# Lipschitz inverse shadowing for nonsingular flows

Dmitry Todorov
Chebyshev laboratory, Saint-Petersburg State University, 14th line of Vasiljevsky Island, 29B, Saint-Petersburg, 199178, Russia; Dmitry Todorov. Email:
###### Abstract

We prove that Lipschitz inverse shadowing for nonsingular flows is equivalent to structural stability.

s

{classcode}

MSC 2010: 37C50, 34D30

## 1 Introduction

The notion of inverse shadowing was introduced by Pilyugin and Corless in [1] and by Kloeden and Ombach in [2]. They defined this notion for diffeomorphisms. Inverse shadowing for flows was first introduced in [3].

It is known that both Lipschitz shadowing and Lipschitz inverse shadowing properties for diffeomorphisms are equivalent to structural stability [4, 5, 6]. In [7] the authors proved that structural stability for flows implies inverse shadowing. In fact, they proved that structural stability implies Lipschitz inverse shadowing although they did not use this term in their paper.

We prove that Lipschitz inverse shadowing for flows without rest points implies structural stability.

## 2 Definitions

Let be a vector field on a Riemannian manifold with metric and let be a flow generated by We will only consider cases when is closed and when .

Let be a (small) positive number. {definition} We say that a mapping is a -method for flow if for any ,

 dist(Ψ(t+s,x),Φ(s,Ψ(t,x)))

and for any

We introduce several classes of -methods, following [7]. Let be a -method. Denote by the set of all functions from to Consider a mapping

 ~Ψ:M→(M)R,

defined as

 (~Ψ(x))(t)=Ψ(t,x),x∈M, t∈R.
• We say that the -method belongs to the class if the mapping is continuous in the pointwise-convergence topology on

• We say that the -method belongs to the class if the mapping is continuous in compact-open topology on

• We say that the -method belongs to the class if it is continuous as a mapping of the form

• We say that the -method belongs to the class if it is a flow of some vector field that satisfies

 d0(X,Y)≤d.

Here is the metric on the space of vector fields on

• We say that the -method belongs to the class if it is smooth as a mapping of the form

{remark}

Our definition of a -method is a definition of a family of pseudotrajectories (see the definition of a -pseudotrajectory of a flow in [4]). If a method belongs to one of the classes defined above, then the dependence of a pseudotrajectory on a point has some continuity (smoothness) properties.

Define as a set of all increasing homeomorphisms with We introduce also the following notation:

 Rep(δ)={α∈Rep ∣∣∣ ∣∣∣α(t)−α(s)t−s−1∣∣∣≤δ,t≠s}.
{definition}

Let be one of the classes of methods defined above. We say that a flow has Lipschitz inverse shadowing property (we write in this case) with respect to the class if for any point there exist constants such that for any -method of the class with there exist a point and a reparametrisation such that

 dist(Φ(t,p),Ψ(α(t),^p))
{remark}

It is easy to see that if is small enough and then

{remark}

Obviously the inclusion with respect to the class implies that with respect to the classes and

{remark}

It is proved in [7] that the inverse shadowing properties with respect to the classes of methods and are equivalent. We do not discuss the class Further away we write just about Lipschitz inverse shadowing without mentioning a class of methods, always meaning the class .

## 3 Main Results

The main result of the work is the following theorem:

{theorem}

Let be a closed manifold and let the flow have no rest points. Then is structurally stable iff

## 4 The idea of the proof

We will use the same idea that the authors of [6] used.

Fix a point Denote and Let be the orthogonal projections with kernels and let be the orthogonal complement to Denote Consider the following system of difference equations

 vk+1=Bkvk+bk+1, k∈Z. (4.1)

Let be the chain-recurrent set of a flow . The following is proved in [6]: {theorem} Let be a closed manifold. Suppose that there exists a constant such that for any point and any bounded sequence with entries from the corresponding equations (4.1) have a solution with the norm bounded by Then

• the set is hyperbolic;

• the strong transversality condition is fulfilled.

It is known ( e.g. [8] ), that the hyperbolicity of the chain-recurrent set implies Axiom In turn, Axiom and strong transversality condition imply structural stability. So the “only if” part of Theorem 3 will be proved if we prove that the conditions of the previous theorem are satisfied. The “if” part is proved in [7].

The reason we consider only vector fields without singularities is that unlike in [6] we cannot prove that singularities are isolated in If we knew that they are than we could easily show that they are hyperbolic and prove Theorem 3 holds.

{definition}

We say that a flow satisfy condition (UB) if

1. the norms of the derivatives of the flow with respect to initial data for are bounded. I.e., there exists

 Q1=maxx∈M, s∈[−1,1]∥∥∥∂Φ(s,x)∂x∥∥∥.
2. The lengths of vectors of the vector field are bounded. I.e., there exists

 Q2=maxx∈Rn|X(x)|.
3. The reminder in the Taylor expansion of the flow is uniformly bounded.

Let the manifold be covered by a countable number of open balls of the same radius each admitting a coordinate chart.

There exists a monotonous function with the following property. If for we fix a coorinate chart of the ball containing the point and denote the representation of the flow in this chart by the same letter , then for any , , , the following estimate is fulfilled

 ∣∣∣Φ(t+h1,x+h2)−Φ(t,x)−h1X(Φ(t,x))−∂Φ(t,x)∂xh2∣∣∣≤g1(|h1|+|h2|),

where tends to uniformly in for

Here we assume that all the points belong to , otherwise we can reparametrize the flow globally.

A flow generated by vector field on a closed manifold satisfies this condition.

At first we prove the solvability of equations that are different from equations (4.1):

###### Statement 4.1

Let the flow have Lipschitz inverse shadowing property and satisfy condition (UB). Then there exists a constant such that for any point and any inhomogeneity that satisfies for any natural there exists a sequence of real numbers such that the system of equations

 xk+1=Akxk+X(pk+1)sk+zk+1,k∈[−N,N−1] (4.2)

has a solution such that

The proof of this statement is the main difficulty and is given later. Now we show how the solvability of equations (4.2) implies the solvability of equations (4.1). The proof of the next corollary is similar to the proof of Lemma 2 from [6].

{corollary}

For any sequence that satisfies and for all integer there exists a solution of system of equations (4.1) such that

{proof}

Take in equations (4.2). Statement 4.1 guarantees that there exists a constant such that for any integer there exists a sequence such that system of equations (4.2) has a solution with the norm bounded by

Fix Note that The definition of the projections implies the inclusion for any Thus Now we have the equality

 Pk+1Ak=Pk+1AkPk.

Multiply equalities (4.2) by

 Pk+1xk+1=Pk+1Akxk+Pk+1X(pk+1)sk+Pk+1zk+1= =Pk+1AkPkxk+bk+1,k∈[−N,N−1].

Thus the sequence is a solution of equations (4.1) for a finite number of indices. Now we can pass to the limit as Due to the boundedness, has a limit whose norm is also bounded by

## 5 Proof of Statement 4.1 for M=Rn

At first we prove Statement 4.1 when and the vector field satisfies condition (UB). This will allow us to demonstrate the idea of the proof with less amount of technical details.

Fix a natural We are going to use a method from [5]. We will “inscribe” equations into a method and use the Lipschitz inverse shadowing property.

Let us construct a method of the class that contains the equations “inside”.

Let be constants from the definition of the and be a positive number which we decrease later.

Fix positive numbers and Assume that satisfies and Let be a function with the following properties (see Fig. 1)

• is smooth as a function of two arguments.

Let be a function such that it tends to zero faster than and satisfies

### 5.1 Construction of the method

Let be either or and be an integer from Suppose that the functions are already defined. Denote

 ~pk=Ψϰ(k,p−N),~Ak=∂Φ(1−τ,Ψϰ(k+τ,p−N))∂x,
 ^pk+1=Φ(1−τ,Ψϰ(k+τ)), ^p−N=p−N.

If define the function in the following way:

 Ωϰ,k+1(s,v)=^pk+1+ϰ~Ak(Ψϰ(k,p−N+v)−~pk+dzk)+ +ϰX(^pk+1)s+dzk+1−ϰd~Akzk.

We also define the interpolation function

 Γ(x,y,v,s)=γ(s,|v|)x+(1−γ(s,|v|))y.

Define the method in the following way (see Fig. 2):

 Ψϰ(t,x)=Φ(t,x),t∈R,x∉Br(p−N); Ψϰ(t,x)=Φ(t,x),t∈(−∞,1−τ],x∈Rn; Ψϰ(k+s,p−N+v)=Γ(Ωϰ,k(s,v),Φ(1−τ+s,Ψϰ(k−1+τ,p−N+v)),v,s), k∈[−N+1,N],s∈[−τ,τ],v∈Br(0); Ψϰ(k+τ+s,p−N+v)=Φ(s,Ψϰ(k+τ,p−N+v)), k∈[−N+1,N−1],s∈[0,1−2τ],v∈Br(0); Ψϰ(N+τ+s,p−N+v)=Φ(s,Ψϰ(N+τ,p−N+v)),s∈[0,∞),v∈Br(0).

Here is the ball of radius centered at .
{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9901883602142334, "perplexity": 399.6834795914369}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882572161.46/warc/CC-MAIN-20220815054743-20220815084743-00641.warc.gz"}