Generalizations of analogs of theorems of Maizel and Pliss and their application in Shadowing Theory

We generalize two classical results of Maizel and Pliss that describe relations between hyperbolicity properties of linear system of difference equations and its ability to have a bounded solution for every bounded inhomogeneity. We also apply one of this generalizations in shadowing theory of diffeomorphisms to prove that some sort of limit shadowing is equivalent to structural stability.


Introduction
In [10] Perron defined property (B) for systems of differential equations. The property is that an inhomogeneous system of differential equations has bounded solution for every bounded inhomogeneity. In [7] A. Maizel proved a theorem that links property (B) on the half-line with the hyperbolicity property. In [16] Pliss characterized an analog of the property (B) on the full line in terms of hyperbolicity on two half-lines. The proof of the Pliss' theorem is based on the Maizel's theorem.
The property (B) if often called admissibility. There exist many papers devoted to study of this property and its analogs. For references see [1,6,5].
We prove generalizations of both theorems for difference equations for the case of spaces of sequences with prescribed decay rate for the case of finite dimensions and bounded coefficients. All similar theorems that were proved so far deal only with spaces having certain homogeneity properties.
The discrete analog of the Pliss' theorem is widely used in shadowing theory (see [14,18,15]). As an application of its generalized version we introduce the notion of Lipschitz two-sided limit shadowing property and prove that this property is equivalent to the structural stability.

Definitions
Let I be either Z + = {k ∈ Z | k ≥ 0 } or Z − = {k ∈ Z | k ≤ 0 } or Z. Let A = {A k } k∈I be a sequence of linear isomorphisms R d → R d indexed by integers from I. Consider homogeneous and inhomogeneous equations associated with this sequence.
x k+1 = A k x k , k ∈ I (2.1) Remark 2.1. For I = Z + we take f k to be defined for k ≥ 0 and f 0 = 0.
Define an analog of fundamental matrix for equations (2.1): We fix ω ≥ 0. We use linear subspaces of the space of sequences of vectors from R d , indexed by integers from I. Denote the Banach space of sequences with bounded norm x ω = sup k∈I |x k | (|k| + 1) ω by N ω (I). Such spaces has been already studied in the similar context (see [2]). It is important to note that these spaces are neither homogeneous in the sense of Baskakov (see [1]) nor translation-Invariant in the sense of Sasu (see [17]). Definition 1. We say that a sequence A has Perron property B ω (I) if for any sequence f ∈ N ω (I) there exists a solution of the inhomogeneous system of difference equations with inhomogeneity f that belongs to N ω (I).
We use the following definition from [11]: Definition 2. We say that a sequence A is hyperbolic on I if there exist constants K > 0, λ ∈ (0, 1) and projections P k , Q k , k ∈ I such that if S k = P k R d and U k = Q k R d then the following holds: (2.4) |Φ k,l v| ≤ Kλ k−l |v|, v ∈ S l , k ≥ l; (2.5) |Φ k,l v| ≤ Kλ l−k |v|, v ∈ U l , k ≤ l; (2.6) estimates hold Also everywhere here we mean that all indices are from I.

Main Results
We prove the following theorem in Section 4.1: Theorem 1 (a generalization of the discrete analog of Maizel Theorem). Let I = Z + and the norms of all matrices A k and A k −1 be bounded by M > 0. A sequence A has property B ω (I) iff it is hyperbolic on Z + .
We prove the following theorem in Section 4.2: Theorem 2 (a generalization of the discrete analog of Pliss Theorem). Let I = Z + and the norms of all matrices A k and A k −1 be bounded by M > 0. A sequence A has property B ω (I) iff it is hyperblic on both Z + and Z − and the spaces B + (A) and B − (A) are transverse. Here

Generalizations of discrete analogs of theorems of Maizel and Pliss
We prove generalizations of theorems of Maizel and Pliss for the case of difference equations. 4.1. Maizel Theorem. Let I = Z + . For brevity we write N ω instead of N ω (I). Assume that the sequence A has property B ω (I). Denote Since our equations are linear and N ω is a linear space, V 1 is also a linear space. Denote the orthogonal complement of V 1 by V 2 and orthogonal projection onto V 1 by P.
It is easy to see that the following holds: For any sequence A the operator T : N ω → N ω from the previous statement is continuous. In particular there exists a positive r such that Proof. Fully analogous to the proof of Statement 4 from [3].
From now on we use the operator T and number r from the previous statement. Also we suppose that r ≥ 1 and that the number M from the statement of Theorem 1 satisfies inequality rM ≥ 1.
It is easy to see that the following holds.
when the series in the second summand converges.
Here we take f 0 equal to 0.
Remark 4.4. We do not care about exact values of the constants and always assume that the value of M is big enough and neglect constants like one that bounds from above the sequence (1 − 1/k) β for real β.
Remark 4.5. The following formula can be interpreted as an analog of the Green's function for difference equations So formula (4.1) can be rewritten in a more compact way: Lemma 4.6. Let k 0 , k 1 , k be nonnegative integers and ξ ∈ R d be a nonzero vector. Then the following inequalities hold Proof. Fix nonnegative integers l 0 , l 1 such that l 0 ≤ l 1 . Consider a sequence f with f i = 0, i > l 1 . Then formula (4.2) looks like: For l ≥ l 1 all the indices u in the sum are less or equal than l 1 and the first string from the definition of G l,u is used. The previous equality turn into the following: Thus the vector y l for l ≥ l 1 is an image of the vector from V 1 that is independent of l. This means that all the sequence y except a finite number of entries is a solution of homogeneous equation (2.1) with initial conditions from V 1 . Thus y belongs to N ω . Using that f 0 = 0 we obtain So y = T f and therefore y ω ≤ r f ω . Let x i = X i ξ. We define the sequence f : Then f ω = 1. Substituting the formula for a solution in the inequality from Statement 4.2 we obtain For l 1 = l = k, l 0 = k 0 from (4.5) we obtain that if k ≥ k 0 then To prove the second inequality from the statement of the lemma we do the similar. The important thing to notion here is that in the second string of the definition of G k,s the inequality is strict. Then for l = k − 1, l 0 = k, l 1 = k 1 from (4.5) we obtain that for 0 < k ≤ k 1 we have Now we prove the second inequality of the statement of the lemma for the case when 0 = k < k 1 using the previous inequality for k = 1 : For k = k 1 = 0 the inequality is obvious.
Lemma 4.7. Let k 0 , k 1 , k, s be nonnegative integers and ξ be a vector.
The following inequalities are satisfied: for k 1 ≥ s ≥ k; Proof. Denote We prove inequality (4.6). Since P ξ = 0, it is easy to see that If we consequently use this inequality enough times, we obtain To prove the second inequality from the statement of the lemma recall that ψ k > 0. Analogous to the proof of the first inequality form the statement of the lemma, Again, if we consequently use this inequality enough times and use Remark 4.4 we get inequality (4.7).

4.1.2.
Proof of the discrete analog of the Maizel theorem.
Theorem 3. The following inequalities holds Proof. Fix a natural s ≥ 1 and a unit vector ξ. Define a sequence y : The sequence y coincides (except a finite number of entries) with a solution of homogenous equation (2.1) with initial conditions from V 1 and therefore y belongs to N ω . Now we define a sequence f in such a way as the sequence y is a solution of inhomogeneous equation (2.2) with inhomogeneity f : It is easy to see that in this case y becomes a solution. This means that y = T f. Thus y ω ≤ r f ω = r(s + 1) ω . We prove the first inequality for the operator norms from the statement of the theorem. Using the definition of the sequence y we can write Since ξ can be any unit vector, this gives us an estimate for the operator norms of X k P X −s . Now we can replace x by the solution of the homogeneous equation x k = X k ξ and substitute in the previous inequality instead of ξ : Let P ξ = 0. Consequently using inequalities (4.3) and (4.6) for k 0 = s and (4.8) for k = s we get If P ξ = 0 then the resulting inequality is obvious. Since x s = X s ξ and X s is an isomorphism, we have an estimate for the operator norm: In this reasoning we have used only the fact that inequality (4.8) is satisfied for s = k. This is also true for s = k = 0 since P ≤ 1. Therefore, we proved the estimate for 0 ≤ s ≤ k.
The proof of the second estimate from the statement for s > k is similar to the proof of the first estimate. The only small differences are due to the fact that now we cannot use an analog of inequality (4.8) for k = s because in the definition of the sequence y the numbers s and k cannot be equal. The following inequality can be proved in a very same manner as one in the proof of the first estimate: For k = s − 1 we multiply the vector inside the norm brackets by A s−1 : After that the proof is fully analogous to the proof of the first estimate.
Lemma 4.8. For any λ ∈ (0, 1) there exists a constant C > 0 depending only on λ and ω such that Proof. We estimate first summand from (4.10). To do this it is enough to estimate the corresponding integral: Now we estimate separately the two integrals from the previous formula. The first one can be estimated in the following way: The second one can be estimated in the following way: Here is the estimate for the second summand from (4.10): Now we prove Theorem 1. We show how property B ω (Z + ) implies hyperbolicity. Let Using Theorem 3 it is easy to check that the first two conditions from the definition of hyperbolicity and inequalities from Remark 2.4 are satisfied. The uniform estimates of the norms of the projectors P k and Q k are due to Remark 2.3. Now we show how hyperbolicity implies B ω (Z + ). Let f ω < R. We define a sequence y k as follows: Let K, λ be the numbers from the definition of hyperbolicity of the sequence A. Then using the Lemma 4.8 we write this estimates:

Pliss Theorem.
Let I = Z and ω ≥ 0. We assume that the norms of A k and A k −1 are bounded. Statement 4.9. If a sequence A have property B ω (Z) then it is hyperbolic on Z + and Z − with corresponding stable and unstable spaces S + k , U + k and S − k , U − k . Proof. Since we have property B ω (Z) for the sequence A, we also have properties B ω (Z + ) for its positive part {A k } ∞ k=0 and B ω (Z − ) for its negative part {A k } 0 k=−∞ . Then the hyperbolicity on Z + follows directly from the Maizel theorem. In particular this means that there exist stable and unstable subspaces S + k and U + k . Hyperbolicity on Z − also follows from Maizel theorem but it should be applied not to equations (2.1) and (2.2) but to the equations with inverted time: Thus hyperbolic sequence {A −1 −k } ∞ k=0 has spaces S k and U k . We denote U − k = S k and S − k = U k keeping in mind the sequence {A k } 0 k=−∞ .
Take a sequence of numbers a k whose entries equal 0 for negative indices and are in (0, 1) for nonnegative. We construct a sequence θ k that will drive us to a contradiction: Vectors f i belong to U + i , i ≥ 0, because Φ i,j maps U + j to U + i by the hyperbolicity definition. The series from the definition of θ k converges: Recall that the sequence {θ k } k∈Z + belongs to N ω (Z + ) : It is easy to see that the sequence θ k is a solution of the inhomogeneous equation (2.2). Moreover, the following equality is satisfied This means that for θ 0 the same thing as for η is true (4.11) θ 0 = y 1 + y 2 , with y 1 ∈ S + 0 , y 2 ∈ U − 0 . Because of Remark 4.10 there existss the only solution {ψ k } ∈ N ω (Z) of inhomogeneous equation (2.2) with inhomogeneity f such that ψ 0 ∈ U − 0 . Every other solution of the inhomogeneous equation can be obtained adding a solution of the homogeneous equation. Thus is a linear space. From this we obtain that the vector ψ 0 − θ 0 belongs to S + 0 . Therefore if we denote y 1 = θ 0 − ψ 0 , y 2 = ψ 0 then we have θ 0 = y 1 + y 2 , what contradicts inequality (4.11). Now we prove Theorem 2. At first we show how the existence of property B ω (Z) follows from the hyperbolicity on Z + and Z − and transversality of B + (A) and B − (A). Fix a sequence f ∈ N ω (Z). Consider its positive and negative parts Since the sequence A is hyperbolic on both Z + and Z − , by the Maizel theorem its positive and negative parts A + = {A k } k∈Z + and A − = {A k } k∈Z − have properties B ω (Z + ) and B ω (Z − ) correspondingly. Thus there exist solutions ψ + ∈ N ω (Z + ) and ψ − ∈ N ω (Z − ) of equations (2.2) for I = Z + and I = Z − with inhomogeneities f + and f − correspondingly. If ψ + 0 = ψ − 0 then the sequence ψ with ψ k = ψ − k , k ≤ 0, ψ k = ψ + k , k > 0 is a solution of the inhomogeneous system (2.2) for I = Z and belongs to N ω (Z). If ψ + 0 = ψ − 0 then the solutions ψ + and ψ − can be modified by solutions of the homogeneous systems: we show that there exist solutions φ + ∈ N ω (Z + ) and φ − ∈ N ω (Z − ) of the homogeneous system (2.1) for I = Z + and I = Z − such that The last condition can be rewritten as (4.12) Recall that B + (A) = S + 0 and B − (A) = U − 0 since for a hyperbolic sequence solutions of the corresponding homogeneous linear system of difference equations are either tend to infinity with exponential speed or or tend to zero with exponential speed.
By assumption the spaces B + (A) and B − (A) are transverse so every vector from R d can be represented as a difference from the right hand sid of (4.12). In particular we can obtain the left hand side of (4.12).
To obtain hyperbolicity and transversality from property B γ we only need to use

Application of The Generalization of Discrete Analog of Pliss Theorem
In this section we apply the generalized version of Pliss theorem for difference equations in shadowing theory.
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 [13,9]). In particular the connections between shadowing and structural stability are interesting.
It is well known that a structurally stable system has shadowing property and this property is Lipschitz (see [13]). Recently it was shown that Lipschitz shadowing implies structural stability (see [14]). Also structural stability follows from Hölder shadowing property under some additional assumptions (see [18]). Moreover it is known that structurally stable system has two-sided limit shadowing property but even the C 1 -interior of the set of diffeomorphisms having two-sided shadowing property (without prescribing the speed of convergence to zero of estimates on each step) not coincide with the set of structurally stable diffeomorphisms (see [12]).
We show that Lipschitz two-sided limit shadowing property is equivalent to structural stability.

5.1.
Definitions. Let f be a homeomorphism of a metric space (M, dist) and consider a dynamical system that is generated by f. Definition 3. We say that a sequence {x k } k∈Z of points of M is a d-pseudotrajectory of the dynamical system f if the following inequalities are satisfied Let γ be a nonnegative real number.
Definition 4. We say that a sequence {x k } k∈Z of points of M is a γ-decreasing dpseudotrajectory of the dynamical system f if the following inequalities are satisfied Definition 5. We say that the homeomorphism f has Lipschitz two-sided limit shadowing property with exponent γ if there exist positive constants d 0 , L such that for any γ-decreasing d-pseudotrajectory {x k } with d ≤ d 0 there exists a point p ∈ M such that dist(x k , f k (p)) ≤ Ld(|k| + 1) −γ , k ∈ Z.
We write f ∈ LT SLmSP (γ) in this case.
Remark 5.1. In [13], where some similar shadowing properties has been studied. It is shown that in the neighborhood of a hyperbolic set both L p -shadowing and weighted shadowing (for a special choice of weights) are present.

Main results.
A diffeomorphism f of a smooth manifold M is said to be structurally stable if there exists a neighborhood U of the diffeomorphism f in the C 1 -topology such that any diffeomorphism g ∈ U is topologically conjugate to f.
Definition 6. We say that for a diffeomorphism f the analytical transversality condition is satisfied at a point p if Theorem (Mañé, [8]). Diffeomorphism f is structurally stable iff the analytical transversality condition is satisfied at every point p of M.
At first we prove one simple lemma Lemma 5.2. If for a sequence {w k } k∈Z from N γ there exists a constant Q such that for any integer N > 0 there exists a sequence {v N k } k∈[−N,N ] of vectors from R d satisfying equalities then there exists a sequence {v k } k∈Z such that it satisfy the same inequalities (5.1) for every integer k and {v k } k∈Z γ ≤ Q.
Proof. To obtain a sequence needed we use a diagonal procedure (we take v N k = 0, k / ∈ [−N, N ]) and pass to a limit in inequalities (5.1). Despite the convergence is not uniform in general the sequence we get as a result has all the necessary properties. Proof. Using Theorem 2 we show that if we have Lipschitz two-sided limit shadowing property then the analytical transversality condition is satisfied at every point. After that we just apply the Mane theorem.
Fix a point p ∈ M, denote p k = f k (p) and define linear isomorphisms A k = Df (p k ) for k ∈ Z. We denote a ball in M with a radius r and a center x by B(r, x) and a ball in T x M with radius r and center 0 by B T (r, x).
The fact that the norms of all A k and A k −1 are bounded follows from the compactness of the manifold. We prove that under our assumptions property B γ (Z) is satisfied for the sequence of matrices A k . After that we will be able to use Theorem 2.
Let exp x : T x M → M be a standard exponential mapping. There exists a r > 0 such that for any point x ∈ M the mapping exp x is a diffeomorphism of a ball B T (r, x) onto its image and exp −1 x is a diffeomorphism of a ball B(r, x) onto its image. Moreover, we may assume that the smallness of r allows us to write the following estimates for relations between distances in the manifold and in a tangent space: if v, w ∈ B T (r, x) then if y, z ∈ B(r, x) then

Consider mappings
From the well-known properties of an exponential mapping we deduce that D exp x (0) = Id; therefore DF k (0) = Df (p k ). Since M is compact for any ε > 0, we can find a δ > 0 such that if |v| ≤ δ, then for Let L, d 0 be the constants from the definition of LT SLmSP (γ). We prove that for any sequence of vectors {z k } k∈Z ∈ N γ satisfying {z k } k∈Z γ < 1 there exists a sequence {v k } k∈Z ∈ N γ that is a solution of equations After this if we use Theorem 2 then we obtain that the analytical transversality condition is satisfied at the point p.
We show that the conditions of Lemma 5.2 are satisfied. We fix natural N, small positive d, and define vectors a k : Now we assume that d is small enough so that all the points of M that appear belong to the corresponding balls B(r, p k ) and all tangent vectors from T p k M that appear belong to the corresponding balls B T (r, p k ).
Using the definition of a k it is easy to see that This means that |g k (t k )| can be made as small as we need only by decreasing of d. Then . Now we show that this is the sequence we have looked for: . The fact that this sequence is a solution of equations (5.5) is obvious. To estimate its norm in the space N γ by a number independent of N we write the following:

5.4.
Structural stability implies Lipschitz two-sided limit shadowing property. We use the method from [13] to prove that structural stability implies Lipschitz two-sided limit shadowing property. Let H k , k ∈ Z be a sequence of subspaces of R d . Consider a sequence of linear mappings A = {A k : H k → H k+1 }.
Definition 7. We say that a sequence A has property (C) with constants N > 1 and λ ∈ (0, 1) if for any integer k there exist projections P k , Q k such that if S k = P k H k and U k = Q k H k then the following conditions are satisfied: Theorem 5. Let γ > 0 and let A have property (C) with constants N > 1 and λ ∈ (0, 1). Consider a sequence of mappings f k : . Suppose that there exist constants κ, ∆ > 0 such that the following inequalities are satisfied: then there exists a sequence v k ∈ H k such that f k (v k ) = v k+1 and {v k } γ ≤ Ld.
Proof. Consider an operator G : We prove that the operator G maps N γ (Z) to N γ (Z) and is bounded. We prove that (|k| + 1) γ (G(x)) k ≤ C for k ≥ 0 (for k ≤ 0 the proof is analogous). We represent G(z) in the following form Lemma 4.8 allows us to estimate (k + 1) γ |(g 2 (z)) k + (g 3 (z)) k | . It remains to estimate only (k + 1) γ |(g 1 (z)) k | : The rest of the proof is fully analogous to the proof of Theorem 1.3.1 from [13].
Theorem 6. Let f be a structurally stable diffeomorphism of the closed Reimannian manifold M. Then f ∈ LT SLmSP (γ) for any γ ≥ 0.
Proof. This theorem is proved in the same way as Theorem 2.2.7 from [13]. We choose T such that µ = Cλ T 1 < 1. Lemma 5.5 shows that to prove that f ∈ LT SLmSP (γ) it is enough to show that f ′ = f T ∈ LT SLmSP (γ). We use Statement 5.4 for C = 1, λ 1 = µ and T = 1.
Let c be a radius such that for each p ∈ M the mapping exp p is a diffeomorphism of the ball E 2c (p) ⊂ T p M onto its image. We take a number d ′ < c such that for all point x, y ∈ M for dist(x, y) ≤ d ′ and y ′ = exp −1 x (y) the following inequalities are satisfied D exp −1 x (y)v ≤ (1 + ν), |D exp x (y ′ )v| ≤ (1 + ν). For β = ν we choose a number d ′′ such that part 5 of Statement 5.4 is satisfied. We assume that d ′ ≤ d ′′ .
Let x k be a γ-decreasing d-pseudotrajectory for d ≤ d ′ .
Consider the mappings f k : T x k → T x k+1 defined as follows . Then the mappings f k are defined on E d ′ (x k ). It is easy to see that |f k (0)| ≤ d(|k| + 1) −γ . Now we introduce the following notations: x k (f −1 (x k+1 )). Denote J k = Df k (0). Then J k = D exp −1 z (p)Df (p) = D exp −1 x k+1 (f (x k ))Df (x k ).
We take ∆ = d ′ and use Theorem 5. Then there exist numbers d 0 , L such that for f k (0) γ ≤ d ≤ d 0 there exists a sequence v k ∈ T x k such that

Acknowledgements
I am grateful to S. Yu. Pilyugin and S. Tikhomirov for their helpful comments and advices.