# Relations between Shadowing and Inverse Shadowing in Dynamical Systems

## Abstract

**:**

## 1. Introduction

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

**Definition**

**7.**

**Theorem**

**1.**

- (1)
- f has the Lipschitz shadowing property;
- (2)
- f has the Lipschitz inverse shadowing property;
- (3)
- f satisfies Axiom A and the strong transversality condition.

**Theorem**

**2.**

- (1)
- f has the shadowing property;
- (2)
- f has the inverse shadowing property;
- (3)
- f satisfies the ${C}^{0}$ transversality condition.

## 2. Shadowing Property Does Not Imply Inverse Shadowing Property

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

- (1)
- f satisfies Axiom A;
- (2)
- $\mathsf{\Omega}\left(f\right)=\{{p}_{1},{p}_{2},{p}_{3},{p}_{4}\}$, where ${p}_{1}$ is a repelling fixed point, ${p}_{4}$ is an attracting fixed point, and ${p}_{2}$ and ${p}_{3}$ are saddle fixed points such that$$dim{W}^{u}\left({p}_{2}\right)=dim{W}^{s}\left({p}_{3}\right)=1;$$
- (3)
- ${W}^{u}\left({p}_{2}\right)\cap {W}^{s}\left({p}_{3}\right)=O(x,f)\ne \varnothing $ (here $O(x,f)$ denotes the trajectory of a point x);
- (4)
- f has the shadowing property.

**Theorem**

**3.**

**Proof.**

- (1)
- for any point $p\in M$, the inequalities$$\mathrm{dist}({g}^{k}\left(p\right),{p}_{3})<\alpha ,\phantom{\rule{1.em}{0ex}}k>0,$$
- (2)
- for any point $p\in M$, the inequalities$$\mathrm{dist}({g}^{k}\left(p\right),{p}_{2})<\alpha ,\phantom{\rule{1.em}{0ex}}k<0,$$

## 3. Inverse Shadowing Property Does Not Imply Shadowing Property

**Example**

**1.**

**Example**

**2.**

- for $x=(t,{\alpha}_{m},{e}_{2m}h\left(t\right))\in {s}_{m}$,$$f\left(x\right)=\left(t/2,{\alpha}_{m+1},{e}_{2(m+1)}h(t/2)\right);$$
- for $x=(t,{\beta}_{m},{e}_{2m+1}h\left(t\right))\in {u}_{m}$,$$f\left(x\right)=\left(2t,{\beta}_{m+1},{e}_{2(m+1)+1}h\left(2t\right)\right).$$

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Petrov, A.A. Relations between Shadowing and Inverse Shadowing in Dynamical Systems. *Axioms* **2019**, *8*, 11.
https://doi.org/10.3390/axioms8010011

**AMA Style**

Petrov AA. Relations between Shadowing and Inverse Shadowing in Dynamical Systems. *Axioms*. 2019; 8(1):11.
https://doi.org/10.3390/axioms8010011

**Chicago/Turabian Style**

Petrov, Alexey A. 2019. "Relations between Shadowing and Inverse Shadowing in Dynamical Systems" *Axioms* 8, no. 1: 11.
https://doi.org/10.3390/axioms8010011