aa r X i v : . [ m a t h . D S ] N ov Shadowing in linear skew products
Sergey Tikhomirov ∗ Abstract
We consider a linear skew product with the full shift in the baseand nonzero Lyapunov exponent in the fiber. We provide a sharpestimate for the precision of shadowing for a typical pseudotrajectoryof finite length. This result indicates that the high-dimensional analogof Hammel-Yorke-Grebogi’s conjecture [1,2] concerning the interval ofshadowability for a typical pseudotrajectory is not correct. The maintechnique is reduction of the shadowing problem to the ruin problemfor a simple random walk. Bibliography 22 titles.
The theory of shadowing of approximate trajectories (pseudotrajectories) ofdynamical systems is now a well-developed part of the global theory of dy-namical systems (see the monographs [3, 4] and [5] for a survey of modernresults). The shadowing problem is related to the following question: un-der which conditions, for any pseudotrajectory of f does there exist a closetrajectory?Let us consider a metric space ( G, dist) and a continuous map f : G → G , d >
0. For an interval I = ( a, b ), where a ∈ Z ∪ {−∞} , b ∈ Z ∪ { + ∞} ,a sequence of points { y k } k ∈ I is called a d -pseudotrajectory if the followinginequalities hold:dist( y k +1 , f ( y k )) < d, k ∈ Z , k, k + 1 ∈ I. ∗ Max Plank Institute for Mathematics in the Sciences, Inselstrasse 22, Leipzig,04103, Germany; Chebyshev Laboratory, Saint-Petersburg State Univeristy, 14th line ofVasilievsky Island, 29B, Saint-Petersburg, 199178, Russia; [email protected] efinition 1. We say that f has the shadowing property if for any ε > d > d -pseudotrajectory { y k } k ∈ Z there existsa trajectory { x k } k ∈ Z such thatdist( x k , y k ) < ε, k ∈ Z . (1)In this case, we say that the pseudotrajectory { y k } is ε -shadowed by { x k } .The study of this problem was originated by Anosov [6] and Bowen [7].This theory is closely related to the classical theory of structural stability.Let G be a smooth compact Riemannian manifold of class C ∞ withoutboundary with metric dist and let f ∈ Diff ( G ). It is well known that a dif-feomorphism has the shadowing property in a neighborhood of a hyperbolicset [6,7] and a structurally stable diffeomorphism has the shadowing propertyon the whole manifold [8, 9]. At the same time, it is easy to give an exampleof a diffeomorphism that is not structurally stable but has shadowing prop-erty (see [10], for instance). Thus, structural stability is not equivalent toshadowing.Relation between shadowing and structural stability was studied in sev-eral contexts. It is known that the C -interior of the set of diffeomorphismshaving the shadowing property coincides with the set of structurally stablediffeomorphisms [11] (see [12] for a similar result for the orbital shadowingproperty). Abdenur and Diaz conjectured that a C -generic diffeomorphismwith the shadowing property is structurally stable; they have proved thisconjecture for the so-called tame diffeomorphisms [13].Analyzing the proofs of the first shadowing results by Anosov [6] andBowen [7], it is easy to see that, in a neighborhood of a hyperbolic set,the shadowing property is Lipschitz (and the same holds in the case of astructurally stable diffeomorphism [4]). Definition 2.
We say that f has the Lipschitz shadowing property if thereexist ε , L > ε < ε and d -pseudotrajectory { y k } k ∈ Z with d = ε/L there exists a trajectory { x k } k ∈ Z such that inequalities (1)hold.Recently [14] it was proved that a diffeomorphism f ∈ C has Lipschitzshadowing property if and only if it is structurally stable (see [10, 15] for asimilar results for periodic and variational shadowing properties).In the present paper, we are interested which type of shadowing is possiblefor non-hyperbolic diffeomorphisms. The following notion will be importantfor us [16]: 2 efinition 3. We say that f has the finite H¨older shadowing property withexponents θ ∈ (0 , ω ≥ θ, ω )) if there exist d , L, C > d < d and d -pseudotrajectory { y k } k ∈ [0 ,Cd − ω ] there exists a tra-jectory { x k } k ∈ [0 ,Cd − ω ] such thatdist( x k , y k ) < Ld θ , k ∈ [0 , Cd − ω ] . S. Hammel, J. Yorke, and C. Grebogi made the following conjecture basedon results of numerical experiments [1, 2]:
Conjecture 1.
A typical dissipative map f : R → R with positive Lya-punov exponent satisfies FinHolSh(1 / , / . In [1, 2], the precise mathematical meaning of word “typical” was notprovided.There are plenty of not structurally stable examples satisfying FinHolSh(1 / , / Theorem 1.
If a diffeomorphism f ∈ C satisfies FinHolSh( θ, ω ) with θ > / , θ + ω > , then f is structurally stable. The paper is organized as follows. In Section 2, we formulate exact state-ments of the results. In Section 3, we formulate a particular problem forrandom walks and prove its equivalence to the shadowing property. In Sec-tion 4, we give a proof of the main result.3
Main Result
Let Σ = { , } Z . Endow Σ with the standard probability measure ν and thefollowing metric:dist( { ω i } , { ˜ ω i } ) = 1 / k , where k = min {| i | : ω i = ˜ ω i } . For a sequence ω = { ω i } ∈ Σ denote by t ( ω ) the 0th element of the sequence: t ( ω ) = ω . Define the “shift map” σ : Σ → Σ as follows:( σ ( ω )) i = ω i +1 . Consider the space Q = Σ × R . Endow Q with the product measure µ = ν × Leb and the maximum metric:dist(( ω, x ) , (˜ ω, ˜ x )) = max(dist( ω, ˜ ω ) , dist( x, ˜ x )) . For q ∈ Q and a > B ( a, q ) the open ball of radius a centeredat q .Fix λ , λ ∈ R satisfying the following conditions0 < λ < < λ , λ λ = 1 . (2)Consider the map f : Q → Q defined as follows: f ( ω, x ) = ( σ ( ω ) , λ t ( ω ) x ) . For q ∈ Q , d > N ∈ N let Ω q,d,N be the set of d -pseudotrajectoriesof length N starting at q = q . If we consider q k +1 being chosen at randomin B ( d, f ( q k )) uniformly with respect to the measure µ , then Ω q,d,N forms afinite time Markov chain. This naturally endows Ω q,d,N with a probabilitymeasure P . See also [20] for a similar concept for infinite pseudotrajectories.For ε > p ( q, d, N, ε ) be the probability of pseudotrajectory in Ω q,d,N to be ε -shadowable. Note that this event is measurable since it forms anopen subset of Ω q,d,N . Lemma 1.
Let q = ( ω, x ) , ˜ q = ( ω, . For any d, ε > , N ∈ N , the followingequality holds: p ( q, d, N, ε ) = p (˜ q, d, N, ε ) . roof. Consider { q k = ( ω k , x k ) } ∈ Ω q,d,N . Put r k := x k +1 − λ t ( ω k ) x k . Considera sequence { ˜ q k = ( ω k , ˜ x k ) } , where˜ x = 0 , ˜ x k +1 = λ t ( w k ) x k + r k . The following holds:1. the correspondence { q k } ↔ { ˜ q k } is one-to-one and preserves the prob-ability measure;2. for any ε > { q k } is ε -shadowed by a trajectory of apoint ( ω, x ) if and only if { ˜ q k } is ε -shadowed by a trajectory of a point( ω, x − x ).These statements complete the proof of the lemma.For d, ε > N ∈ N define p ( d, N, ε ) := Z ω ∈ Σ p (( ω, , d, N, ε )d ν. Note that the integral exists since for fixed d , N , ε , the value p (( ω, , d, N, ε )depends only on a finite number of entries of ω . The quantity p ( d, N, ε ) canbe interpreted as the probability of a d -pseudotrajectory of length N to be ε -shadowed.The main result of the paper is the following: Theorem 2.
For any λ , λ ∈ R satisfying (2) there exist ε > , < c < ∞ such that for any ε < ε , the following holds:1. If c < c , then lim N →∞ p ( ε/N c , N, ε ) = 0 ;2. if c > c , then lim N →∞ p ( ε/N c , N, ε ) = 1 . Remark 1.
Later (Lemma 2) we prove that for any N ∈ N , L > ε , ε ∈ (0 , ε ), the equality p ( ε /L, N, ε ) = p ( ε /L, N, ε ) holds. Hence the resultof Theorem 2 actually does not depend on the value of ε . Remark 2.
Due to Remark 1 analog of the Hammel-Grebogi-Yorke conjec-ture for map f suggests that p ( ε/N, N, ε ) is close to 1. Hence, if c >
1, thenHammel-Grebogi-Yorke conjecture is not satisfied. For an example of suchparameters see Remark 3. 5
Equivalent Formulation
Let a = ln λ , a = ln λ . Consider the following random variable: γ = ( a with probability 1/2 ,a with probability 1/2 . Fix
N >
0. Consider the random walk { A i } i ∈ [0 , ∞ ) generated by γ andindependent uniformly distributed in [ − ,
1] variables { r i } i ∈ [0 , ∞ ) . Define asequence { z i } i ∈ [0 ,N ] as follows: z = 0 , z i +1 = z i + r i +1 e A i +1 . (3)For given sequences ( { A i } i ∈ [0 ,N ] , { r i } i ∈ [0 ,N ] ) define B ( k, n ) := e A k + A n e A k + e A n | z n − z k | = e A n e A k + e A n (cid:12)(cid:12) e A k z n − e A k z k (cid:12)(cid:12) ,K ( { A i } , { r i } ) := max ≤ k There exist ε > , L > such that for any d ≥ , L > L , N ∈ N satisfying Ld < ε the following equality holds: p ( d, N, Ld ) = s ( N, L ) . Proof. Let us choose ε , L > ω, ˜ ω ) < ε , then t ( ω ) = t (˜ ω )and the map σ satisfies the Lipschitz shadowing property with constants ε , L .Fix d < d , N > L > L satisfying Ld < ε . Let us choose ω at random according to the probability measure ν and a pseudotajectory { q k } = { ( ω k , x k ) } ∈ Ω ( ω, ,d,N according to the measure P (see Section 2).Consider the sequences γ k = a t ( ω k ) , A k = k X i =0 γ i , r k = ( x k − λ t ( ω k − ) x k − ) /d. r k are independent uniformly distributed in [ − , 1] and γ k areindependent and distributed according to γ .Below we prove that the sequence { q k } can be Ld -shadowed if and onlyif L ≥ K ( { A i } , { r i } ) . (4)Assume that the pseudotrajectory ( ω k , x k ) is Ld -shadowed by an exacttrajectory ( ξ k , y k ). By the choice of ε , the following equality holds: t ( ω k ) = t ( ξ k ) . (5)Now let us study the behavior of the second coordinate. Note that y k +1 = λ t ( ξ k ) y k = e γ k y k , y n = e A n − A k y k , (6) x n = e A n − A k x k + e A k ( z n − z k ) , where z k are defined by (3). Hence,( y n − x n ) = e A n − A k ( y k − x k ) + e A k ( z n − z k ) . From this equality it is easy to deduce thatmax( | y k − x k | , | y n − x n | ) ≥ B ( k, n )and the equality holds if ( y k − x k ) = − ( y n − x n ). Hence, inequality (4) holds.Now let us assume that (4) holds and prove that ( w k , x k ) can be Ld -shadowed. Let us choose a sequence { ξ k } which Ld -shadows { w k } , thenequalities (5) hold.For y ∈ R define y k by relations (6) and consider function F : R → R defined as follows: F ( y ) = max ≤ k ≤ N | y k − x k | . Since the function F is continuous, it is easy to show that it attains a min-imum for some y . Denote L ′ := min y ∈ R F ( y ) and let y be such that L ′ = F ( y ). Let D = { k ∈ [0 , N ] : | y k − x k | = F ( y ) } . Let us consider twocases. Case 1. For all k ∈ D the value y k − x k has the same sign. Without lossof generality, we can assume that these values are positive. Then for smallenough δ > 0, the inequality F ( y − δ ) < F ( y ) holds, which contradicts thechoice of y . 7 ase 2. There exists indices k, n ∈ D such that the values y k − x k and y n − x n have different signs. Then ( y k − x k ) = − ( y n − x n ), and hence L ′ = B ( k, n ) ≤ K ( { A i } , { z i } ). Note that shadowing problems for the maps f and f − are equivalent (up toa constant multiplier at d ). In what follows, we assume that λ λ > 1. Put v := E ( γ ) = ( a + a ) / > , M := (ln N ) , w := v/ . In the proof of Theorem 2, we use the following statements. Lemma 3 (Large Deviation Principle, [22, Secion 3]) . There exists an in-creasing function h : (0 , ∞ ) → (0 , ∞ ) such that for any ε > and δ > andfor large enough n , the following inequalities hold: P (cid:18) A n n − E ( γ ) < − ε (cid:19) < e − ( h ( ε ) − δ ) n .P (cid:18) A n n − E ( γ ) < − ε (cid:19) > e − ( h ( ε )+ δ ) n . Lemma 4 (Ruin Problem, [21, Chapter XII, § 4, 5]) . Let b be the uniquepositive root of the equation (cid:0) e − ba + e − ba (cid:1) = 1 . For any δ > and for large enough C > , the following inequalities hold: P ( ∃ i ≥ A i ≤ − C ) ≤ e − C ( b − δ ) , (7) P ( ∃ i ≥ A i ≤ − C ) ≥ e − C ( b + δ ) , (8)Put c = 1 /b . Due to Lemma 2, it is enough to prove the following:(S1) If c < c , then lim N →∞ s ( N, N c ) = 0.(S2) If c > c , then lim N →∞ s ( N, N c ) = 1. Remark 3. For λ = 1 / λ = 3 the inequalities b < c > c > λ = 1 / λ = 2.Below we prove items (S1) and (S2).8 .1 Proof of (S1) Assume that c < /b . Let us choose c ∈ ( c, /b ) and δ > c ( b + δ ) < . (9)Consider the following events: I = {∃ i ∈ [0 , M ] : A i ≤ − c ln N ; and A M ≥ } ,I = {∃ i ∈ [0 , M ] : A i ≤ − c ln N } ,I = {∃ i ∈ [0 , M ] : A i ≤ − wM } ,I = { A M − A M ≤ wM } . The following holds: P ( I ) ≥ P ( I ) − P ( I ) − P ( I ) , (10) P ( I ) ≥ P ( ∃ i ≥ A i ≤ − c ln N ) − P ( ∃ i > M : A i ≤ − c ln N ) ≥ e − c ln N ( b + δ ) − N X i = M +1 P ( A i ≤ ≥ N − c ( b + δ ) − N X i = M +1 e − ih ( v ) ≥ N − c ( b + δ ) − − e − h ( v ) e − ( M +1) h ( v ) ≥ N − c ( b + δ ) + o ( N − ) . (11)Similarly P ( I ) ≤ ∞ X i = M +1 P ( A i ≤ 0) = o ( N − ) , (12) P ( I ) ≤ e − Mh ( v − w ) = o ( N − ) . (13)From inequalities (10)-(13) we conclude that P ( I ) ≥ N − c ( b + δ ) + o ( N − ) . (14)Assume that the event I has happened and let i ∈ [0 , M ] be one of theindices satisfying the inequality A i < − c ln N . Note that the followingevents are independent: J = { r i ∈ [1 / 2; 1] } , J = n z M − z ≥ r i e A i o . P (cid:18) z M − z ≥ e A i (cid:19) ≥ P ( J ) P ( J ) = 1 / · / / P ( B (0 , M ) > N c / ≥ P ( I ) = 18 N − c ( b + δ ) + o ( N − ) . Note that for large enough N , the inequality N c < N c / P ( B (0 , M ) > N c ) ≥ N − c ( b + δ ) + o ( N − ) . Similarly, for any k ∈ [0 , N − M ], P ( B ( k, k + 2 M ) > N c ) ≥ N − c ( b + δ ) + o ( N − ) . Note that the events in the last expression for k = 0 , M, · M, . . . ([ N/ (2 M )] − M are independent, and hence P ( ∃ k ∈ [0 , N − M ] : B ( k, k + 2 M ) > N c ) ≥ − (cid:18) − (cid:18) N − c ( b + δ ) + o ( N − ) (cid:19)(cid:19) [ N/ (2 M )] . (15)Using (9), we conclude that (cid:18) N − c ( b + δ ) + o ( N − ) (cid:19) [ N/ (2 M )] ≥ (cid:18) N − c ( b + δ ) + o ( N − ) (cid:19) (cid:18) N N ) − (cid:19) = 116(ln N ) N − c ( b + δ ) + o ( N − ) −−−→ N →∞ ∞ and hence (cid:18) − (cid:18) N − c ( b + δ ) + o ( N − ) (cid:19)(cid:19) [ N/ (2 M )] −−−→ N →∞ . (16)Relations (15), (16) imply that P ( K ( { A i } i ∈ [0 ,N ] , { r i } i ∈ [0 ,N ] ) > N c ) −−−→ N →∞ . Hence, lim N →∞ s ( N, N c ) = 0 . .2 Proof of (S2) Let c > /b . 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