounds measure-theoritic entropy, Proc. Amer. Math. Soc. 23 (969), [9] A. Katok: Lyapunov exponents,entropy and periodic orbits for diffeomorphisms, Publ. Math. I.H.E.S. 5 (980), [0] W. de Melo and S. van Strien: One-Dimensional Dynamics, Springer-Verlag, Berlin-Heidelberg, 993. [] L. Mendoza: Topological entropy of homoclinic closures, Trans. Amer. Math. Soc. 3 (989), [2] M. Misiurewicz: Horseshoes for mappings of an interval, Bull. Acad. Pol. Soc. Sér. Sci. Math. 27 (979), [3] M. Misiurewicz and W. Szlenk: Entropy of piecewise monotone mappings, Studia. Math. 67 (980), [4] Ya.B. Pesin: Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR Izvestija, 0 (976), [5] Ya.B. Pesin: Characteristic Lyapunov exponents and smooth ergodic theory, Russian Math. Surveys, 32 (977), [6] F. Przytycki: On Đ-stability and structural stability of endomorphisms satisfying Axiom A, Studia Math. 60 (977), [7] D. Ruelle: An inequality for the entropy of differentiable maps, Bol. Soc. Bras. Math. 9 (978), [8] A. Katok and A. Mezhirov: Entropy and growth of expanding periodic orbits for onedimensional maps, Fund. Math. 57 (998), Department of Applied Mathematics Faculty of Engineering Hiroshima University Higashi-Hiroshima , Japan chung@amath.hiroshima-u.ac.jp