# American Institute of Mathematical Sciences

December  2021, 41(12): 5537-5549. doi: 10.3934/dcds.2021087

## Hearing the shape of right triangle billiard tables

 School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author: Jiazhong Yang

Received  December 2020 Revised  April 2021 Published  December 2021 Early access  May 2021

Fund Project: The paper is supported by NSFC-12071006

In this paper, we give a positive answer to the problem that whether one can identify the shape of a right triangle billiard table by a single bounce sequence. Moreover, a convenient method to calculate the shape of polygons is given in this paper, too.

Citation: Yang Shen, Jiazhong Yang. Hearing the shape of right triangle billiard tables. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5537-5549. doi: 10.3934/dcds.2021087
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##### References:
After a reflection, $P_{n}$ becomes $P_{n+1}$ in Case ($i$) and $P_{n-1}$ in Case ($ii$)
From $B(\gamma^+) = ({r}, {b}, {r}, {g}, {b}, {r}, {b}, {r}, {b}, {g}, {r}, \cdots)$ to get the return sequence $R(\gamma^+) = (1, 2, 1, 2, \cdots)$ and then to the get the level sequence $L(\gamma^+) = (0, 1, 0, -1, 0, \cdots)$
The visual diagram of $\theta_{n}$, $\theta_{n+1}$ and $\theta_{n+2}$
Cases ($i$) and ($ii$) correspond to $LM_n = ({r}, {g}, {r})$ and $\theta_{n}\in(\alpha, \pi-\alpha)$; Cases ($iii$) and ($iv$) correspond to $LM_n = ({r}, {b}, {r})$ and $\theta_{n}\in[0, \alpha)\bigcup\, (\pi-\alpha, \pi] = (-\alpha, \alpha)\, (\bmod\, \pi)$
In Case ($i$), $\theta_{n_0} = \pi-\alpha$. In Case ($ii$), $\theta_{n_0} = \alpha$
The left-hand part is a right triangle billiard table $Rt$ and a periodic orbit: $p_{0}$ $\longrightarrow$ $p_{1}$ $\longrightarrow$ $p_{2}$ $\longrightarrow$ $\cdots$ $\longrightarrow p_{9}$ $\longrightarrow$ $p_{10}$ $\longrightarrow$ $\cdots$ in $Rt$. The right-hand part is the unfolding floors of the corresponding rhombus of $Rt$ along the periodic orbit
The left-hand part is a right triangle billiard table $Rt$ and a periodic orbit: $p_{0}$ $\longrightarrow$ $p_{1}$ $\longrightarrow$ $p_{2}$ $\longrightarrow$ $\cdots$ $\longrightarrow$ $p_{6}$ $\longrightarrow$ $\cdots$ in $Rt$. The right-hand part is the unfolding floors of the corresponding rhombus of $Rt$ along the periodic orbit
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