Brin, M. I.

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AstérisqueM. I. BRINNonwandering points of Anosov diffeomorphismsAstérisque, tome 49 (1977), p. 11-18<http://www.numdam.org/item?id=AST_1977__49__11_0>© Société mathématique de France, 1977, tous droits réservés.L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/Publications/Asterisque/) implique l’accord avec les conditions générales d’uti-lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ouimpression systématique est constitutive d’une infraction pénale. Toute copieou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programmeNumérisation de documents anciens mathématiqueshttp://www.numdam.org/Société Mathématique de France Astérisque 49 (1977) p.11-18 HONWANDERING POINTS OF MOSOV DIFFEOMORPHISMS. M.I.Brin. There are two well-known conjectures in the Anosov diffeomor-phism (C-diffeomorphism) theory. Let f be a C-diffeomorphism of a smooth compact Riemannian manifold M n. Then (seep] and [ 2 ] ) : 1 . The set NW(f) of nonwandering points of f is M n . 2 . The covering manifold M is homeomorphic to R n. Many of the results in the C-diffeomorphism theory were got assuming that ail the points are nonwandering. The condition NW(f) = M n implies for instance that a C-diffeomorphism f is topo-logically transitive (or simply "transitive"), the set Per(f) of ail the periodic points of f is dense in M n, every stable or un-stable layer is dense in M n. Ail the existing examples of C-diffeomorphisms act on nilma-nifolds (in particular on tori) and on their generalizations - in-franilmanifolds. That 1s why proving the second statement (or find-ing the conditions under which it is valid) is the natural fist step in the classification of C-diffeomorphisms. The sufficient conditions for the set NW(f) to coincide with M n and for the covering manifold M to be homeomorphic to R11 are stated in this paper. Every diffeomorphism f of a compact Riemannian manifold M in­duces a linear operator fm in the space of continuous vector fields: — 1 (f^v)(x) = df f-1 xv(f x ) . C-diffeomorphisms are characterized (see f33) by the fact that the spectrum S of the complexification of the 11 M.I. BRIN operator f̂ . doesn't intersect the unit circle. So S is contained in the interiors of two rings with the radii 0<T^<T2<Î and 1 < R 2 < R 1 < oo • There exist such a constant 0 <:C1 <: co that for every positive integer n q 1 r n l|v|| < l|dfnv(I < C 1 r£ ||v|| if v € E s C' 1 R^ ||v|l < |(dfnv[| ^ C 1 Rf ||v|| if v ^ Eu here E s and E u are respectively the stable and unstable subbundles of the tangent bundle TM. Let's say that the correspondence mapping for the stable W s and unstable W u foliations can be infinitely extended if for every three points X É M , y € W s ( x ) , Z6ïlfu(x) there exists such a continuous 2 mapping g of the ufilt square I into M that: 1 ) g ( 0 , 0 ) = x, g ( 0 , 1 ) = y, g ( 1 , 0 ) = z; 2 ) g(t,.) is a continuous curve on a stable layer for every fixed t e l ; 3 ) g(.,t) is a continuous curve on an unstable layer for every fixed t é l . The existence of such a map-ping is obvious if the distancies between x,y,z measured along the corresponding layers are small enough. According to the given de-finition this mapping can be extended beyond the boundaries of a small neibourhood. , p ln r.. In r 0 In R,. PROPOSITION 1 . If 1 + g - | 2 > 1 H ^ W o r 1 + i r ^ > ^ ( * * ) then the correspondence mapping for the foliations W s and W u can be infinitely extended. Let d 0 and d„ dénote the distancies induced by the internai s u metrics of stable and unstable layers, and l(c) be the length of a piecewise smooth curve c. LEMMA. 2 . Let c be a smooth curveonan unstable layer Wu(x,j) (respec-tively W8(x.j)) Connecting the points x^ and y^, and let x 2 € Ws ( x 1 ) (W u(x 1)). Suppose that there exists such a continuous mapping g(c,x 2) = g of the unit square I into M that: 1 ) g(t ,0 ) = c(t), 12 NONWANDERING POINTS g(0 , l ) = x 2 ; 2) the points g C t . t - j ) and g ( t , t 2 ) belong to the stable (unstable) layer; 3 ) g(t,1)€ W u ( x 2 ) ( Ws ( x 2 ) ) ; 4) the restriction of g to the set I * £ o J i s bijective. Then there are such constants b 1 , b 2 > 0 that i f m 0(g) = max _ c l C g C t * * - ) , g ( t , t p ) ) < b 1 -min(l(c),b p) s 0 £ t <1, 0 4 ^ < t 2 ^ 1 1 (respectively for d y) then there is such a curve c*Connecting x 2 and y 2= g(1,1) on the layer Wu ( x 2 ) (Ws ( x 2 ) ) that l(c') < 2 l(c). Proof. If l(c) is sufficiently small then the statement of the lemma follows from the transversality and continuity of the stable and unstable foliations. L e , there are such b^ > 0 and q > 0 that the statement of lemma i s true i f l(c) < q and m ( g ) < b 1 l ( c ) . Let s i i b 2= T̂ q. The curve c can be divided into segments c^ with the ends g(z^,0) and g ( z^ +^,C), «^q^l(c^)<q. Let ' s consider for each segment c^ in the capacity of c^ the shortest géodésie on the layer W u ( x 2 ) (Ws ( x 2 ) ) Connecting g ( z i,1) and g ( z i + 1 , 1 ) , Since l(c|) <2-l(c i) we have l(cf ) < 2* l(c). Q.e.d. Let f s say that two unstable layers W u ( x 1 ) and Wu ( x 2 ) are ( £ ,r)-close at points x^ and x 2 € Ws ( x ^ ) i f there exists such a continuous mapping h:B x I —* M of the direct product of the unit k-ball and unit segment into M that: 1) h(0,0) = x^, h(0,1) = x 2; 2) h(0 , t ) is the géodésie segment Connecting x^ and x 2 on the layer W s ( x 1 ) ; 3 ) h(B k,C) is the r-ball on the layer Wu ( x 1 ) with the centre x < 1; 4) h ( Bk , t ) C W u(h(0, t ) ) ; 5) h(y,I)e W s(h(y,0) ) and d s ( h(y , t 1),h(y , t 2 ) ) ^ 6 for every y e B k , t r t 2 < = I; 6) the restric-tion of h on either (B k x{û}) or ( { o j x I) is bijective. 13 M.I. BRIN Let d(r,£ ) be tiie supremum of those d that for every x^ and x 2 é Ws ( x 1 ) with d s(x 1,x 2)^ïï the layers Wu(x 1 ) and Wu ( x 2 ) are U,r) -close at the points x 1 a and x 2 . It follows from the continuity of the stable and unstable foliations that d(r, £ ) > 0 . Let 1 s déno-te by £ 0 the diameter of the neighbourhood with product structure for f (see [4j ). LEMMA 3* Let r be greater than L0 and £ = b^minC 60 > b 2 ) . Then there is such a constant C > 0 independent of r that d(r, £ ) ̂  C exp (-ln( ^ ) ). Proof. It follows from the compactness of M and continuity of the foliations that there are such points x^, x 2 é Ws ( x ^ ) , y. é W u ( x , ) , i=1,2 that 1) d s ( x r x 2 ) = d(r, £ ) = d; 2) the layers Wu(x 1 ) and W u ( x 2 ) are ( £,r)-close at the points x 1 and x 2 ; 3) d u ( x 1 , y 1 ) ^ r , d s ( y r y 2 ) = £ Let c g= c g(x 1,x 2) be the géodésie segment Connecting x^ and x 2 on the layer Ws(x^) and let c u= c^x^y^j) be the géodésie seg-ment Connecting x^ and y^ on the layer Wu(x.j). The points , y^, h(0,t), the curve c (x^y^) and the restriction of h on the set e x c, satisfy the condition of Lemma 2 for every t € 1 . Let s u h(z,C) = y^. It follows then from Lemma 2 that d u(h(0, t) ,h(z, t) )<c2r for every tel. Suppose n is the minimal integer for which d u(f"nh(0,t),f" nh(z,t)) ^ b ^ é = ei , t é l . There is such a constant C f= C f ( £ 1 ) that n <(ln r)-(ln R 2 ^ ~ 1 + C'* The points f ^ x ^ f""nx2, f"ny-| and the curve c = f~ n(c g) satisfy the condition of Lemma 2. That 1s why there's a pieeewise smooth curve c f, l(c l)<2»l(c) Connecting the points f""ny,| and f" ny 2 and c C W s ( f n y i ) . Now l(c')>C~1 r 2n d s ( y r y 2 ) and l(c)^ C 1r~nd g(x 1 , x 2 ) . 14 NONWANDERING POINTS Thèse inequalities imply the statement of Lemma 3. Proof of Proposition 1. Let y<£W s(x), Z é W u ( x ) , d o(x,y) = a, s d u(x,z) = b. The distancies between fnx, f ûy, and f nz satisfy the following inequalities: d g(fnx, f ny) ̂  C ^ a , d u(fnx,f nz) ̂  C ^ b , Let 1s verify that for sufficiently large n one has d(d u(fnx,f nz), i ) z d s(fnx,f ny). Indeed in accordance with Lemma 3 d ( d u ( fn x , f n z ) , £ ) C e x p ( - l n ^ l n C C ^ b H l n R 2)~ 1) ^ 1 ^ d g ( fn x , f n y ) C ^ a c c 2 * C7a" exp (-n rl n R1 'ln R 2 r 2 ln r 2 ) ) . It is clear now that the statement of Proposition 1 follows from (*). The inequality is treated by analogy. Q.e.d. REMARK 4. If the correspondence mapping can be infinitely extended then every stable layer W 8(x) intersects every unstable layer W u(y). Indeed let 1 s connect x and y by a smooth curve c and divide this curve into arcs (with the ends x i and Kc^,) ̂  ^ 6 C ( <5C is the diameter of the product structure neighbourhood). If the intersection W u(x i)D Ws(x) isn't empty then neither is the inter-section W u ( x i + 1 ) H Ws(x)• So the induction argument shows that W u(y)/lW s(x) * 0. THEOREM 5. Let f:M n-*M n be a C-diffeomorphism. Suppose that the correspondence mapping for the stable and unstable foliations can be infinitely extended, Then the set NW(f ) of nonwandering points coincides with M n and the universel covering manifold M is homeo-morphic to R n. Proof. In accordance with the Smale spectral theorem (see [2] ) the 15 M.I. BRIN set NW(f) can be represented as a finite union of disjoint closed sets called basic sets. Let B^ be a repeller and B 2 - an attractor. It's well known that B^ consists of stable layers and B 2 consists of unstable layers. If x 6? B and y é B 2 then Ws ( x ) c B 1 and Wu(y)c B 2 But W s(x)f|W u(y) £ 0 (see Remark 4 ) , hence B^/1B2 ji 0. It follows that there exists only one basic set B = M n. Now let x ^ M n . There are (see [1]) two diffeomorphisms F s: R k - » W s ( x ) and F u: R n ~ k - * W u ( x ) . If y*W u(x) and z*W s(x) then according to the infinité extendability of the correspondence map-ping there's a continuous function g:I —* M • Let fs dénote g O , 1 ) = q(y>z). It's easy to verify that although xx g isn't uniquely defined the point q(y,z) is independent of the choise of g. This statement is obvious if the distancies d(x,y) and d 0(x,z) u s are sufficiently small. Indeed q(y,z) is the unique point of inter-section of the local layers in this case. Almost the same argument shows that q(y,z) is independent of g if either d (x,y) or d 0(x,z) u s is small enough, and to achieve such a situation it is sufficient-ly to apply the itérations of f• Let 1 s define the mapping p: R n=R kx R11"*—* M n by the formula p(v,w) = q(P u(w),F s(v)). It follows from the continuity and trans-versality of the stable and unstable foliations that p is a local homeomorphism. I.e. for every point x e R n there's such a neighbour-hood U(x) that the restriction p|U(x) maps it homeomorphically on-to a neighbourhood of p(x). Let 1s show that for every curve c(t), te [0,1] on M n and for every point u p " 1 ( c ( 0 ) ) there is the uni-que curve c(t) on R n with p(c(t)) = c(t) and c(0) = x. Since p is a local homeomorphism it is sufficient to prove this statement for 16 NON WANDERING POINTS piecewise smooth curves with the length less than £ 0 • Let c be such a curve, c(C) = x, p(x) = x, x = (x x̂" ), x \ e R k , x é R n " k . S U S U Since c is contained in the product structure neighbourhood with the centre x there are such curves c e W s(x) and c c W u(x) that s u the intersection point of the local layers wV" (c (t)) and 10C s W^ Q C(c u(t)) coincides with c(t) for every t £ [0,1]. To construct the curve c it is sufficient toapply the property of the infinité correspondence mapping extension to the points c(t), c Q(t) and p(0,x u). The uniqueness of this curve follows from the fact that p is a local homeomorphism. So p is the covering mapping. Q.e.d. COROLLARY 6. Let f be a C-diffeomorphism of M n and ln R p ln r 1 ln r 9 ln R 1 e i t h e r 1 + _ ^ > _ _ i ( ^ ) o r 1 + _ 2 > _ i ( ^ > ) . Then NW(f) = M n and the covering manifold for M n is homeomorphic to R n. Let A be a hyperbolic automorphism of a nilpotent Lie algebra N inducing a hyperbolic diffeomorphism of a compact nilmanifold. The eigenvalues of A are contained in two rings with the radii C < < r 2 < 1 and 1 < R^ < R^ < co . PROPOSITION 7. If either 1. a) 1 + ln R 2 ln ln R 1 > ln r 2 ' b) r 2R 1 ^ 1 or 2. a) 1 + ln r 2 ln R 1 ln r 1 > ln R 2 ; b ) r-,R2 s 1 then N is a commutative algebra. Proof. Let 1 s dénote V q=[x 6 N/ 3 k: (A-qE)k= OJ. It is easy to show that [ V Q»V--JÇV-- . Let condition 1 be true and N be noncommuta-tive. The intersection of the uniform discrète subgroup and the 17 M.I. BRIN commutator group is a uniform discrète subgroup of the commutator group (see [ 5 ] )• That 1s why among the eigenvalues of the restric-tion of A on the commutator algebra there are those numbers of mo-dulo less and greater than 1. It is easy to show that there exists such a non-zero vector x £ V r > jr|< 1 that x =fy»zj» y 6' V^, zc-V-, There is an alternative: either Jq^jqf^l or one of thèse num-2 bers is greater than 1. In the first case we get r^ <: which con-tradicts condition 1. Let's consider the second case. Let )q|>1. Since lnjr/ = lnfq| + ln/qf then ln r 2 > ln r^ + ln R 2 which also contradicts condition 1. Condition 2 is treated on the analogy. The proposition is proven. It seems that the conditions r^R^ ^ 1 and r^R^ ^1 are inessen-tial. A.Katok noted that conditions 1a and 2a aren ft valid for the well-known Smale examples of C-diffeomorphisms of nilmanifolda CONJECTURE. If a C-diffeomorphism f:M—>M possesses the property (*) or (*#) (see proposition 1 and corollary 6) then M is a torus. I wish to express my gratitude to A.Katok and D.Anosov. REFERENCES 1. Anosov L., Géodesic Flows on Closed Riemann Manifolds with Ne-gative Curvature, Proc. Steklov Inst.of Math., No 90 (1967). 2. Smale S., Differentiable Dynamical Systems, Bull.Amer.Math.Soc., 73 (1967), 747-817. 3. Mather I., Characteristic of Anosov Diffeomorphisms, Proc.Honi¬ pelijke Niderlandske Academie Oneweter schappen, Amsterdam, s.A., 71 (1968), 479-483. 4. Hirsch M., Palis J., Shub M., Neighbourhoods of Hyperbolic Sets, Inventions Math., 9 (1970), 121-134. 5. Malcev A., On a Class of Homogeneous Spaces, Amer.Math.Soc Transi., 39 ( 1 9 4 9 ) . M.I. Brin Prospekt Kalinina g 33 kb 55 Moscow 121099 - USSR 18 

Nonwandering points of Anosov diffeomorphisms

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Nonwandering points of Anosov diffeomorphisms

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