Lipschitz shadowing implies structural stability Sergei Yu. Pilyugin Sergei B. Tihomirov Abstract We show that the Lipschitz shadowing property of a diffeomorphism is equivalent to structural stability. Bibliography: 16 titles. 1 Introduction The theory of shadowing of approximate trajectories (pseudotrajectories) of dynamical systems is now a well developed part of the global theory of dynamical systems (see, for example, the monographs [1,2]). This theory is closely related to the classical theory of structural stability. It is well nown that a diffeomorphism has the shadowing property in a neighborhood of a hyberbolic set [3, 4] a structurally stable diffeomorpism has the shadowing property on the whole manifold [5 7]. Analyzing the proofs of the first shadowing results by Anosov [3] Bowen [4], it is easy to see that, in a neighborhood of a hyperbolic set, the shadowing property is Lipschitz ( the same holds in the case of a structurally stable diffeomorpism, see [1]). At the same time, it is easy to give an example of a diffeomorphism that is not structurally stable but has the shadowing property (see [8], for example). Thus, structural stability is not equivalent to shadowing. One of possible approaches in the study of relations between shadowing structural stability is the passage to C 1 -interiors. At present, it is nown that the C 1 -interior of the set of diffeomorphisms having the shadowing property coincides with the set of structurally stable diffeomorphisms [9]. Later, 1 Faculty of Mathematics Mechanics, St. Petersburg State University, University av. 28, 198504, St. Petersburg, Russia 2 Department of Mathematics, National Taiwan University, No. 1, Section 4, Roosevelt Road, Taipei 106, Taiwan 3 The research of the second author is supported by NSC (Taiwan) 98-2811-M-002-061 1
a similar result was obtained for the orbital shadowing property (see [10] for details). Here, we are interested in the study of the above-mentioned relations without the passage to C 1 -interiors. Let us mention in this context that Abdenur Diaz conjectured that a C 1 -generic diffeomorphism with the shadowing property is structurally stable; they have proved this conjecture for so-called tame diffeomorphisms [11]. Recently, the first author has proved that the so-called variational shadowing is equivalent to structural stability [8]. In this short note, we show that the Lipschitz shadowing property is equivalent to structural stability. 2 Main result Let us pass to exact definitions statements. Let f be a diffeomorphism of class C 1 of an m-dimensional closed smooth manifold M with Riemannian metric dist. Let Df(x) be the differential of f at a point x. For a point p M, we denote p = f (p), Z. Denote by T x M the tangent space of M at a point x; let v, v T x M, be the norm of v generated by the metric dist. We say that f has the shadowing property if for any ɛ > 0 there exists d > 0 with the following property: for any sequence of points X = {x M} such that dist(x +1, f(x )) < d, Z, there exists a point p M such that dist(x, p ) < ɛ, Z (1) (if inequalities (1) hold, one says that the trajectory {p } ɛ-shadows the d- pseudotrajectory X). We say that f has the Lipschitz shadowing property if there exist constants L, d 0 > 0 with the following property: For any d-pseudotrajectory X as above with d d 0 there exists a point p M such that dist(x, p ) Ld, Z. (2) The main result of this note is the following statement. Theorem. The following two statements are equivalent: (1) f has the Lipschitz shadowing property; (2) f is structurally stable. 2
Proof. The implication (2) (1) is well nown (see [1]). In the proof of the implication (1) (2), we use the following two nown results (Propositions 1 2). First we introduce some notation. For a point p M, define the following two subspaces of T p M: B + (p) = {v T p M : Df (p)v 0, B (p) = {v T p M : Df (p)v 0, + } }. Proposition 1 (Mañé, [12]). The diffeomorphism f is structurally stable if only if B + (p) + B (p) = T p M for any p M. Consider a sequence of linear isomorphisms A = {A : R m R m, Z} for which there exists a constant N > 0 such that A, A 1 N. Fix two indices, l Z denote A 1 A l, l < ; Φ(, l) = Id, l = ; A 1 A 1 l 1, l >. Set B + (A) = {v R m : Φ(, 0)v 0, B (A) = {v R m : Φ(, 0)v 0, + } }. Proposition 2 (Pliss, [13]). The following two statements are equivalent: (a) For any bounded sequence {w R m, Z} there exists a bounded sequence {v R m, Z} such that v +1 = A v + w, Z; (b) the sequence A is hyperbolic on any of the rays [0, + ) (, 0] (see the definition in [11]), the spaces B + (A) B (A) are transverse. 3
Remar 1. In fact, Pliss considered in [13] not sequences of isomorphisms but homogeneous linear systems of differential equations; the relation between these two settings is discussed in [14]. Remar 2. Later, a statement analogous to Proposition 2 was proved by Palmer [15, 16]; Palmer also described Fredholm properties of the corresponding operator {v R m : Z} {v A 1 v 1 }. We fix a point p M consider the isomorphisms A = Df(p ) : T p M T p+1 M. Note that the implication (a) (b) of Proposition 2 is valid for the sequence {A } with R m replaced by T p M. It follows from Propositions 1 2 that our main theorem is a corollary of the following statement (indeed, we prove that the Lipschitz shadowing property implies the validity of statement (a) of Proposition 2 for any trajectory {p } of f, while the validity of statement (b) of this proposition implies the structural stability of f by Proposition 1). Lemma 1. If f has the Lipschitz shadowing property with constants L, d 0, then for any sequence {w T p M, Z} such that w < 1, Z, there exists a sequence {v T p M, Z} such that v 8L + 1, v +1 = A v + w, Z. (3) To prove Lemma 1, we first prove the following statement. Lemma 2. Assume that f has the Lipschitz shadowing property with constants L, d 0. Fix a trajectory {p } a natural number n. For any sequence {w T p M, [ n, n]} such that w < 1 there exists a sequence {z T p M, Z} such that z 8L + 1, Z, (4) z +1 = A z + w, [ n, n]. (5) Proof. First we locally linearize the diffeomorphism f in a neighborhood of the trajectory {p }. Let exp be the stard exponential mapping on the tangent bundle of M let exp x be the corresponding mapping T x M M. 4
We introduce the mappings F = exp 1 p +1 f exp p : T p M T p+1 M. It follows from the stard properties of the exponential mapping that D exp x (0) = Id; hence, DF (0) = A. Since M is compact, we can represent where F (v) = A v + φ (v), φ (v) v 0 as v 0 uniformly in. Denote by B(r, x) the ball in M of radius r centered at a point x by B T (r, x) the ball in T x M of radius r centered at the origin. There exists r > 0 such that, for any x M, exp x is a diffeomorphism of B T (r, x) onto its image, exp 1 x is a diffeomorphism of B(r, x) onto its image. In addition, we may assume that r has the following property. If v, w B T (r, x), then dist(exp x (v), exp x (w)) v w 2; if y, z B(r, x), then exp 1 x (y) exp 1 x (z) 2. dist(y, z) Now we pass to construction of pseudotrajectories; every time, we tae d so small that the considered points of our pseudotrajectories, points of shadowing trajectories, their lifts to tangent spaces etc belong to the corresponding balls B(r, p ) B T (r, p ) ( we do not repeat this condition on the smallness of d). For brevity, for v B T (r, p ) we write i(v ) instead of exp p (v ), for y B(r, p ) we write l(y ) instead of exp 1 p (y ) (thus, the index of the argument shows at which point p, the exponential mapping or its inverse is applied). Consider the sequence { T p M, [ n, n + 1]} defined as follows: { n = 0, (6) +1 = A + w, [ n, n]. 5
Fix a small d > 0 construct a pseudotrajectory {ξ } as follows: ξ = i(d ), [ n, n + 1], ξ l = f l+n (ξ n ), l n 1, ξ l = f l n 1 (ξ n+1 ), l > n + 1. Denote Then η +1 = f(ξ ), ζ +1 = l(η +1 ), ζ +1 = l(ξ +1 ). ζ +1 = exp 1 p +1 f(exp p (d )) = F (d ) = A d + g (d), where g (d) = o(d), Hence, for small d, ζ +1 = exp 1 p +1 (ξ +1 ) = d +1 = A d + dw. ζ +1 ζ +1 dw + o(d) 2d dist(f(ξ ), ξ +1 ) 4d. Let us note that the required smallness of d is determined by the chosen trajectory {p }, the sequence w, the number n. The Lipschitz shadowing property of f implies that there exists an exact trajectory {y } such that dist(ξ, y ) 4Ld, [ n, n + 1]. (7) Consider the finite sequence {b T p M, [ n, n + 1]} defined as follows: db n = l(y n ) (8) b +1 = A b, [ n, n]. (9) Define t T p M by y = i(t ). Since {y } is a trajectory of f, t +1 = F (t ). Hence, t +1 = A t + h (t ), [ n, n], where h (t ) = o( t ). Let us note that dist(y, p ) dist(y, ξ ) + dist(ξ, p ) 4Ld + 2d. 6
Hence, t 8Ld + 4d 4(2L + N 2n + N 2n 1 + + 1)d, [ n, n + 1], where N = sup A. We see that h (t ) = o(d), [ n, n + 1]. (10) Consider the sequence c = t db. Note that c n = 0 by (8). Clearly, c +1 = A c + h (t ), [ n, n + 1], it easily follows from (10) that for small enough d, the inequalities c = t db < d, [ n, n + 1], (11) hold. We note that l(ξ ) = d l(y ) = t get the following estimate (using (7) (11)): b = 1 d (l(ξ ) l(y ))+(t db ) 1 (8Ld+d) = 8L+1, [ n, n+1]. d (12) Consider the sequence {z T p M, Z} defined as follows: { z = b, [ n, n + 1], z = 0, / [ n, n + 1]. Inequality (12) implies estimate (4), while equalities (6) (9) imply relations (5). Lemma 2 is proved. Proof of Lemma 1. Fix n > 0 consider the sequence { w (n) w (n) = w, [ n, n], = w (n) = 0, > 0. By Lemma 2, there exists a sequence {z (n) T p M, Z} such that z (n) 8L + 1, Z, (13) z (n) +1 = A z (n) + w (n), [ n, n]. (14) Passing to a subsequence of {z (n) }, we can find a sequence {v T p M, Z} such that v = lim z (n) n, Z. (Let us note that we do not assume uniform convergence.) Passing to the limit in estimates (13) equalities (14) as n, we get relations (3). Lemma 1 our theorem are proved. 7
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