Asymptotic orbital shadowing property for diffeomorphisms

Lemma 2.1

Let f ∈ Diff(M). f is chain transitive if and only if it does not contain a proper attractor.

We also recall that the Hausdoroff distance between two compact subsets A and B of M is given by:

Lemma 2.2

If f has the asymptotic orbital shadowing property then it is chain transitive.

Proof. Suppose that there is a proper attractor Λ ⊂ M. Since Λ is a proper attractor, there is a neighborhood U of Λ such that f n ( U ¯ ) ⊂ U for all n ≥ 1 and Λ = ⋂ n ≥ 0 f n ( U ¯ ) . Λ Since Λ is compact, there is ϵ₁ > 0 such that Λ ⊂ B ϵ 1 ( Λ ) ⊂ U . Then we have d H ( Λ , U ¯ ) > ϵ 1 / 2. Now we construct an asymptotic pseudo orbit. For this, we consider two points a and b such that a ∈ Λ a n d b ∈ M ∖ U . Now we consider sequences x_i = fⁱ(a) for all i ≤ 0 and x_i = fⁱ(b) for all i > 0. It is clear that the sequence { x i } i ∈ Z is an asymptotic pseudo orbit of f. If there are z ∈ M and N₁, N ₂ ∈ N such that f − N 1 ( z ) ∈ B ϵ 1 / 4 ( a ) then according to the proper attracting property, f − N 1 + i ( z ) ∈ U for all i ≥ 0. Then we have

where O r b + ( f − N 1 ( z ) ) = { f − N 1 + i ( z ) : i ≥ 0 } , and O r b + ( f ( b ) ) = { f i ( b ) : i > 0 } . Thus we have O r b + ( f − N 1 ( z ) ) ∩ U ¯ = ∅ . Let ϵ = ϵ 1 / 4. Then by the asymptotic orbital shadowing property, there exist z ∈ M and N 1 , N 2 ∈ N such that f must have

Since O r b + ( f − N 1 ( z ) ) ⊂ Λ = ⋂ i ≥ 0 f i ( U ) a n d d H ( Λ , U ¯ ) > ϵ 1 / 2 , this is a contradiction. Thus if f has the asymptotic orbital shadowing property then there is no proper attractor, and so, by Lemma 2.1 it is chain transitive.

Let p be a hyperbolic periodic point of f. We define the sets

are C¹ injectively immersed submanifolds of M, where π(p) is the period of p. We denote i n d e x ( p ) = d i m W s ( p ) . Note that for any hyperbolic p, q ∈ P(f), if W s ( p ) ⋔ W u ( q ) / = ∅ and W u ( p ) ⋔ W s ( q ) / = ∅ then i n d e x ( p ) = i n d e x ( q ) . For any ϵ > 0 , l e t W ϵ ( x ) s ( x ) = { y ∈ M : d ( f n ( x ) , f n ( y ) ) < ϵ , for all n ≥ 0}, and W ϵ ( x ) u ( x ) = { y ∈ M : d ( f − n ( x ) , f − n ( y ) ) < ϵ , for all n ≥ 0} be the local stable set and the local unstable set of x, respectively. Note that for a hyperbolic periodic point p, we know W ϵ ( p ) s ( p ) ⊂ W s ( p ) and W ϵ ( p ) u ( p ) ⊂ W u ( p ) .

The concept of the asymptotic orbital shadowing property can be rewritten as follows : a diffeomorphism f has the asymptotic orbital shadowing property if for each asymptotic pseudo orbit { x i } i ∈ Z there is a point y ∈ M such that for all ϵ > 0 there exist N 1 , N 2 ∈ N such that

1. (i) f − N 1 − i ( z ) ∈ B ϵ ( { x − N 1 − i } ) and f N 2 + i ( z ) ∈ B ϵ ( { x N 2 + i } ) for i ≥ 0,

2. (ii) x − N 1 − i ∈ B ϵ ( f − N 1 − i ( z ) ) and x N 2 + i ∈ B ϵ ( f N 2 + i ( z ) ) for i ≥ 0.

Lemma 2.3

If f has the asymptotic orbital shadowing property then for any hyperbolic periodic points p and q, we have W s ( p ) ∩ W u ( q ) / = ∅ and W u ( p ) ∩ W s ( q ) / = ∅ .

Proof. Let p and q be hyperbolic periodic points of f. By Lemma 2.2, f is chain transitive. Then by [27, Lemma 2.1], f does not contain sinks nor sources. Thus p and q are saddles. For simplicity, we assume that f(p) = p and f(q) = q. Since f is chain transitive, for any δ > 0 there is a sequence { x i } i = 0 n ( n ≥ 1 ) such that x₀ = p, x_n = q, and d ( f ( x i ) , x i + 1 ) < δ f o r i = 0 , … , n − 1. Then the sequence { x i } i = 0 n ( n ≥ 1 ) is a finite δ-pseudo orbit of f. Substituting x_i = fⁱ(p) for all i ≥ 0 and x_i = fⁱ(q) for all i ≥ n, the sequence { … , p ( = x − 1 ) , p ( = x 0 ) , x 1 , x 2 , … , q ( = x n ) , q ( = x n + 1 ) , … } = { x i } i ∈ Z is an asymptotic pseudo orbit of f. Let ϵ = min { ϵ ( p ) , ϵ ( q ) } . Then base on the asymptotic orbital shadowing property, there are z ∈ M and N 1 , N 2 ∈ N such that

Then we know that d ( f − i ( f − N 1 ( z ) ) , p ) < ϵ and d ( f i ( f N 2 ( z ) ) , q ) < ϵ for all i ≥ 0. Therefore we know f N 2 ( z ) ∈ W ϵ s ( q ) ⊂ W s ( q ) and f − N 1 ( z ) ∈ W ϵ u ( p ) ⊂ W u ( p ) , and so, z ∈ W u ( p ) ∩ W s ( q ) . Thus W u ( p ) ∩ W s ( q ) / = ∅ . The other case is similar.

We say that f is Kupka-Smale if the periodic points of f are hyperbolic, and if p, q ∈ P(f), then W^s(p) is transversal to W u ( q ) . We denote KS as the set of all Kupka-Smale diffeomorphsms. It is well-known that the set of all Kupka-Smale diffeomorphisms is C¹ residual in Diff(M) (see [35]).

Proposition 2.4. If f has the C¹ robustly asymptotic orbital shadowing property then there is C¹ neighborhood U ( f ) such that for any g ∈ U ( f ) and any hyperbolic p, q ∈ P(g) we have

where P(g) is the set of all periodic points of g.

Proof. Let U ( f ) be a C¹ neighborhood of f. Suppose, by contradiction, that there are g ∈ U ( f ) such that g has two periodic points p and q with i n d e x ( p ) / = i n d e x ( q ) . Then we know that d i m W s ( p ) + d i m W u ( q ) < d i m M or d i m W u ( p ) + d i m W s ( q ) < d i m M . We may assume that d i m W s ( p ) + d i m W u ( q ) < d i m M . Note that for any hyperbolic periodic points p and q, there is a C¹ neighborhood U ( f ) such that for any g ∈ U ( f ) , we know that d i m W s ( p ) = d i m W s ( p g ) and d i m W u ( q ) = d i m W u ( q g ) , where p_g is the continuation of p and q_g is the continuation of q.

Since f has the C¹ robustly asymptotic orbital shadowing property, there is g 1 ∈ U ( g ) ∩ K S such that d i m W s ( p g 1 ) + d i m W u ( q g 1 ) < d i m M , where p g 1 is the continuation of p and q g 1 is the continuation of q. Since g 1 ∈ K S , W s ( p g 1 ) ∩ W u ( q g 1 ) = ∅ . However, g₁ has the asymptotic orbital shadowing property, and so, by Lemma 2.2, g₁ is chain transitive. By Lemma 2.3, we have W s ( p g 1 ) ∩ W u ( q g 1 ) / = ∅ . This is a contradiction.

The following is called Franks’ lemma [14] which is very useful in our proofs.

Lemma 2.5

Let U(f) be any given C¹ neighborhood of f. Then there exist ϵ > 0and a C¹ neighborhood U₀(f) ⊂ U ( f ) of f such that for given g ∈ U 0 ( f ) a finite set {x₁, x₂, · · · , x_N}, a neighborhood U of {x₁, x₂, . . . , x_N} and linear maps L i : T x i M → T g ( x i ) M satisfying ∥ L i − D x i g ∥ ≤ ϵ for all 1 ≤ i ≤ N, there exists g ^ ∈ U ( f ) such that g ^ ( x ) = g ( x ) i f x ∈ { x 1 , x 2 , ⋯ , x N } ∪ ( M ∖ U ) and D x i g ^ = L i f o r a l l 1 ≤ i ≤ N .

Lemma 2.6

If a point p ∈ P(f) is not hyperbolic then there is g C¹ close to f such that g has two hyperbolic periodic points q, r ∈ P(g) with index(q) ̸= index(r).

Proof. Suppose that p ∈ P(f) is not hyperbolic. Then D p f π ( p ) has eigenvalues whose modulus are 1, where π(p) is the period of p. Then by Lemma 2.5, there is g C¹ close to f such that D p g g π ( p g ) has only one eigenvalue λ with |λ| = 1, where π(p_g) is the period of p_g. For simplicity, we may assume that g π ( p g ) ( p g ) = g ( p g ) = p g . We denote E p g c as the eigenspace corresponding to λ. If λ ∈ R then dim E p g c = 1 and if λ ∈ C then dim E p g c = 2 .

First, if λ ∈ R then λ = 1 or λ = −1. In the proof, we consider that λ = 1. The case of λ = −1 is similar. Let U ( g ) be a C¹ neighborhood of g. By Lemma 2.5, there are α > 0 and g 1 ∈ U ( g ) such that (i) g 1 ( p g ) = g ( p g ) = p g , (ii) g 1 ( x ) = e x p p g ∘ D p g g ∘ e x p p g − 1 ( x ) i f x ∈ B α ( p g ) , and g ( x ) = g 1 ( x ) i f x ∉ B 4 α ( p g ) . We can consider nonzero vector v associated λ such that ∥ v ∥ ≤ α / 2. Then we have

If we set I v = { t ⋅ v : − α / 4 ≤ t ≤ α / 4 } , then e x p p g ( J v ) is a closed small arc, and by (2), we have g 1 ( e x p p g ( J v ) ) = e x p p g ( J v ) . Thus g 1 | e x p p g ( J v ) : e x p p g ( J v ) → e x p p g ( J v ) is the identity map. Taking two endpoints q , r ∈ e x p p g ( J v ) it is clear that D q g 1 | E q c = D r g 1 | E r c = D p g g 1 | E p g c = 1. By Lemma 2.5, there is h C¹ close to g 1 ( h ∈ U ( g ) ) such that q_h and r_h are hyperbolic with i n d e x ( q h ) / = i n d e x ( r h ) , where q_h is the continuation of q and r_h is the continuation of r.

Finally, we consider λ ∈ C . We assume that g(p) = p. As in the previous case, by Lemma 2.5, there are α > 0 and g 1 ∈ U ( g ) such that (i) g 1 ( p g ) = g ( p g ) = p g , (ii) g 1 ( x ) = e x p p g ∘ D p g g ∘ e x p p g − 1 ( x ) i f x ∈ B α ( p g ) , and g ( x ) = g 1 ( x ) i f x ∉ B 4 α ( p g ) . Then there is k > 0 such that D p g g 1 k ( v ) = v for any nonzero v ∈ e x p p − 1 ( E p g c ( α ) ) . Taking w ∈ e x p p g − 1 ( E p g c ( α ) ) such that ∥ w ∥ = α / 4 , we set

Then L p g is a closed small arc such that (i) g1i(Lpg)∩g1j(Lpg)=0 if 0≤i/=j≤k−1, (ii) g1k(Lpg)=Lpg and (iii) g1k|Lpg is the identity map. As in the previous case, we consder two endpoints q,r∈Lpg. Then by Lemma 2.5, there is g₂ C¹ close to g₁ such that g₂ has two hyperbolic periodic points qg2 and rg2 with index(qg2)/=index(rg2), where qg2 is the continuation of q and rg2 is the continuation of r.

We say that f satisfies the star condition if there is a C¹ neighborhood U(f) of f such that for any g∈U(f), each p ∈ P(g) is hyperbolic. F(M) denotes the set of all diffeomorphisms satisfying the star condition. Hayashi [15] proved that if a diffeomorphism f∈F(M) then f satisfies Axiom A. We will show that if a diffeomorphism f has the C¹ robustly asymptotic orbital shadowing property then f satisfies Axiom A.

Proposition 2.7

If f has the C¹ robustly asymptotic orbital shadowing property then f∈F(M).

Proof. Suppose, by contradiction, that f/∈F(M). Then there is g C¹ close to f such that g has a nonhyperbolic periodic point p. By Lemma 2.6, there is h C¹ close to g (h C¹ close to f) such that h has two hyperbolic periodic points q, r with index(q)/=index(r). Since f has the C¹ robustly asymptotic orbital shadowing property, by Proposition 2.4 this is a contradiction.

A diffeomorphism f is robustly chain transitive if there is C¹ neighborhood U(f) of f such that for any g∈U(f), g is chain transitive. Lee [27] proved that if a diffeomorphism f is robustly chain transitive then f admits a weak hyperbolic, that is, a closed f -invariant set Λ admits a dominated splitting if the tangent bundle TΛM has a Df-invariant splitting E ⊕ F and there exist constants C > 0 and λ ∈ (0, 1) such that

for all x ∈ Λ and n ≥ 0. It is clear that if a closed f -invariant set Λ ⊂ Mis hyperbolic then it admits a dominated splitting for f.

Proof of Theorem A. Suppose that f has the C¹ robustly asymptotic orbital shadowing property. Since f has the asymptotic orbital shadowing property then by Lemma 2.2 f is chain transitive. Since f has the C¹ robustly asymptotic orbital shadowing property, f is robustly chain transitive. By Proposition 2.7, each periodic points of f is hyperbolic. By Proposition 2.4, the index of each periodic points of f is the same. Thus by [27, Theorem 1.1], f is Anosov.

Lemma 3.1

There is a residual set G1⊂Diff(M) such that for any f∈G1 and if f has the asymptotic orbital shadowing property then index(p) = index(q), for any points p, q ∈ P(f).

Proof. Let f∈G1=KS have the asymptotic orbital shadowing property. Since f is Kupka-Smale, each periodic points of f is hyperbolic. Suppose, by contradiction, that index p / = index q . Then d i m W s ( p ) + d i m W u ( q ) < d i m M o r d i m W u ( p ) + d i m W s ( q ) < d i m M . Assume that dim Ws(p)+dimWU(q)<dimM Other case is similar). Since f ∈ KS, we have

Since f has the asymptotic orbital shadowing property, by Lemma 2.3,

This is a contradiction. Thus index(p) = index(q).

Lemma 3.2. [30, Lemma 2.2] There is a residual set G2⊂Diff(M) such that for any f∈G2, and if for any C¹ neighborhood U(f) of f there is g∈U(f) such that g has two periodic points p and q with index p / = index q then f has two periodic points p_f and q_f with index p f / = index q f

For any δ > 0, we say that a hyperbolic p ∈ P(f) has a δ weak eigenvalue if there is an eigenvalue λ of D_pf^π(p) such that

where π(p) is the period of p.

Lemma 3.3

There is a residual set G3⊂Diff(M) such that for any f∈G3, if f has the asymptotic orbital shadowing property then there is δ > 0 such that for any p ∈ P(f), p does not have a δ weak eigenvalue.

Proof. Let f∈G3=G1∩G2 have the asymptotic orbital shadowing property. Suppose that for any δ > 0 there is p ∈ P(f) such that p has a δ weak eigenvalue. Then there is g C¹ close to f such that Dpggπ(pg) has an eigenvalue λ with |λ| = 1. By Lemma 2.6, there is g₁ C¹ close to g( g₁ C¹ close to f) such that g₁ has two hyperbolic periodic points q, r with index(q)/=index(r). By Lemma 3.2 f has two periodic points q_f and r_f with index(qf)/=index(rf). This is a contradiction since f∈G1 and f has the asymptotic orbital shadowing property.

Lemma 3.4

[3, Lemma 5.1] There is a residual set G4⊂Diff(M) such that for any f∈G4, for any δ > 0and any C¹ neighborhood U(f) of f, if there is g∈U(f) and a hyperbolic p ∈ P(g) such that p has a δ weak eigenvalue then there is a hyperbolic p _f ∈ P(f) with a 2δ weak eigenvalue.

Lemma 3.6

[11, Theorem 7] There is a residual set G6⊂Diff(M) such that for any f∈G6, f is chain transitive if and only if f is transitive.

Proof of Theorem B. Let f∈G5∩G6 have the asymptotic orbital shadowing property. Then by Proposition 3.5, f∈F(M). Thus the nonwandering set Ω(f)=P(f)¯ hyperbolic. Since f∈G6,  Ω(f)=M is hyperbolic. Thus f is transitive Anosov.