# Uniform Hyperbolicity of a Scattering Map with Lorentzian Potential

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Scattering Map

## 3. Horseshoe Condition

**Proposition**

**1.**

**Proof.**

## 4. Non-Wandering Set and the Filtration Property

**Lemma**

**1.**

**(a)**- $U\left({\mathcal{O}}^{+}\right)\subset {\mathcal{O}}^{+}$ and $U\left({\mathcal{O}}^{-}\right)\subset {\mathcal{O}}^{-}$.
**(b)**- ${q}_{n}\in {\mathcal{O}}^{+}$ is strictly increasing, and ${q}_{n}\in {\mathcal{O}}^{-}$ is strictly decreasing under forward iteration of the map U.
**(c)**- ${U}^{-1}\left({\mathcal{I}}^{+}\right)\subset {\mathcal{I}}^{+}$ and ${U}^{-1}\left({\mathcal{I}}^{-}\right)\subset {\mathcal{I}}^{-}$.
**(d)**- ${q}_{n}\in {\mathcal{I}}^{+}$ is strictly increasing, and ${q}_{n}\in {\mathcal{I}}^{-}$ is strictly decreasing under backward iteration of the map U.

**Proof.**

**Remark.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 5. Sector Condition

**Definition**

**1.**

**Lemma**

**4.**

**Proof.**

**Proposition**

**3.**

**Proof.**

## 6. Sufficient Condition for the Sector Condition

#### 6.1. Numerical Observation for the Sector Condition

#### 6.2. Preliminary for the Division of the Phase Space

#### 6.3. Division of the Phase Space

#### 6.4. Sufficient Conditions for the Sector Condition

**(Case 1)**${q}_{n},{q}_{n+1}\in \overline{\mathcal{Y}}$.

**(Case 2)**${q}_{n},{q}_{n+1}\in \overline{\mathcal{X}}\cup \overline{\mathcal{Z}}$.

**(Case 3)**${q}_{n}\in \overline{\mathcal{X}}\cup \overline{\mathcal{Z}}$ and ${q}_{n+1}\in \overline{\mathcal{Y}}$, or ${q}_{n}\in \overline{\mathcal{Y}}$ and ${q}_{n+1}\in \overline{\mathcal{X}}\cup \overline{\mathcal{Z}}$.

#### 6.4.1. Case 1

#### 6.4.2. Case 2

#### 6.4.3. Case 3

#### 6.5. The Condition for Case 2

#### 6.6. The Condition for Case 3

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The potential function $V\left(q\right)$ with $({q}_{b},{q}_{f},\kappa )=(1.0,1.5,3.0)$. The central valley and two side hills are formed by the function ${f}_{1}\left(q\right)$ and ${f}_{2}(q\pm {q}_{b})$, respectively.

**Figure 2.**Stable (red) and unstable (blue) manifolds associated with the fixed points (green dots). The set of parameters is chosen as $({q}_{b},{q}_{f},\kappa )=(1.0,1.5,3.0)$.

**Figure 3.**The region R and its boundaries $U\left({l}_{1}\right),{l}_{2},{l}_{3},{U}^{-1}\left({l}_{4}\right)$ (black curves). The set of parameters is given as $({q}_{b},{q}_{f},\kappa )=(1.0,1.5,3.0)$.

**Figure 4.**(

**a**) The region R (gray) and (

**b**) its image (gray). The intersection $R\cap U\left(R\right)$ is composed of three disjointed regions ${\mathcal{X}}_{1},{\mathcal{Y}}_{1}$, and ${\mathcal{Z}}_{1}$. The regions R and $U\left(R\right)$ are schematically displayed in panels (

**c**,

**d**). (

**c**,

**d**) The square represents the region R, and the upper left and lower right corners (black dots) correspond to the fixed points.

**Figure 5.**(

**a**) The region R (gray) and (

**b**) its inverse image(gray). The intersection $R\cap {U}^{-1}\left(R\right)$ is composed of three disjointed regions ${\mathcal{X}}_{0},{\mathcal{Y}}_{0}$, and ${\mathcal{Z}}_{0}$. The region R and ${U}^{-1}\left(R\right)$ are schematically displayed in panels (

**c**,

**d**). (

**c**,

**d**) The square represents the region R, and the upper left and lower right corners (black dots) correspond to the fixed points.

**Figure 6.**The boundary curves for the region R (black curves) and its inverse image ${U}^{-1}\left(R\right)$ (black dashed curves). The curves $U\left({l}_{1}\right)$ and ${U}^{-1}\left({l}_{3}\right)$ are specifically shown in cyan and orange, respectively. The horseshoe is realized if the intersection points (red dots) exist.

**Figure 7.**The set $R\cap {U}^{-1}\left(R\right)$ (gray) and the decomposition of its complement. The region ${\mathcal{C}}_{1}^{+}$ (resp. ${\mathcal{C}}_{1}^{-}$) is mapped to the region ${\mathcal{O}}^{+}$ (resp. ${\mathcal{O}}^{-}$) under forward iteration. The region ${\mathcal{C}}_{2}$ is mapped to the region $(R\cap {U}^{-1}\left(R\right))\cup {\mathcal{C}}_{1}^{+}\cup {\mathcal{C}}_{1}^{-}$ under forward iteration, meaning that the points contained in the set ${\mathcal{C}}_{2}$ either stay in $R\cap {U}^{-1}\left(R\right)$ or go out to ${\mathcal{O}}^{\pm}$ under more than one-step forward iteration.

**Figure 8.**The set $R\cap {U}^{-1}\left(R\right)$ (gray) and the decomposition of its complement. The region ${\mathcal{D}}_{1}^{+}$ (resp. ${\mathcal{D}}_{1}^{-}$) is mapped to the region ${\mathcal{I}}^{+}$ (resp. ${\mathcal{I}}^{-}$) under backward iteration. The region ${\mathcal{D}}_{2}$ is mapped to the region $(R\cap {U}^{-1}\left(R\right))\cup {\mathcal{D}}_{1}^{+}\cup {\mathcal{D}}_{1}^{-}$ under backward iteration, meaning that the points contained in the set ${\mathcal{D}}_{2}$ either stay in $R\cap {U}^{-1}\left(R\right)$ or go out to ${\mathcal{I}}^{\pm}$ under more than one-step backward iteration.

**Figure 9.**The regions R (black cuve) and its inverse image ${U}^{-1}\left(R\right)$ (black dashed curve). The set of parameters is chosen as $({q}_{b},{q}_{f},\kappa )=(1.0,1.5,3.0)$.

**Figure 10.**The region satisfying (dark gray) and not satisfying (light gray) the condition (58). Notice that there exist regions not satisfying the condition (58) in $R\cap {U}^{-1}\left(R\right)$, in which the non-wandering set is contained. The set of parameters is chosen as $({q}_{b},{q}_{f},\kappa )=(1.0,1.5,3.0)$.

**Figure 12.**The boundary of the region R (black) and the function $L\left(q\right)$ (blue) for $\kappa =10.0$.

**Figure 13.**The zeros ${\omega}_{1},{\omega}_{2}$ of the function $L\left(q\right)$ for $\kappa =10.0$ are illustrated. The black curve represents the boundary of R.

**Figure 14.**Three disjointed regions $\overline{\mathcal{X}},\overline{\mathcal{Y}}$ and $\overline{\mathcal{Z}}$ (gray). The boundaries are specified by the lines $q=\pm {q}_{f},\pm {\omega}_{1},$ and $\pm {\omega}_{2}$.

**Figure 15.**The region ${U}^{-1}(\overline{\mathcal{X}}\cup \overline{\mathcal{Y}}\cup \overline{\mathcal{Z}})\cap (\overline{\mathcal{X}}\cup \overline{\mathcal{Y}}\cup \overline{\mathcal{Z}})$ (gray). The solid and broken curves show the boundaries of R and ${U}^{-1}\left(R\right)$, respectively.

**Figure 16.**Stable (red) and unstable (blue) manifold associated with fixed points (green dots). The set of parameters is chosen as $({q}_{b},{q}_{f},\kappa )=(1.0,1.5,1.8)$.

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**MDPI and ACS Style**

Yoshino, H.; Kogawa, R.; Shudo, A.
Uniform Hyperbolicity of a Scattering Map with Lorentzian Potential. *Condens. Matter* **2020**, *5*, 1.
https://doi.org/10.3390/condmat5010001

**AMA Style**

Yoshino H, Kogawa R, Shudo A.
Uniform Hyperbolicity of a Scattering Map with Lorentzian Potential. *Condensed Matter*. 2020; 5(1):1.
https://doi.org/10.3390/condmat5010001

**Chicago/Turabian Style**

Yoshino, Hajime, Ryota Kogawa, and Akira Shudo.
2020. "Uniform Hyperbolicity of a Scattering Map with Lorentzian Potential" *Condensed Matter* 5, no. 1: 1.
https://doi.org/10.3390/condmat5010001