# Diffeomorphisms with Shadowable Measures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

- (1)
- the non-wandering set $\mathsf{\Omega}\left(g\right)$ of g is the union of 4 hyperbolic fixed points, that is, $\mathsf{\Omega}\left(g\right)=\{{p}_{1},\phantom{\rule{4pt}{0ex}}{p}_{2},\phantom{\rule{4pt}{0ex}}{p}_{3},\phantom{\rule{4pt}{0ex}}{p}_{4}\}$, where ${p}_{1}$ is a source, ${p}_{4}$ is a sink, and ${p}_{2},\phantom{\rule{4pt}{0ex}}{p}_{3}$ are saddles;
- (2)
- with respect to coordinates $(v,w)\in [-2,2]\times [-2,2]$, the following conditions hold:
- (2.1)
- ${p}_{1}=(1,2),\phantom{\rule{4pt}{0ex}}{p}_{2}=(1,0),\phantom{\rule{4pt}{0ex}}{p}_{3}=(-1,0),\phantom{\rule{4pt}{0ex}}{p}_{4}=(-1,2)$,
- (2.2)
- ${W}^{u}\left({p}_{2}\right)\cup \left\{{p}_{3}\right\}={W}^{s}\left({p}_{3}\right)\cup \left\{{p}_{2}\right\}=[-2,2]\times \left\{0\right\}$,

**Claim**

**1.**

**Claim**

**2.**

## 3. Proof of Theorem 1

**Lemma**

**1.**

**Proof.**

**End of the proof of Theorem 1**. Suppose $f\in \mathrm{int}\mathcal{PS}$, and let ${\mu}_{L}$ be the Lebesgue measure on M. Then, since $\mathrm{supp}\left({\mu}_{L}\right)=M$, the proof of Theorem 1 quickly follows from Lemma 1, so that Theorem 1 is proved. □

## 4. Proof of Theorem 2

**Lemma**

**2.**

- (a)
- $g\left(x\right)=f\left(x\right)$ if $x\in M\backslash U$, and
- (b)
- $g\left(x\right)={exp}_{f\left({x}_{i}\right)}\circ {L}_{i}\circ {exp}_{{x}_{i}}^{-1}\left(x\right)$ if $x\in {B}_{{\u03f5}_{0}}\left({x}_{i}\right)$ for all $1\le i\le N$.

**End of the proof of Theorem 2**. Suppose $f\in \mathrm{int}\left(\mathcal{PIS}\right)$, and we show that $f\in {\mathcal{F}}^{1}\left(M\right)$. By contradiction, assume that $f\notin {\mathcal{F}}^{1}\left(M\right)$ and we shall derive a contradiction. Since $f\in \mathrm{int}\left(\mathcal{PIS}\right)$, there exists a ${C}^{1}$-neighborhood $\mathcal{U}\left(f\right)$ of f such that for any $g\in \mathcal{U}\left(f\right)$ and any $\mu \in {\mathcal{M}}_{g}\left(M\right)\backslash \mathcal{A}\left(M\right)$, g is $\mu $-shadowable. On the other hand, it follows from the assumption $(f\notin {\mathcal{F}}^{1}\left(M\right))$ that there are $g\in \mathcal{U}\left(f\right)$ and non-hyperbolic periodic point p of g.

- ${\phi}^{i}\left({\mathcal{L}}_{p}\right)\cap {\phi}^{j}\left({\mathcal{L}}_{p}\right)=\varnothing $$(0\le i\ne j\le \pi (p)-1)$,
- ${\phi}^{\pi \left(p\right)}\left({\mathcal{L}}_{p}\right)={\mathcal{L}}_{p}$, and
- ${\phi}^{\pi \left(p\right)}{|}_{{\mathcal{L}}_{p}}$ is the identity map.

- (i)
- ${x}_{0}=p$,
- (ii)
- ${B}_{\frac{{\delta}_{1}}{2}}\left({x}_{k}\right)\cap {B}_{\frac{{\delta}_{1}}{2}}\left({x}_{k+1}\right)\ne \varnothing $ for $0\le k\le m-2$,
- (iii)
- $d({x}_{0},{x}_{k})<\frac{3{\u03f5}_{1}}{4}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}(k=1,2,\cdots ,m-1),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}d({x}_{0},{x}_{m-1})>\frac{3{\u03f5}_{1}}{8}$.

- ${\phi}^{i}\left({\mathcal{J}}_{p}\right)\cap {\phi}^{i}\left({\mathcal{J}}_{p}\right)=\varnothing $$(0\le i\ne j\le l-1)$,
- ${\phi}^{l}\left({\mathcal{J}}_{p}\right)={\mathcal{J}}_{p}$, and
- ${\phi}^{l}{|}_{{\mathcal{J}}_{p}}$ is the identity map.

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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Moriyasu, K.; Sakai, K.; Sumi, N.
Diffeomorphisms with Shadowable Measures. *Axioms* **2018**, *7*, 93.
https://doi.org/10.3390/axioms7040093

**AMA Style**

Moriyasu K, Sakai K, Sumi N.
Diffeomorphisms with Shadowable Measures. *Axioms*. 2018; 7(4):93.
https://doi.org/10.3390/axioms7040093

**Chicago/Turabian Style**

Moriyasu, Kazumine, Kazuhiro Sakai, and Naoya Sumi.
2018. "Diffeomorphisms with Shadowable Measures" *Axioms* 7, no. 4: 93.
https://doi.org/10.3390/axioms7040093