H¨older shadowing and structural stability Sergey Tikhomirov∗ November 29, 2010

Abstract We show that C 2 -diffeomorphisms satisfying H¨older shadowing property with exponent greater than 1/2 are in fact structurally stable. We give a simple example of a C ∞ -diffeomorphism which has H¨older shadowing property with exponent 1/3 and is not structurally stable.

Keywords: H¨older shadowing, hyperbolicity, structural stability

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 known that a diffeomorphism has shadowing property in a neighborhood of a hyperbolic set [3, 4] and a structurally stable diffeomorphism has shadowing property on the whole manifold [5, 6, 7]. Analyzing the proofs of the first shadowing results by Anosov [3] and Bowen [4], it is easy to see that, in a neighborhood of a hyperbolic set, the shadowing property is Lipschitz ∗

Departamento de Matematica, Pontificia Universidade Catolica do Rio de Janeiro, Rua Marques de Sao Vicente, 225, Gavea - Rio de Janeiro - Brazil CEP 22453-900. [email protected]

1

(and the same holds in the case of a structurally stable diffeomorphism, see [1]). At the same time, it is easy to give an example of a diffeomorphism that is not structurally stable but has shadowing property (see [8], for example). Thus, shadowing does not imply structural stability. Under several additional assumptions shadowing is equivalent to structural stability. In [9] it was shown that C 1 -robust shadowing property is equivalent to structural stability (later [10] this result was generalized for the orbital shadowing property). Abdenur and Diaz conjectured [11] that shadowing C 1 -generically is equivalent to structural stability and proved this for tame diffeomorphisms. Recently in a series of works it was proved the equivalence between some forms of shadowing and structural stability without robust or generic assumptions. In a joint work with Pilyugin [13] we showed that Lipschitz shadowing property is equivalent to structural stability. In [8] it was proved that variational shadowing is equivalent to structural stability. In [12] it was proved that Lipschitz periodic shadowing is equivalent to Ω-stability. In this work we consider H¨older shadowing property and concentrate on the following Question 1. Is H¨older shadowing equivalent to structural stability? One of the motivation to study H¨older shadowing is a question suggested by Katok: Question 2. Does any diffeomorphism H¨older conjugated to Anosov must be Anosov by itself ? It is easy to show that diffeomorphisms H¨older conjugated to Anosov satisfy H¨older shadowing property. Hence a positive answer to question 1 implies positive answer to question 2. Answer to question 2 depends on regularity of considered diffeomorphisms. Recently it was shown that in general the answer to question 2 is negative. In [19] it was constructed an example of not Anosov diffeomorphism f : T2 → T2 of smoothness up to C 3−ε H¨older conjugated to linear Anosov automorphism. At that same time the following positive result was proved in [19, 20]. Theorem 1. A C 2 -diffeomorphism that is conjugate to an Anosov diffeomorphism via H¨older conjugacy h is Anosov itself provided that the product of H¨older exponents for h and h−1 is greater than 1/2. 2

The proof of theorem 1 is based on H¨older shadowing property. Note that those results shows that assumptions on regularity of involved diffeomorphisms are critical. The main result of this article is theorem 3, where we prove that C 2 diffeomorphisms satisfying H¨older shadowing property with exponent greater than 1/2 are in fact structurally stable (see section Main Results for exact statements). This result generalizes theorem 1. We also give a simple example of a C ∞ -diffeomorphism which has H¨older shadowing with exponent 1/3 and is not structurally stable. This example shows that strong generalizations of theorem 3 with lower H¨older exponents and higher regularity is not possible.

2

Main Results

Let M be a smooth compact manifold of class C ∞ without boundary with the Riemannian metric dist. Consider a diffeomorphism f ∈ Diff 1 (M ). For any d > 0 a sequence {yk } is called a d-pseudotrajectory if the following inequalities hold dist(yk+1 , f (yk )) < d,

k ∈ Z.

Definition 1. We say that f has the standard shadowing property (StSh) if for any ε > 0 there exists d > 0 such that for any d-pseudotrajectory {yk } there exists an exact trajectory {xk } such that dist(xk , yk ) < ε,

k ∈ Z.

(1)

If inequalities (1) hold we say that pseudotrajectory {yk } is ε-shadowed by {xk }. Definition 2. We say that f has the Lipschitz shadowing property (LipSh) if there exist constants d0 , L > 0 such that for any d < d0 and dpseudotrajectory {yk } there exists an exact trajectory {xk } such that inequalities (1) hold with ε = Ld. Theorem 2. [13] Diffeomorphism f ∈ C 1 has Lipschitz shadowing property if and only if it is structurally stable. In the present article we study H¨older shadowing property which corresponds to H¨older dependency between ε and d in the definition of StSh. 3

Definition 3. We say that f has the H¨older shadowing property with exponent θ ∈ (0, 1) (HolSh(θ)) if there exist constants d0 , L > 0 such that for any d < d0 and d-pseudotrajectory {yk } there exists an exact trajectory {xk } such that dist(xk , yk ) < Ldθ , k ∈ Z. (2) For any θ ∈ (0, 1) the following inclusions hold SS = LipSh ⊂ HolSh(θ) ⊂ StSh, where SS denote the set of structurally stable diffeomorphisms. The main result of present article is the following. Theorem 3. If a diffeomorphism f ∈ C 2 satisfy HolSh(θ) with θ > 1/2 then f is structurally stable. Remark 1. In fact the following statement is correct: if a diffeomorphism f ∈ C 1+δ with δ ∈ (0, 1] satisfy HolSh(θ) with θ > 1/(1 + δ) then f is structurally stable. The proof is similar to the proof of theorem 3 but notation is heavy. We leave details to the reader. In section 5 we give an Example 1. There exists a non-structurally stable diffeomorphism f : S 1 → S 1 , such that f ∈ C ∞ and f ∈ HolSh(1/3).

3

Proof of Theorem 3

Now let us explain the main ideas of the proof of the main theorem. First we introduce some notation. Definition 4. We say that diffeomorphism f has Property A if for any trajectory {pk } of f and any sequence {wk ∈ Tpk M } there exists a bounded sequence {vk ∈ Tpk M } such that vk+1 = Ak vk + wk+1 , where Ak = D f (pk ). In [8, 13] it was proved 4

k ∈ Z,

(3)

Lemma 1. The following two statements are equivalent: (i) f satisfy Property A, (ii) f is structurally stable. For completeness of the exposition let us describe the main ideas of the proof of lemma 1. For a point p ∈ M , define the following two subspaces of Tp M : B + (p) = {v ∈ Tp M : |Df k (p)v| → 0, k → +∞} and B − (p) = {v ∈ Tp M : |Df k (p)v| → 0,

k → −∞}.

Proposition 1. [14] The diffeomorphism f is structurally stable if and only if B + (p) + B − (p) = Tp M for any p ∈ M . Consider a sequence of linear isomorphisms A = {Ak : Rm → Rm , k ∈ Z} for which there exists a constant N > 0 such that kAk k, kA−1 k k ≤ N . Fix two indices k, l ∈ Z and denote   Ak−1 ◦ · · · ◦ Al , l < k, Φ(k, l) = Id, l = k,   −1 −1 Ak ◦ · · · ◦ Al−1 , l > k. Set B + (A) = {v ∈ Rm : |Φ(k, 0)v| → 0,

k → +∞}

B − (A) = {v ∈ Rm : |Φ(k, 0)v| → 0,

k → −∞}.

and Proposition 2. [15] The following two statements are equivalent: (a) For any bounded sequence {wk ∈ Rm , k ∈ Z} there exists a bounded sequence {vk ∈ Rm , k ∈ Z} such that vk+1 = Ak vk + wk+1 , 5

k ∈ Z,

(b) the sequence A is hyperbolic on any of the rays [0, +∞) and (−∞, 0] (see the definition in [16]), and the spaces B + (A) and B − (A) are transverse. Remark 2. In fact, Pliss considered in [15] not sequences of isomorphisms but homogeneous linear systems of differential equations; the relation between these two settings is discussed in [16]. Remark 3. Later, a statement analogous to proposition 2 was proved by Palmer [17, 18]; Palmer also described Fredholm properties of the corresponding operator {vk ∈ Rm : k ∈ Z} 7→ {vk − Ak−1 vk−1 }. We fix a point p ∈ M and consider the isomorphisms Ak = Df (pk ) : Tpk M → Tpk+1 M. Note that the implication (a)⇒(b) of proposition 2 is valid for the sequence {Ak } with Rm replaced by Tpk M . It is easy to see that Property A is equivalent to statement (a) of proposition 2 for any trajectory {pk } of f , while the validity of statement (b) of this proposition is equivalent to the structural stability of f by proposition 1. See [8] for the detailed proof. So, to prove theorem 3 it is enough to prove Lemma 2. If a diffeomorphism f ∈ C 2 satisfies HolSh(θ) with θ > 1/2 then f satisfies Property A.

4

Proof of lemma 2

Let exp be the standard exponential mapping on the tangent bundle of M and let expx : Tx M → M be the corresponding exponential mapping at a point x. Denote by B(r, x) the ball in M of radius r centered at a point x and by BT (r, x) the ball in Tx M of radius r centered at the origin. There exists ε > 0 such that, for any x ∈ M , expx is a diffeomorphism of BT (ε, x) onto its image, and exp−1 x is a diffeomorphism of B(ε, x) onto its image. In addition, we may assume that ε has the following property. 6

If v, w ∈ BT (ε, x), then dist(expx (v), expx (w)) ≤ 2; |v − w|

(4)

−1 | exp−1 x (y) − expx (z)| ≤ 2. dist(y, z)

(5)

if y, z ∈ B(ε, x), then

Let L, d0 > 0 and θ ∈ (1/2, 1) be constants from the definition of HolSh. Denote α = θ − 1/2. Since M is compact and f ∈ C 2 there exists R > 0 such that dist(f (expx (v)), expf (x) (D f (x)v)) < R|v|2 ,

x ∈ M , v ∈ Tx M ,|v| < ε, (6)

(we additionally decrease ε, if necessarily). For an arbitrarily sequence {wk ∈ Tpk M } with |wk | ≤ 1 consider equations vk+1 = Ak vk + wk+1 ,

k ∈ Z.

(7)

Below we prove that there exists a bounded sequence {vk } satisfying (7). First let us prove that there exists a constant Q such that for any N > 0 there exists a sequence {vk } satisfying equalities (7) and inequalities |vk | ≤ Q for k ∈ [−N, N ]. Fix N > 0. For any sequence {vk ∈ Tpk M }k∈[−N,N +1] denote k{vk }k = maxk∈[−N,N +1] |vk |. For any sequence {wk ∈ Tpk M }k∈[−N,N +1] consider the set © ª E(N, {wk }) = {vk∈[−N,N +1] } satisfy (7) for k ∈ [−N, N ] . Denote F (N, {wk }) =

min

{vk }∈E(N,{wk })

k{vk }k.

(8)

Since k · k ≥ 0 is a continuous function on the space of all {vk } and set E(N, {wk }) is closed the value F (N, {wk }) is well-defined. Note that sequence {vk } ∈ E(N, {wk }) is defined by the value v−N . Consider the sequence {vk } corresponding to v−N = 0. It is easy to see that |vk | ≤ 1 + B + B 2 + · · · + B k+N −1 for k ∈ [−N + 1, N + 1], where B = maxx∈M k D f (x)k. Hence F (N, {wk }) ≤ 1 + B + B 2 + · · · + B 2N for any {|wk | ≤ 1}. It is easy to see that F (N, {wk }) is continuous with respect to {wk } and hence Q = Q(N ) =

max

{wk }, |wk |≤1

7

F (N, {wk })

(9)

is well defined. Let us prove that Q(N ) is uniformly bounded over all N > 0. By the definition of Q and linearity of equation (7) for any sequence {wk0 }k∈[−N,N +1] there exists a sequence {vk0 }, satisfying (7) and k{vk0 }k ≤ Q(N )k{wk0 }k.

(10)

Let us choose sequences {wk } and {vk } ∈ F (N, {wk }) such that Q(N ) = F (N, {wk }), Consider d=

ε Q2+β

,

F (N, {wk }) = k{vk }k. α . 1/2 − α

(11)

(2 + β)(1/2 − α) < 1 − α.

(12)

with 0 < β <

Note that the following inequalities hold (2 + β)(1/2 + α) > 1 + β,

Without loss of generality we can assume that Q > 4 and ε < 1/2. Consider sequence   k ∈ [−N, N + 1], exppk (dvk ), k+N yk = f (y−N ), k < −N,   k−(N +1) f (yN +1 ), k > N + 1. Let us show that {yk } is (R + 2)d-pseudotrajectory. For k ∈ [−N, N ] equations (4), (6) and inequalities |dvk | < ε, (dQ)2 < d imply the following dist(f (yk ), yk+1 ) = dist(f (exppk (dvk )), exppk+1 (d(Ak vk + wk+1 )) ≤ ≤ dist(f (exppk (dvk )), exppk+1 (dAk vk ))+ + dist(exppk+1 (dAk vk ), exppk+1 (d(Ak vk + wk ))) ≤ ≤ R|dvk |2 + 2d ≤ (R + 2)d. (13) For k ∈ / [−N, N ] holds equality f (yk ) = yk+1 . We can assume that Q > ((R + 2)ε/d0 )1/(2+β) . 8

(14)

Indeed, the right side of (14) does not depend on N , and if Q is smaller than the right side of (14) then we already proved an upper bound of Q which does not depend on N . In the text below we will make several times similar remarks to ensure that Q is big enough. Inequality (14) implies that (R + 2)d < d0 . Since f ∈ HolSh(1/2 + α) the pseudotrajectory {yk } can be L((R+2)d)1/2+α shadowed. Let it be shadowed by an exact trajectory {xk }. By reasons similar to (14) we can assume that L((R + 2)d)1/2+α < ε/2. Inequalities (4) and (14) imply that dist(pk , xk ) ≤ dist(pk , yk ) + dist(yk , xk ) ≤ ≤ 2d|vk | + L((R + 2)d)1/2+α < ε/2 + ε/2 = ε. Hence ck = exp−1 pk (xk ) is well-defined. Denote L1 = L(R + 2)1/2+α . Since dist(yk , xk ) < L1 d1/2+α , inequalities (5) imply that |dvk − ck | < 2L1 d1/2+α . (15) Hence |ck | < Qd + L1 d1/2+α .

(16)

By the reasons similar to (14) we can assume that |ck | < ε. Since f (xk ) = xk+1 inequalities (5) and (6) imply that |ck+1 − Ak ck | < 2 dist(exppk+1 (ck+1 ), exppk+1 (Ak ck )) = = 2 dist(f (exppk (ck )), exppk+1 (Ak ck )) ≤ 2R|ck |2 . (17) Inequalities (11), (12), (16) imply that |ck | < L2 Qd for some some L2 > 0 which does not depend on N . Let ik = ck+1 − Ak ck . From inequality (17) it follows that |ik | ≤ 2R|ck |2 ≤ L3 (Qd)2 . for some L3 > 0 which does not depend on N . Inequality (10) implies that there exists a sequence {˜ ck ∈ Tpk M } satisfying c˜k+1 − Ak c˜k = ik+1 ,

|˜ ck | ≤ QL3 (Qd)2

for k ∈ [−N, N ]. Consider the sequence rk = ck − c˜k . Obviously it satisfies the following conditions rk+1 = Ak rk ,

|rk − ck | ≤ QL3 (Qd)2 . 9

(18)

Consider the sequence ek = d1 (dvk − rk ). Equations (15) and (18) imply that ek+1 = Ak ek + wk ,

k ∈ [−N, N ]

(19)

and

¯ ¯ ¯1 ¯ ¯ |ek | = ¯ ((dvk − ck ) − (rk − ck ))¯¯ ≤ L1 d−1/2+α + L3 Q3 d, d

k ∈ [−N, N ].

By the definition of Q for any sequence satisfying (19) there exists k ∈ [−N, N ] such that |ek | ≥ Q. Hence L1 d−1/2+α + L3 Q3 d ≥ Q. Recall that d = ε/Q2+β . The last inequality is equivalent to L4 Q−(2+β)(−1/2+α) + L5 Q1−β ≥ Q, where L4 , L5 > 0 are some constants which do not depend on N . This inequality and (12) imply that L4 Q1−α + L5 Q1−β ≥ Q. Hence L4 Q1−α ≥ Q/2 or L5 Q1−β ≥ Q/2, and Q ≤ max((2L4 )1/α , (2L5 )1/β ). We proved that there exists K > 0 such that Q(N ) < K for any N . (N ) Hence for any N > 0 there exists a sequence {vk ∈ Tpk M, k ∈ [−N, N + 1]} such that (N ) |vk | ≤ K, k ∈ [−N, N + 1] (20) and

(N )

(N )

vk+1 = Ak vk

+ wk+1 ,

k ∈ [−N, N ].

(21)

(N )

Define vk = 0 for k ∈ / [−N, N + 1]. (N ) Passing to a subsequence of {vk }, we can find a sequence {vk ∈ Tpk M, k ∈ Z} such that (N )

vk = lim vk , N →∞

k ∈ Z.

(Let us note that we do not assume uniform convergence.) Passing to the limit in the inequalities (20) and equalities (21) as N → ∞, we get the relations (3). Lemma 2 and theorem 3 are proved. 10

5

Example 1

Consider a diffeomorphism f : S 1 → S 1 constructed as the following. (i) The nonwandering set of f consists of two fixed points s, u ∈ S 1 . (ii) For some neighborhood Us of s there exists a coordinate system such that f |Us (x) = x/2. (iii) For some neighborhood Uu of u there exists a coordinate system such that f |Uu (x) = x + x3 . (iv) In S 1 \(Us ∪Uu ) the map is chosen to be C ∞ and to satisfy the following: there exists N > 2 such that f N (S 1 \ Uu ) ⊂ Us ,

f −N (S 1 \ Us ) ⊂ Uu ,

f 2 (Uu ) ∩ Us = ∅.

Theorem 4. If f : S 1 → S 1 satisfies properties (i)–(iv) then f ∈ HolSh(1/3). Proof. First let us prove a technical statement. Lemma 3. Denote g(x) = x + x3 . If |x − y| ≥ ε then |g(x) − g(y)| ≥ ε + ε3 /4. Proof. Using inequality x2 + xy + y 2 > (x − y)2 /4 we deduce that |g(x) − g(y)| = |x + x3 − y − y 3 | = |(x − y)(1 + x2 + xy + y 2 )| ≥ ≥ |(x − y)||1 + (x − y)2 /4| ≥ ε(1 + ε2 /4).

We divide proof of theorem 4 into several steps. Step 1. Note that conditions (ii), (iii) imply that there exists d1 > 0 such that B(d1 , f (S 1 \ Uu )) ⊂ S 1 \ Uu . (22) Since f |Us is hyperbolic contracting, standard reasoning shows that there exist L > 0 and d2 ∈ (0, d1 ) such that for any d-pseudotrajectory {yk } with d < d2 and y0 ∈ S 1 \ Uu the following holds B(d1 , f (Us )) ⊂ Us ,

B(d1 , f −1 (Uu )) ⊂ Uu ,

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• {yk }k≥0 ⊂ S 1 \ Uu , • dist(f k (x0 ), yk ) < Ld, for x0 ∈ B(d, y0 ), k ≥ 0, • if {yk }k∈Z ⊂ S 1 \ Uu then {yk }k∈Z can be Ld shadowed by an exact trajectory. Step 2. For any d-pseudotrajectory {yk }k≤0 with d < d1 and y0 ∈ Uu the following holds dist(yk , f k (y0 )) < 2d1/3 , k ≤ 0. (23) Indeed, step 1 implies that yk ∈ Uu for k < 0. Assume (23) does not hold. Let l = maxk≤0 (dist(yk , f k (y0 )) ≥ 2d1/3 ). Note that l < 0. Lemma 3 implies that dist(f (yl ), f l+1 (y0 )) > 2d1/3 + 2d. Hence dist(yl+1 , f l+1 (y0 )) > 2d1/3 which contradicts to the choice of l. Step 3. If {yk }k∈Z ⊂ Uu is a d-pseudotrajectory with d < d1 then dist(yk , u) < 2d1/3 ,

k ∈ Z.

(24)

To prove this associate yk with its coordinate in system introduced in (iii) and consider Y = supk∈Z |yk |. Assume that Y > 2d1/3 , then there exists k ∈ Z such that |yk | > max(2d1/3 , Y − d/2). Without loss of generality we can assume that yk > 0. Since yk ∈ Uu the following holds f (yk ) − yk = yk3 > 2d. Hence yk+1 −yk > (f (yk )−yk )−d > d and yk+1 > Y +d/2, which contradicts to the choice of Y . Inequalities (24) are proved. Now we are ready to finish the proof of theorem 4. Consider arbitrarily d-pseudotrajectory {yk }k∈Z with d < d1 . Let us prove that it can be Ld1/3 shadowed by an exact trajectory. If {yk } ⊂ Uu then by step 3 it can be 2d1/3 shadowed by {xk = u}. If {yk } ⊂ S 1 \ Uu then by step 1 it can be Ld shadowed. In other cases there exists l such that yl ∈ Uu and yl+1 ∈ / Uu . By step 2 dist(yk , f k−l (yl )) < 2d1/3 ,

k ≤ l.

By step 1 dist(yk , f k−l (yl )) < Ld,

k ≥ l + 1.

Hence {yk } is Ld1/3 shadowed by trajectory {xk = f k−l (yl )}. 12

Acknowledgement: Author would like to thanks Anatole Katok for introduction to problem of H¨older shadowing and Andrey Gogolev for fruitful discussions.

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