Measures with maximum total exponent and generic properties of C 1 expanding maps

Transcription

1 Hiroshima ath. J. 43 (2013), easures with maximum total exponent and generic properties of C 1 expanding maps Takehiko orita and Yusuke Tokunaga (Received August 11, 2011) (Revised September 12, 2012) Abstract. We show that a generic C 1 expanding map on a compact Riemannian manifold has a unique measure of maximum total exponent which is fully supported and of zero entropy. We also show that for r b 2 a generic C r expanding map does not have fully supported measures of maximum total exponent. 1. Introduction Let be an N-dimensional, compact, connected, smooth Riemannian manifold without boundary and let T be a C 1 expanding map of. Recall that a C 1 map T :! is called expanding if there exist c > 0 and l > 1 such that kdt n xþvk T n x b cln kvk x holds for any x A, v A T x and nonnegative integer n, where DT : T! T; x; vþ 7! DTxÞv is the tangent map of T and kvk x is the norm of v induced by the Riemannian metric of. Clearly such a map T is surjective since it is an open map and its image is compact. We denote by TÞ the totality of invariant Borel probability measures of T. Since the absolute value of the determinant of the matrix representation of DTxÞ : T x! T Tx is independent of the choice of orthonormal bases of T x and T Tx, we can define the determinant jdet DTÞÞj :! R of the tangent map DT. Let JTÞxÞ ¼jdet DTÞxÞj and consider the quantity lt; mþ ¼ log JTÞxÞdm: We may call lt; mþ total exponent of T with respect to m A TÞ since the following formula holds X sxþ log JTÞxÞdm ¼ k j; xþw j; xþdm; j¼1 First author partially supported by the Grant-in-Aid for Scientific Research (B) , Japan Society for the Promotion of Science athematics Subject Classification. Primary 137D35; Secondary 37C20, 37C40. Key words and phrases. Expanding ap, Total Exponent, Optimization easure.

2 352 Takehiko orita and Yusuke Tokunaga where fw1; xþ < < wsxþ; xþg is the totality of distinct Lyapunov exponents of T at x and k j; xþ is the multiplicity of w j; xþ for each j. Put ltþ ¼supflT; mþ : m A TÞg: An element m in TÞ is called a measure with maximum total exponent if lt; mþ¼ltþ. We denote by LTÞ the set of measures with maximum total exponent. It is easy to show that LTÞ is not empty since TÞ is compact in the weak topology and the map m 7! lt; mþ is continuous. For any nonnegative integer r, C r ; Þ denotes the space of C r maps on endowed with the C r topology. Note that a sequence T n converges in C r ; Þ if and only if all the derivatives of T n of order less than or equal to r converge uniformly on. Let E r ; Þ be the space of C r expanding maps of. Then it is easy to see that it is an open subspace in C r ; Þ. Recall that a subset of a topological space X is called residual if it contains a set expressed as a countable intersection of open dense subsets in X. A topological space X is called a Baire space if any residual subset is dense in X. For a Baire space, consider a property P with respect to elements in X. We say that the property P is generic or a generic element satisfies P if there exists a residual subset each member of which satisfies P. It is well known that the topological space C r ; Þ is a Baire space, consequently, so is E r :Þ. The purpose of this paper is to prove the following theorems. Theorem 1. Each of the following properties for element T in E 1 ; Þ is generic. (1) T has a unique measure with maximum total exponent. (2) Any measure with maximum total exponent for T has zero entropy. (3) Any measure with maximum total exponent for T is fully supported. On the contrary, we have the following when r b 2. Theorem 2. For r b 2, a generic element in E r ; Þ has no fully supported measures with maximum total exponent. The same kind of theorems are first proved by Jenkinson and orris in [8] for expanding maps on the circle. Afterward, inspired by their results, the second author of this paper extended those to expanding maps of the n-torus in his dissertation [14]. We should note that these results are similar in spirit to some theorems in ergodic optimization. So we explain about some preceding results briefly (see the survey by Jenkinson in [7] for the further discussion and references). Consider a continuous map T on a compact metric space

3 easures with maximum total exponent 353 X and a continuous function f : X! R. The main interests in ergodic Ð optimization are invariant probability measures m which maximize the integral X fdm. Such measures are referred to maximizing measures for f. Bousch and Jenkinson showed that for a fixed expanding map of the circle, a generic continuous function has a unique maximising measure with full support in Theorem C of [2] (cf. Proposition 9 in [1]). On the other hand Brémont proved in [4] that for any fixed continuous map on a compact metric space, any maximizing measure for a generic continuous function has zero entropy. Therefore we see that for any expanding map on the circle, a generic continuous function has a unique maximizing measure with full support and zero entropy. One of the main results in Jenkinson and orris [8] asserts that for a generic but not fixed C 1 expanding map T on the circle, log JTÞ has a unique maximizing measure with full support and zero entropy. In other words we generalize the results in [8] on the unit circle to those on a compact manifold admitting C 1 expanding maps. The ideas of our proofs are essentially the same as those in [8]. But the technique in [8] seems to have some di culties to be applied to the general expanding maps. So we need to make modifications of lemmas in [8] so that they can work in our general case. In particular, the crucial step of constructing an auxiliary perturbation of C 1 map along a periodic orbit is given with full generality. In Section 2, we summarize some fundamental results on expanding maps. Section 3 is devoted to the construction of an appropriate perturbation. Finally proofs of Theorem 1 and Theorem 2 are given in Section Preliminaries In this section we summarize the results which are needed in the proof of those theorems in Introduction. ost of them are well know facts for expanding maps, so we just give references or sketch the proof. First we need the following lemma. For the proof consult Lemma 20 of Section 3 in [15] (see also Section 7.26 Section 7.30 in [10] and Section 3 in Bowen [3]). Lemma 1. Let T be an element in E 1 ; Þ. Then for any b > 0 there exists a arkov partition for T with diameter less than b, i.e. we can construct a finite cover fr 1 ;...; R q g of by closed sets satisfying the following conditions. (1) int R i ¼ R i for each i. (2) int R i V int R j ¼ q for i 0 j. (3) 6 q i¼1 int R i is dense in. (4) T6 q i¼1 qr iþ H6 q i¼1 qr i.

4 354 Takehiko orita and Yusuke Tokunaga (5) If Tint R i Þ V int R j 0q, then R j H TR i. (6) max 1aiaq diam d R i Þ < b, where diam d RÞ is the diameter of R H with respect to the distance d induced by the Riemannian metric on. For T A E 1 ; Þ, choose b > 0 so small that T maps any ball of radius less than b homeomorphically onto its image. Let R ¼fR 1 ;...; R q g be a arkov partition of T with diameter less than b. Now we define a subshift of finite type in the usual way as follows. Put S ¼fx ¼x i Þ ib0 : Ax i x iþ1 Þ¼1 for each i b 0g; where A ¼AijÞÞ is a q q matrix given by AijÞ ¼ 1 if Tint R iþ V int R j 0q; 0 otherwise: Now we consider the shift transformation s : S! S satisfying sxþ i ¼ x iþ1 for any i b 0 and x A S. We choose c > 0 and l > 1 such that kdt n xþvk T n x b cl n kvk x for any x A, v A T x and nonnegative integer n. Putting y ¼ 1=l, we define d y : S S! R by d y x; hþ ¼y nx; hþ, where nx; hþ ¼inffi b 0 : x i 0 h i g. Here we regard inf q and y y as þy and 0, respectively. For x A S, we see that 7 n 1 i¼0 T i R xi is nonempty and diam d 7 n 1 i¼0 T i R xi Þ a 1=cÞy n diam d Þ by the choice of b. Therefore 7 y i¼0 T i R xi consists of a single point and we can define a map p : S! so that pxþ is the single point in 7 y i¼0 T i R xi. oreover we have the following. Lemma 2. (1) p is a Lipschitz continuous map from S; d y Þ to ; dþ. (2) 7 y j¼0 T j 6 q i¼1 int R iþ is dense in. (3) For any x A 7 y j¼0 T j 6 q i¼1 int R iþ, p 1 fxg is a single point set. (4) p is surjective. (5) p s ¼ T p holds. (6) The subshift S; sþ of finite type is topological mixing. i.e. A n 0 > 0 for some positive integer n 0. Proof. The assertion (1) is obvious from the fact that dpx; phþ a nx; hþ 1 diam d 7i¼0 T i R xi Þ a 1=cÞy nx; hþ diam d Þ holds. The assertions (2) (5) are a sort of exercises of general topology. To prove the assertion (6), we have only to show that T is topological mixing. But this is a well known fact. For example, by using the lifting of T to the universal covering space of, we can easily see that for any nonempty open set U H, there exists an integer n > 0 such that T n U ¼. As a corollary we obtain the following lemma which is a modification of Lemma 1 in [8] (see also [9] and [12]).

5 easures with maximum total exponent 355 Lemma 3. If T is an element in E 1 ; Þ, then we obtain the following. (1) Let p TÞ be the set of invariant probability measures each of which is supported on a periodic point. Then p TÞ is dense in TÞ in the weak topology. (2) Let Y be a proper closed subset of. Then for any m A TÞ with supp m H Y, there exists a sequence m n A TÞ converging to m in the weak topology such that m n is supported on a periodic orbit and supp m n V 7 y i¼0 T i YÞ¼q for each n. Proof. Since the subshift S; sþ of finite type constructed in Lemma 2 is topological mixing, it satisfies the specification property (see [6]). Consequently so does T. Therefore the proofs of Lemma 1 and Lemma 2 for the closed set A ¼ 7 y i¼0 T i Y in [12] do work. Thus we obtain (1) and (2). Shub proved the conjugacy theorem of expanding maps in [13] via Contraction Principle. His proof leads us to the following statement which corresponds to Lemma 2 in [8]. Lemma 4. Let T k be a sequence of elements in E 1 ; Þ satisfying the conditions: (i) there exist c > 0 and l > 1 independent of k such that kdtk nxþvk T n k x b cln kvk x for any x A, v A T x and nonnegative integer n; (ii) T k converges to T A E 1 ; Þ in the C 0 topology. Then for su ciently large k, there exists a homeomorphism h k :! such that h k T ¼ T k h k, and both h k and hk 1 converge to the identity id in the C 0 topology. Proof. By virtue of Theorem aþ in [13] and its proof, we see that any two homotopic expanding endomorphisms T and S of a compact manifold, there exists a unique homeomorphism h :! such that h T ¼ S h. On the other hand the condition (ii) of the lemma implies that there exists an integer k 0 such that T k is homotopic to T for any k b k 0. Therefore we conclude that for any k b k 0, there exists a unique homeomorphism h k :! such that h k T ¼ T k h k. Thus it remains to verify that both h k and hk 1 converge to the identity id in the C 0 topology. To this end we recall the strategy in [13]. We denote by p : ~! the universal covering space of endowed with the Riemannian metric which is the pull-back of the Riemannian metric on by the natural projection p. Let V be the set of continuous maps F : ~! ~ which are lifting of continuous maps of into itself such that g F ¼ F g holds for any covering transformation g. We define a function D : V V! R defined by DF; GÞ ¼sup x A ~ dpfx; pgxþ, where d is the distance function on induced by the Riemannian metric. We easily see that V; DÞ is a complete metric space. Consider homotopic elements T; S A E 1 ; Þ. We assume that S satisfies the condition (i) in the statement of

6 356 Takehiko orita and Yusuke Tokunaga the lemma. Let ~T and S ~ be the liftings of T and S, respectively. We note that the lifting of an expanding map on the universal covering space becomes a di eomorphism. Since T and S are homotopic, we see that S ~ 1 F ~T A V holds for any F A V. Therefore we can consider a mapping F : V! V by F 7! S ~ 1 F ~T. Then it is not hard to see that DF n F; F n GÞ a c 1 l n DF; GÞ for any F; G A V and n b 0. Therefore Contraction Principle yields that there exists a unique H A V with FH ¼ H. In fact, we can see that H is the lifting of the desired homeomorphism h satisfying h T ¼ S h in the same way as in the proof of Theorem 3 in [13]. Now we have DH; id ~ Þ a DH; Fn id ~ ÞÞ þ DFn id ~ Þ; id ~ Þ a c 1 l n DH; id ~ ÞþD ~ S n ~T n ; id ~ Þ a c 1 l n DH; id ~ ÞþXn k¼1 D ~ S k ~T k ; ~ S k 1Þ ~T k 1 Þ ¼ c 1 l n DH; Id ~ ÞþXn 1 DF k S ~ 1 ~TÞ; F k id ~ Þ a c 1 l n DH; Id ~ Þþ l cl 1Þ D S ~ 1 ~T; id ~ Þ: Thus if we choose n satisfying c 1 l n < 1, we have DH; id ~ Þ a l=l 1ÞÞl n =cl n 1ÞÞD S ~ 1 ~T; id ~ Þ. Consequently, we have d 0h; id Þ a l=l 1ÞÞl n =cl n 1ÞÞd 0 S; TÞ, where d 0 is the C 0 metric defined by d 0 f ; gþ ¼sup x A d f xþ; gxþþ for f ; g A C 0 ; Þ. By virtue of the above argument, we see that d 0 h k ; id Þ!0 if d 0 T k ; TÞ!0. By the definition the metric d 0, we also have d 0 hk 1; id Þ!0. Next we summarize the results corresponding to Lemma 3 and Lemma 4 in [8]. Lemma 5. Let T k be a sequence of elements in E 1 ; Þ satisfying the conditions (i) and (ii) in Lemma 4. Then we have the following. (1) Every m A TÞ is the weak limit of a sequence of Borel probability measures fm k g with m k A T k Þ for each k. (2) Let fm k g be a sequence of Borel probability measures with m k A T k Þ for each k. Then any weak accumulation point of the sequence belongs to TÞ. (3) Let fm k g be a sequence of Borel probability measures with m k A T k Þ for each k. If m k converges to m in the weak topology, we have lim sup k!y k¼0 ht k ; m k Þ a ht; mþ:

7 easures with maximum total exponent 357 Proof. By virtue of Lemma 4, we can prove the assertions (1) and (2) in the same way as the assertions (a) and (b) in Lemma 3 in [8]. If we verify that the entropy map TÞ C m 7! ht; mþ A R is upper semicontinuous for T A E 1 ; Þ, we obtain the assertion (3) in the same way as Lemma 4 in [8] by using Lemma 4. It remains to show the upper semicontinuity of entropy map for T. NotethatT is forward expansive, i.e. there exists b > 0 such that if dt n x; T n yþ < b for any n b 0, then x ¼ y. Thus it is easy to establish the upper semi-continuity of the entropy map in the same way as the proof of Theorem 8.2 in [16]. Finally we need the following. Lemma 6. If T k is a sequence of elements in E 1 ; Þ and converges to T A E 1 ; Þ in the C 1 topology, then for su ciently large k, T k satisfies the conditions (i) and (ii) in Lemma 4. oreover, any weak accumulation point of a sequence m k with m k A LT k Þ belongs to LTÞ. Proof. We can easily see that the first assertion is true. Since JT k Þ converges JTÞ uniformly on, the second assertion follows from the assertions (1) and (2) in Lemma 5 in the same way as Lemma 5 in [8]. 3. Construction of an auxiliary perturbation along a periodic orbit In this section we construct an appropriate perturbation of an given C 1 map along its periodic point. Before giving them we need some definitions and notation. Let U and V be neighborhoods of the origin in N-dimensional Euclidean space R N. Consider a C 1 map F : U! V which is locally di eomorphic around each point of U and F0Þ ¼0. We denote by e i Þ the standard orthonormal basis of R N with respect to the Euclidean metric. For our convenience we write qf xþ for the matrix representation of the tangent map DFxÞ with respect to the standard orthonormal basis e i Þ, i.e. the usual Jacobian matrix of F. We assume that U and V are endowed with Riemannian metrics g U and g V, respectively. Applying the Gram-Schmidt orthonormalization to the standard basis e i Þ, we obtain matrix valued smooth functions PÞ : U! GLN; RÞ and QÞ : V! GLN; RÞ such that a i xþ ¼ P N j¼1 P jiþxþe j and b i yþ ¼ P N j¼1 Q jiþyþe j form orthonormal frames axþ ¼ a i xþþ and byþ ¼b i yþþ of R N ¼ T x U and R N ¼ T y V with respect to the metrics g U and g V, respectively. The matrix representation of DFxÞ : T x U! T FxÞ V is given by QFxÞÞ 1 qfxþpxþ. Now we define JFÞ ¼jdet DFj : U! R by JFÞxÞ ¼jdet QFxÞÞ 1 qfxþpxþj for each x A U. Then it is

8 358 Takehiko orita and Yusuke Tokunaga easy to see that JF Þ is independent of the choice of orthonormal frames a and b. We can prove the following. Lemma 7. There exists a positive number C depending only on F such that for any e (0 < e < 1) and g > 0, there exists d 0 > 0 such that for any d with 0 < d < d 0, we can find a C 1 map F d : U! V satisfying the following properties with G d xþ ¼logJF d ÞxÞ=JFÞxÞÞ. (1) F d 0Þ ¼0. (2) sup x A U kf d xþ FxÞk < Cd, where kk denotes the Euclidean norm on R N. (3) sup x A U kqf d xþ qfxþk GL < Ce, where kk GL denotes the operator norm on GLN; RÞ induced by the Euclidean norm on R N. (4) F d xþ ¼FxÞ and G d xþ ¼0 if kxk b d. (5) G d 0Þ ¼e (6) sup x A U G d xþ < g þ e. Proof. Consider the functions exp 1=tÞ if t > 0; utþ ¼ 0 if t a 0 and vtþ ¼tanh t: For a; d > 0 small, we define a C y map D : R N! R N vanishing for x with kxk > d by D 1 xþ vax 1 =ud 2 ÞÞ DxÞ ¼B. A ¼ ud 2 kxk 2 Þ. B A ; 3:1Þ D N xþ vax N =ud 2 ÞÞ where the choice of a will be specified later. of the Jacobian matrix qdxþ is given as Observe that the i; jþ-th element qd i qx j xþ ¼ 2x j u 0 d 2 kxk 2 Þvax i =ud 2 ÞÞ þ a ud2 kxk 2 Þ ud 2 v 0 ax j =ud 2 ÞÞdijÞ; Þ 3:2Þ where dijþ is the Kronecker delta. jd i xþj a exp 1=d 2 Þ; and if d 2 a 1=2. Define F d by Therefore, we easily see that qd i xþ qx a 2 j d 3 exp 1=d2 ÞþadijÞ 3:3Þ F d xþ ¼FxÞþqF0ÞDxÞ: 3:4Þ

9 easures with maximum total exponent 359 Then the Jacobian matrix qf d is given by qf d xþ ¼qFxÞþqF0ÞqDxÞ: 3:5Þ From the first inequality in (3.3), there exists d 1 such that F d UÞ H V holds if d < d 1. Since D0Þ ¼0 by the definition (3.1), this yields the assertion (1). Using the first inequality in (3.3) again, we can find C 1 > 0 and d 2 > 0 with d 2 < d 1 depending only on F, sup x A U kf d xþ FxÞk < C 1 d holds for d < d 2. This yields the assertion (2). The assertion (4) is obvious since DxÞ ¼0 for x with kxk b d. In order to verify the other assertions, we consider the matrix representation of DF d xþ with respect to the orthonormal frames a and b. We have JF d ÞxÞ ¼jdetQF d xþþ 1 qfxþpxþþqf d xþþ 1 qf0þqdxþpxþþj: 3:6Þ Therefore we obtain JF d Þ0Þ ¼jdetQ0Þ 1 qf0þp0þþq0þ 1 qf0þqd0þp0þþj ¼jdetQ0Þ 1 qf0þp0þþj jdeti N þ qd0þþþj ¼ JFÞ0Þ1 þ aþ N : 3:7Þ Note that we have used the fact qd0þ ¼aI N, where I N is the identity matrix. Now we are in a position to specify the choice of a. If we put a ¼ expe=nþ 1, we have G d 0Þ ¼e. Then we can choose a number C 2 b C 1 depending only on F and a positive number d 3 depending only on F and e such that sup x A U kqf d xþ qfxþk GL < C 2 e holds for any d < d 3. Thus the assertion (3) is valid. It remains to show the last assertion (6). By virtue of the assertion (4), we have only to evaluate G d xþ for x with kxk a d. We will use an elementary inequality det L 2 det L a expnkl 1 1 k GL kl 2 L 1 k GL Þ 3:8Þ 1 for L 1 ; L 2 A GLN; RÞ. This is verified as follows. From L1 1L 2 ¼ I N þ L1 1L 2 L 1 Þ, we have kl1 1L 2k GL a 1 þkl1 1L 2 L 1 Þk GL. Since it is easy to see that jaj a kak GL holds for any matrix A A GLN; RÞ and for any eigenvalue a of A, we obtain jdet L 2 =det L 1 j¼jdet L1 1 L 2j a kl1 1 L 2k N GL a 1 þkl 1 1 L 2 L 1 Þk GL Þ N : Now the inequality (3.8) is an easy consequence of the inequality 1 þ l a e l for l b 0.

10 360 Takehiko orita and Yusuke Tokunaga First using the equation (3.6) we have det QFxÞÞ JF d ÞxÞ ¼JFÞxÞ det QF d xþþ deti N þ qfxþ 1 qf0þqdxþþ deti N þ qdxþþ jdeti N þ qdxþþj: 3:9Þ Since Q : V! GLN; RÞ is C y and kxk a d, we see that kqf d xþþ 1 k GL kqf d xþþ QFxÞÞk GL < C 3 d 3:10Þ for a positive constant C 3 depending only on F. see that In addition, it is not hard to ki N þ qdxþþ 1 k GL kqfxþ 1 qf0þ I N ÞqDxÞk GL < C 4 sup kqfxþ qf0þk GL x:kxkad 3:11Þ for a positive constant C 4 depending only on F. Next we evaluate jdeti N þ qdxþþj as follows. By virtue of the second inequality in (3.3), its i; jþ-th element satisfies ji N þ qdxþþijþj < dijþ expe=nþþc 5 d, where C 5 is a large constant depending only on F. Thus by using the definition of the determinant, we have jdeti N þ qdxþþj < e e=n þ C 5 dþ N þ XN 1 N! N jþ!j! ee=n þ C 5 dþ j C 5 dþ N j N jþ! j¼0 < e e 1 þ C 6 dþ 3:12Þ for large C 6 depending only on F. (3.12), we obtain Combining (3.8), (3.9), (3.10), (3.11) and JF d ÞxÞ JFÞxÞ < 1 þ C 6dÞ exp C 3 d þ C 4 sup kqfxþ qf0þk GL þ e x:kxkad This yields G d xþ < C 3 þ C 6 Þd þ C 4 sup x:kxkad kqfxþ qf0þk GL þ e: Hence, putting C ¼ C 2, we can find a positive number d 0 < d 3 such that all the assertions in the lemma are valid. The following theorem is the main result in this section.! :

11 easures with maximum total exponent 361 Theorem 3. Let T be an element in C 1 ; Þ such that JTÞxÞ 0 0 holds for every point x A. Assume that T has a periodic point x 0 with least period p. Then for any e with 0 < e < 1 and g > 0, there exists a positive number d 0 > 0 such that for each d with 0 < d < d 0, we can find an open neighborhood Ud i of T i x 0 for each i ¼ 0; 1;...; p 1. and an element T d of C 1 ; Þ satisfying the following. (1) T i x 0 ¼ Td ix 0 for each i ¼ 0; 1;...; p 1. (2) Ud i V U j d ¼ q if i 0 j. (3) Tx ¼ T d x for any x A n6 p 1 i¼0 U d i. (4) If 0 < d 0 < d, then we have U i d H U i 0 d. (5) For any charts U; jþ, V; cþ with TU H V and any compact set K H U, T d U H V we have sup x A K kc T d xþ ctxþk < C j; c; K d and sup x A K kqc T d j 1 ÞjxÞÞ qc T j 1 ÞjxÞÞk GL < C j; c; K e, where C j; c; K is a positive constant depending only on T, K, j, and c. (6) Define G d :! R by G d xþ ¼logJT d ÞxÞ=JTÞxÞÞ. Then we have G d Td ix 0Þ¼e for each i ¼ 0; 1;...; p 1. (7) sup x A G d xþ < g þ e holds. In particular, we can choose fud ig 0<d<d 0 so that diam d Ud i < d and 7 d:0<d<d0 Ud i ¼fTi x 0 g for each i ¼ 0; 1;...; p 1, where diam d AÞ is the diameter of A H with respect to the distance d of as before. Proof. For each i ¼ 0; 1;...; p 1, choose a chart V i ; j i Þ around T i x 0 and an open set U i such that T i x 0 A U i H U i H V i, TU i H V iþ1, and j i T i x 0 Þ¼0, where we regard U p and V p ; j p Þ as U 0 and V 0 ; j 0 Þ respectively. oreover we assume that V i s are mutually disjoint. Consider T i; iþ1 ¼ j iþ1 T ji 1 : j i U i Þ!j iþ1 V iþ1 Þ. Each j i V i Þ is endowed with the Riemannian metric g i which is the push-forward of the Riemannian metric of the manifold by j i. Obviously we can apply Lemma 7 to the case when U ¼ j i U i Þ, V ¼ j iþ1 V iþ1 Þ, F ¼ T i:iþ1, g U ¼ g i and g V ¼ g iþ1. Note that there exist a positive constant c depending only on and d 0 > 0 such that 0 < d < d 0 yields that B c d0þ H j i U i and diam d ji 1 B c d0þþ < d for each i ¼ 0; 1;...; p 1, where B r 0Þ ¼fxAR N : kxk < rg. We denote by T i; iþ1; d the map corresponding to F d. Define a map T i; d : U i! V iþ1 by T i; d ¼ jiþ1 1 i; iþ1; d j i. Now put Ud i i B c d0þþ and define a map T d by ( T d x ¼ T i; dx if x A U i for 0 a i a p 1; Tx if x A n6 p 1 i¼0 U i: We just verify that Ud i s and T d satisfy the assertions in the theorem. The assertions (1) (4) are obvious from the definition of Ud i s and T d. T d is of class

12 362 Takehiko orita and Yusuke Tokunaga C 1 on U i and coincides T on U i nud i by definition. Therefore T d is of C 1 on. The assertion (5) of the theorem immediately follows from the assertions (2), (3), and (4) in Lemma 7. By virtue of the assertions (5) and (6) in Lemma 7, we easily see the validity of the other assertions concerned with G d if we notice that JTÞxÞ ¼JT i; iþ1 Þj i xþþ and JT d ÞxÞ ¼JT i; iþ1; d Þj i xþþ by definition. From the definition of Ud i above, the last assertion of the theorem is clearly valid. As a corollary we obtain a modification of Lemma 7 in [8] which plays a crucial role in our argument. Corollary 1. Let T be an element in E 1 ; Þ and let x 0 be a periodic point of T with least period p. Then there exists e 0 > 0 such that for any e with 0 < e < e 0 and g > 0, there exists a positive number d 0 > 0 such that for each d with 0 < d < d 0, we can find an open neighborhood Ud i of T i x 0 for each i ¼ 0; 1;...; p 1 and an element T d of E 1 ; Þ satisfying (1) (7) in the statement of Theorem 3. In particular, we can choose fud ig 0<d<d 0 so that diam d Ud i < d and 7 d:0<d<d 0 Ud i ¼fTi x 0 g for each i ¼ 0; 1;...; p 1. Proof. Let us consider the case when the map T in Theorem 3 is an element in E 1 ; Þ. Note that if the map T d obtained in Theorem 3 could be an element in E 1 ; Þ for any e < 1, there would be nothing to be proved. On the other hand the second inequality in the assertion (5) of Theorem 3 guarantees that there exists e 0 > 0 depending only on T such that if e < e 0,thenT d is expanding. 4. Proof of theorems As we have constructed the perturbation in the previous section, the arguments in this section are almost the same as those in Section 3 and Section 4 in [8]. It is well known that one can define a distance function r on the set Þ of Borel probability measures on such that it induces the weak topology on Þ and satisfies the condition r1 lþm þ ln; mþ a l 4:1Þ For each k > 0 we consider the fol- for every m; n A Þ and l A 0; 1Þ. lowing sets. R k ¼fT A E 1 ; Þ : diam r LTÞÞ < kg ( ) S k ¼ T A E 1 ; Þ : sup m A LTÞ ht; mþ < kh top TÞ ;

13 easures with maximum total exponent 363 where h top TÞ denotes the topological entropy of T. We need the following to show that the properties (1) and (2) in Theorem 1 is generic. Proposition 1. For each k > 0, both R k and S k are open and dense subsets of E 1 ; Þ in the C 1 topology. Proof. First we show that E 1 ; ÞnR k and E 1 ; ÞnS k are closed in E 1 ; Þ in the C 1 topology. Assume that T k A E 1 ; ÞnR k converges to T A E 1 ; Þ in the C 1 topology. Since LT k Þ is compact, we can find m k ; n k A LT k Þ satisfying rm k ; n k Þ b k. By Lemma 6, their weak accumulation points are in LTÞ. Choosing subsequences if necessary, we may assume that there exist m; n A Þ such that rm k ; mþ, rn k ; nþ converge to 0. Thus we have diam r LTÞÞ b k. Hence we have T A E 1 ; ÞnR k. Next assume that T k A E 1 ; ÞnS k converges to T A E 1 ; Þ in the C 1 topology. By Shub s theorem in [13], we may assume each T k is topologically conjugate to T. Consequently h top T k Þ¼h top TÞ. Since LT k Þ is compact and the entropy map for an expanding map is upper semi-continuous (see Proof of Lemma 5 (3)), we can find n k such that ht k ; n k Þ¼ sup m A LTk Þ ht k ; mþ. Again choosing a subsequence if necessary, we may assume that rn k ; nþ converges to 0. Lemma 6 yields n A LTÞ. oreover, by Lemma 5 (3), we have ht; nþ b lim sup k!y ht k ; n k Þ b kh top TÞ: Hence we have T A E 1 ; ÞnS k. Now we have only to prove R k V S k is dense in E 1 ; Þ. Choose any T A E 1 ; Þ and e > 0 with 0 < e < e 0, where, e 0 > 0 is the positive number appearing in Corollary 1 depending only on T. By virtue of Lemma 3, there exists a periodic point x 0 with least period p such that the measure m 0 ¼1=pÞ P p 1 i¼0 d T i x 0 satisfies log JTÞdm 0 > ltþ ke 8 : 4:2Þ Let T d be the perturbation obtained by applying Corollary 1 to T and x 0 with g ¼ ke=8. Note that since e < e 0, T d A E 1 ; Þ. Then (6) in Theorem 3 and the inequality (4.2) yields that lt d Þ b log JT d Þdm 0 ¼ log JTÞdm 0 þ G d dm 0 e: 4:3Þ > ltþþ 1 k 8

14 364 Takehiko orita and Yusuke Tokunaga Take any strictly decreasing sequence d k a d 0 of positive numbers converging to 0, where d 0 is as in the statement in Theorem 3. For the sake of simplicity, we write T dk and G dk as T k and G k, respectively. Choose n k A LT k Þ. The first inequality in Theorem 3 (5), T k converges to T in the C 0 topology. Therefore, by taking a subsequence if necessary, we may assume that n k converges to n A TÞ. Thus we have G k dn k ¼ log JT k Þdn k ¼ lt k Þ log JTÞdn k log JTÞdn k > lt k Þ ltþ ke 8 4:4Þ for any k su ciently large. Combining (4.3) with (4.4) we have G k dn k > 1 k e: 4:5Þ 4 Recalling the open sets Ud i in Theorem 3, we see that U k ¼ 6 p 1 i¼0 U d i k satisfies that U kþ1 H U k and 7 y k¼1 U k ¼ O T x 0 Þ, where O T x 0 Þ¼fx 0 ; Tx 0 ;...; T p 1 x 0 g. Now we evaluate n k U k Þ as follows. n k U k Þ b 1 þ k 1 e 1 G k dn k ¼ 1 þ k 1 e 1 G k dn k 8 U k 8 > 1 þ k 1 1 k > 1 3k : 4:6Þ In the above, the first inequality follows from (7) in Theorem 3 with g ¼ ke=8, the equality in the first line follows from (3) in Theorem 3, and the first inequality in the second line follows from (4.5). Using the well known fact that n k converges to n in the weak topology if and only if lim sup k!y n k FÞ a nfþ holds for any closed set F, we can easily see from (4.6) that nu k Þ b 1 3=8Þk for each k. Thus we obtain no T x 0 ÞÞ b 1 3=8Þk. Hence we can write n as n ¼1 3k=8Þm 0 þ3k=8þ^n for some ^n A TÞ. From the condition (4.1), this yields lim k!y rn k; m 0 Þ¼rn; m 0 Þ a 3k 8 : 4:7Þ In addition, by Lemma 5 (3), we have lim sup ht k ; n k Þ a ht; nþ ¼ 1 3k k!y 8 ht; m 0 Þþ 3k ht; ^nþ 8 a 3k 8 h toptþ: 4:8Þ

15 easures with maximum total exponent 365 Hence we have shown that the inequalities (4.7) and (4.8) hold for any accumulation point as d! 0 of the sets LT d Þ. This implies that for any e with 0 < e < e 0 Þ, there exists d 1 such that T d A R k V S k whenever d < d 1. Since the second inequality in (5) in Theorem 3 holds, we can choose e so that T d belongs to a given neighborhood of T in the C 1 topology. To prove that the property (3) in Theorem 1 is generic, we show the following. set Proposition 2. For a nonempty closed proper subset Y of, consider the 1 YÞ ¼fT A E 1 ; Þ : supp m H Y holds for some m A LTÞg: Then 1 YÞ is a closed and nowhere dense subset of E 1 ; Þ in the C 1 topology. Proof. First we show that 1 YÞ is closed in E 1 ; Þ. Assume that T k A 1 YÞ converges to T A E 1 ; Þ in the C 1 topology. Note that the sequence T k satisfies the conditions (i) and (ii) in Lemma 4. Let m k A LT k Þ satisfy supp m k H Y and let m be an accumulation point of them. We may assume that m k converges to m in the weak topology. From Lemma 6 m turns out to be an element in LTÞ. oreover, since Y is closed, we have myþ b lim sup k!y m k YÞ ¼1. Consequently we see myþ ¼1 and 1 YÞ is closed. Next we show that 1 YÞ is nowhere dense in E 1 ; Þ. If 1 YÞ is not empty, take any T A 1 YÞ and Y y ¼ 7 y j¼0 T j Y. Choose any e with 0 < e < e 0, where e 0 is the same as in Corollary 1 as before. Note that if m A TÞ satisfies myþ ¼1, we also have my y Þ¼1. By Lemma 3, we can find a periodic point x 0 of T with least period p satisfying Y y V O T x 0 Þ¼q and log JTÞdm 0 > ltþ e; 4:9Þ where m 0 denotes the T-invariant probability measure supported on O T x 0 Þ as before. We can find a positive integer N 0 such that Y N0 ¼ 7 N 0 j¼0 T j Y satisfies Y N0 V O T x 0 Þ¼q. For each k > 0 consider the set Y N0 ; k ¼ 6 S A E 1 ; Þ:d 0 S; TÞak N 0 7 S j Y; j¼0 where d 0 S; TÞ is the usual C 0 -metric defined by d 0 S; TÞ ¼sup x A dsx; TxÞ.

16 366 Takehiko orita and Yusuke Tokunaga It is not hard to see that 7 k>0 Y N0 ; k ¼ Y N0. Now applying Theorem 3 to T and x 0 with g ¼ 1, we construct C 1 map T d. As stated in Corollary 1, there exists e 0 > 0 depending only on T such that T d is expanding for any d su ciently small. Note that by virtue of the first inequality in (5) in Theorem 3, there exists C T > 0 depending only on T such that d 0 T d ; TÞ < C T d. Choose d > 0 so that inf inf dt i x 0 ; yþ > d: y A Y N0 ; C T d 0aiap 1 Since the diameter of each Ud i in Theorem 3 is less than d, the assertion (3) in Theorem 3 yields that T ¼ T d on the set Y N0 ; C T d. Consequently T ¼ T d on the set Y N0 ; C T d and O T x 0 Þ is also a periodic orbit of T d by Theorem 3 (1). Note that if m A T d Þ satisfies myþ ¼1, we have my N0 ; C T dþ¼1. In particular, for a Borel probability measure with my N0 ; C T dþ¼1, m is T-invariant if and only if it is T d -invariant. We show that m A T d Þ with myþ ¼1 cannot be an element of LT d Þ. Assume that m A T d Þ satisfies myþ ¼1. Then we have log JT d Þdm ¼ log JTÞdm a ltþ 4:10Þ since T ¼ T d on Y N0 ; C T d. 3 (6) yields ltþ < On the other hand, the inequality (4.9) and Theorem log JTÞdm 0 þ e ¼ Combining this with (4.10), we arrive at log JT d Þdm ¼ log JTÞdm a ltþ < log JT d Þdm 0 : log JT d Þdm 0 a lt d Þ: Choose any T in 1 YÞ and consider any neighborhood of T in the C 1 topology. If e > 0 and d > 0 are small enough, the map T d constructed in Corollary 1 can be found in the neighborhood since the assertion (5) in Theorem 3 holds. The argument above implies that if d > 0 is su ciently small, we see that T d B 1 YÞ. Hence 1 YÞ has no interior points. Proof of Theorem 1. We can easily verify that the set of T A E 1 ; Þ satisfying the property (1) and the set of T A E 1 ; Þ satisfying the property (2) are given by 7 y n¼1 R 1=n and 7 y n¼1 S 1=n, respectively. Thus, properties (1) and (2) are generic by Proposition 1. Since is a compact metric space, we can find a countable family fy n g of closed proper subsets of such that any closed proper subset Y of turns

17 easures with maximum total exponent 367 out to be a subset of Y n for some n. Indeed, let fb n g be a countable open base of consisting of open balls. We may assume that the radius of each B n is so small that B n is a proper subset of. Putting Y n ¼ nb n, we obtain the desired family of closed proper subsets of. We easily see that the set T A E 1 ; Þ satisfying the property (3) is given by 7 y n¼1 E1 ; Þn 1 Y n ÞÞ. Since each 1 Y n Þ is a closed and nowhere dense subset of E 1 ; Þ by Proposition 2, we arrive at the desired result. r Finally we prove Theorem 2. To this end we consider the symbolic dynamics S; sþ in Lemma 2 and a function V : S! R which is d y -Lipschitz continuous. Denote by sþ the set of s-invariant Borel probability measures on S. Put ls; V; nþ ¼ Vdn; S ls; VÞ ¼sup S Vdn : n A sþ and denote by Ls; V Þ the set of measures in sþ satisfying ls; V Þ¼ ls; V; nþ. Since sþ is compact with respect to the weak topology, the continuity of V yields that Ls; V Þ is nonempty. We need the following fact that can be find in Savchenko [11]. We state it with proof for the reader s convenience. Lemma 8. There exists a unique nonnegative d y -Lipschitz continuous function j : S! R satisfying the following properties. (1) V a j s j þ ls; VÞ on S. (2) For any nonnegative function c : S! R satisfying V a c s c þ ls; VÞ on S, we have j a c on S. (3) For n A Ls; VÞ, we have V ¼ j T j þ ls; VÞ on supp n. Proof. We just follow the same lines as the proof of Proposition 11 in [5]. We may assume that ls; V Þ¼0. Define j by jxþ ¼supfS n VhÞ : n b 0 and s n h ¼ xg; where S n VhÞ ¼ P n 1 j¼0 Vs j hþ for n b 1 and S 0 ¼ 0. First we show that 0 a jxþ < þy for each x A S. Since S 0 V ¼ 0, we have only to show that the set fs n VhÞ : n b 0 and s n h ¼ xg is bounded from above. From Lemma 2 (6), There exists an integer n 0 > 0 such that for any h A S and n b 0 we have s nþn 0 Zh½0; n 1ŠÞÞ ¼ S, where h½0; n 1Š is the word h 0 h 1...h n 1 and Zh½0; n 1ŠÞ is the cylinder set fz : z j ¼ h j for 0 a j a n 1g. Thus for each

18 368 Takehiko orita and Yusuke Tokunaga h with s nþn 0 h ¼ x, we can find a fixed point h 0 of s nþn 0 in Zh½0; n 1ŠÞ. Note that ls; VÞ ¼0 yields S nþn0 Vh 0 Þ a 0. On the other hand we have js nþn0 VhÞ S nþn0 Vh 0 Þj a ½VŠ y nþn X0 1 j¼0 d y s j h; s j h 0 Þ a 1 1 y þ n 0 ½VŠ y ; where ½VŠ y is the Lipschitz constant of V. Thus we easily see that jxþ a 1=1 yþþn 0 Þ½VŠ y. Next we show that j is d y -Lipschitz continuous. Note that if x and h in S satisfy x 0 ¼ h 0, then z 0 z 1...z n 1 x A S yields z 0 z 1...z n 1 h A S for a word z 0 z 1...z k 1, where w w 0 denotes the concatenation of words w and w 0. Therefore, we have js n Vz 0 z 1...z n 1 xþ S n Vz 0 z 1...z n 1 hþj a 1=1 yþþ½vš y d y x; hþ. Thus j is continuous. oreover, we see easily that jjxþ jhþj a max max jjzþj; ½VŠ y d y x; hþ z A S 1 y holds for any x; h A S. By definition the inequality V a j s j is valid on S. Now proof of (1) is complete. Next, let c be a nonnegative function satisfying V a c s c on S. Then for any pair x; hþ A S S with s n h ¼ x, we have cxþ b S n VhÞþchÞ. This clearly implies that the assertion (2) is valid. Finally, let n A Ls; VÞ. Combining the fact that Ð Sj s j VÞdn ¼ 0 with the assertion (1), we obtain j s j V ¼ 0 n-a.e. The continuity of V and j implies that this equality holds everywhere on supp n. Proof of Theorem 2. Put V ¼ log JTÞp. Note that since T is of class C 2, it is easy to see that V is a d y -Lipschitz continuous function on S. First we verify that p Ls; VÞÞ ¼ LTÞ as follows, where p is the pushforward of p defined by fdp n ¼ f p dn for f A CÞ. Note that p is surjective since so is p. oreover, we have p sþþ ¼ TÞ by Lemma 2 (5) (see, for example, Proposition 3.2 and Proposition 3.11 in [6]). Thus for any m A LTÞ, there exists n A sþ such that p n ¼ m and ltþ ¼ls; V; nþ. Clearly we have ltþ ¼ log JTÞdm ¼ Vdnals; VÞ: 4:11Þ S S

19 easures with maximum total exponent 369 On the other hand, for any n 0 A sþ, we have Vdn 0 ¼ log JTÞdp n 0 a ltþ: S Thus we obtain ls; VÞ a ltþ. Combining this with (4.11), we conclude that Vdn¼ls; VÞ ¼lTÞ: S Next we show that if there exists an element in LTÞ with support, then TÞ coincides LTÞ, namely, ltþ ¼ Ð log JTÞdm holds for any m A TÞ. Let m be an element in LTÞ with support. Then from the argument above, we find n A Ls; VÞ such that p n ¼ m. We can show that supp n ¼ S. Indeed, choose any x A S, then we see that 7 n 1 k¼0 T j int R xj 0q for any positive integer n since the arkov partition satisfies (3), (4), and (5) in Lemma 1. Therefore Zx½0; n 1ŠÞ I p 1 7 n 1 j¼0 T j int R xj Þ holds. Thus we have nzx½0; n 1ŠÞÞ b np 1 7 n 1 j¼0 T j int R xj ÞÞ > 0 for all n b 0. This yields supp n ¼ S. Now by virtue of Lemma 8, there exists a d y -Lipschitz continuous function j : S! R satisfying V ¼ j s j þ ls; VÞ on S. This implies that ls; VÞ ¼ Ð S Vdnholds for any n A sþ. Consequently, we have ltþ ¼ Ð log JTÞdm for any m A TÞ. For any neighborhood of T in the C r topology, it is not hard to construct an element S A E r ; Þ such that there exists a fixed point x 0 and a periodic point y 0 with least period p b 2 such that log JSÞx 0 Þ 0 1=pÞ log JS p Þy 0 Þ. Thus the set F r ; Þ of the maps T such that LTÞ ¼TÞ is nowhere dense. In addition clearly it is closed. Therefore we arrive at the desired result. r Acknowledgement The authors would like to thank the anonymous referee for helpful comments and suggestions. References [ 1 ] T. Bousch, La condition de Walters, Ann. Sci. École. Norrm. Sup. 34 (2001) [ 2 ] T. Bousch and O. Jenkionson, Cohomology classes of dynamically non-negative C k functions, Invent. ath. 148 (2002) [ 3 ] R. Bowen, Equilibrium states and the ergodic theory of Anosov di eomorphisms, Lecture Note in ath. 470, Springer Verlag, Berlin-Hedelberg-New York [ 4 ] J. Brémont, Entropy and maximizing measures of generic continuous functions, C. R. ath. Acad. Sci. Paris 346 (2008)

20 370 Takehiko orita and Yusuke Tokunaga [ 5 ] G. Contreras, A. Lopes, and Ph. Thieullen, Lyapunov minimizing measures for expanding maps of the circle, Ergod.Th. and Dynam. Sys. 21 (2001) [ 6 ]. Denker, Ch. Grillenberger, and K. Sigmund, Ergodic theory on compact spaces, Lecture Note in ath. 527, Springer Verlag, Berlin-Hedelberg-New York [ 7 ] O. Jenkinson, Ergodic optimization, Discrete Contin. Dyn. Syst. 15 (2006) [ 8 ] O. Jenkinson and I. D. orris, Lyapunov optimizing measures for C 1 expanding maps, Ergod.Th. and Dynam. Sys. 28 (2008) [ 9 ] K. R. Parthasarathy, On the category of ergodic measures, Illinois J. ath. 5 (1961) [10] D. Ruelle, Thermodynamic formalism, 2nd. ed. Cambridge Univ. Press, Cambridge [11] S. V. Savchenco, Homological inequalities for finite topological arkov chains, Funktsional. Anal. i Prilozhen 33 (1999) [12] K. Sigmund, Generic properties of invariant measures for Axiom A di eomorphisms, Invent. ath. 11 (1970) [13]. Shub, Endomorphisms of compact di erentiable manifolds, Amer. J. ath. 91 (1969) [14] Y. Tokunaga, Lyapunov optimizing measures for C 1 expanding maps of the n-torus, aster Dissertation of Hiroshima University 2010 (in Japanses). [15] P. Walters, Invariant measures and equilibrium states for some mappings which expand distances, Trans. Amer. ath. Soc. 236 (1978) [16] P. Walters, Introduction to ergodic theory, Springer Verlag, Berlin-Hedelberg-New York Takehiko orita Department of athematics Graduate School of Science Osaka University Toyonaka Osaka , Japan take@math.sci.osaka-u.ac.jp Yusuke Tokunaga Department of athematics Graduate School of Science Osaka University Toyonaka Osaka , Japan y-tokunaga@cr.math.sci.osaka-u.ac.jp