# Spectral form factor in the Hadamard-Gutzwiller model: orbit pairs
contributing in the third order

Huynh M. Hien
Department of Mathematics and Statistics,
Quy Nhon University,
170 An Duong Vuong, Quy Nhon, Vietnam
e-mail<EMAIL_ADDRESS>
###### Abstract

In this paper we consider orbit pairs contributing in the third order of the
spectral form factor in the Hadamard-Gutzwiller model. We prove that periodic
orbits including two 2-encounters in certain structures have partner orbits.
The action differences are estimated at $\ln(1+u_{1}s_{1})(1+u_{2}s_{2})$ with
explicit error bounds, where $(u_{1},s_{1})$ and $(u_{2},s_{2})$ are the
coordinates of the piercing points. A new symbolic dynamics for orbit pairs
via conjugacy classes is also provided.

Keywords.  Hadamard-Gutzwiller model, Spectral form factor, Third order, Orbit
pair, 2-encounter.

##  1 Introduction

In quantum chaos, there is considerable interest in understanding statistics
associated to periodic orbits since these are related to eigenvalue statistics
through trace formulae. Special attention has been given to the spectral form
factor, which is expressed by a double sum over periodic orbits

$K(\tau)=\Big{\langle}\frac{1}{T_{H}}\sum_{\gamma,\gamma^{\prime}}A_{\gamma}A_{\gamma^{\prime}}^{*}e^{\frac{i}{\hbar}(S_{\gamma}-S_{\gamma^{\prime}})}\delta\Big{(}\tau
T_{H}-\frac{T_{\gamma}+T_{\gamma^{\prime}}}{2}\Big{)}\Big{\rangle},$ (1.1)

where $\langle\cdot\rangle$ abbreviates the average over the energy and over a
small time window, $T_{H}$ denotes the Heisenberg time and $A_{\gamma}$,
$S_{\gamma}$, and $T_{\gamma}$ are the amplitude, the action, and the period
of the orbit $\gamma$, respectively.

The diagonal approximation $\gamma=\gamma^{\prime}$ to (1.1) studied by
Hannay/Ozorio de Almeida [9] and Berry [2] in the 1980’s contributes to the
first order term $2\tau$; see also [15]. The efforts of researchers have been
to understand higher order effects. To the next orders, as $\hbar\to 0$, the
main term from (1.1) arises owing to those orbit pairs
$\gamma\neq\gamma^{\prime}$ for which the action difference
$S_{\gamma}-S_{\gamma^{\prime}}$ is ‘small’. In 2001, an influential heuristic
work of Sieber and Richter [22] who predicted that a given periodic orbit with
a small-angle self-crossing in configuration space will admit a partner orbit
with almost the same action. The original orbit and its partner are then
called a Sieber-Richter pair. In phase space, a Sieber-Richter pair contains a
region where two stretches of each orbit are almost mutually time-reversed and
one addresses this region as a $2$-encounter or, more strictly, a
$2$-antiparallel encounter; the ‘2’ stands for two orbit stretches which are
close in configuration space, and ‘antiparallel’ means that the two stretches
have opposite directions. It was shown in [22] that Sieber-Richter pairs
contribute to the spectral form factor (1.1) the second order term
$-2\tau^{2}$, and it turned out that the result agreed with what is obtained
using random matrix theory [5], for certain symmetry classes. The work by
Sieber and Richter has led to the important and difficult problem of
understanding this phenomenon is more detail and more rigorously in particular
classes of systems. Until 2012, Gutkin and Osipov [7] analysed Sieber-Richter
pairs for the Baker map, which admits very transparent symbolic dynamics, in a
combinatorial way.

Most contribution in this subject matter is Müller et al. In a series of works
[11, 17, 18, 19], the authors provided an expansion to all orders in $\tau$

$K(\tau)=2\tau-\tau\,\ln(1+2\tau)=2\tau-2\tau^{2}+2\tau^{3}+\ldots$

for the symmetry class relevant for time-reversal invariant systems, by
including the higher-order encounters also; see also [8, 16]. It was shown in
[11] that there are five families of pairs of orbits responsible for the third
order $\tau^{3}$, namely three families of orbit pairs differing in two
2-encounters and two families of orbit pairs differing in one single
3-encounter. Periodic orbits with encounters have partners obtained by
reconnections stretches inside encounter area owing to the hyperbolicity.
However, the existence of partner orbits and estimates of the action
differences are still missing.

To establish a more detailed mathematical understanding, it is necessary to
consider the classical side and try to prove the existence of partner orbits
and derive good estimates for the action differences of the orbit pairs. For
$2$-antiparallel encounters this was done in [12, 14], where the authors
considered the geodesic flow on compact factors of the hyperbolic plane; in
this case the action of a periodic orbit is half of its length/period. It was
shown in [12] that a $T$-periodic orbit of the geodesic flow crossing itself
in configuration space at a time $T_{1}$ has a unique partner orbit that
remains $9|\sin(\phi/2)|$-close to the original one and the action difference
between them is approximately equal
$\ln(1-(1+e^{-T_{1}})(1+e^{-(T-T_{1})})\sin^{2}(\phi/2)))$ with the error
bound $12\sin^{2}(\phi/2)e^{-T}$, where $\phi$ is the crossing angle, and this
proved the accuracy of Sieber/Richter’s prediction in [22] mentioned above.
For higher-order encounters, Huynh [13] shows that there exist $(L-1)!-1$
partner orbits for a given periodic orbit with an $L$-parallel encounter such
that any two piercing points are not too close and provided estimates for the
action differences.

In the present paper we continue considering the geodesic flow on compact
factor of the hyperbolic plane, which is a compact Riemann surface of constant
curvature of genus at least two. In the physics community this system is often
called the Hadamard-Gutzwiller model, and it has frequently been studied [4,
11, 21]; further related work includes [8, 19, 23]. We prove the existence of
the partner orbit which differs in both encounters for a given periodic orbit
including two 2-encounters with piercing points having coordinates
$(u_{1},s_{1}),(u_{1},u_{2})$ in certain distributions. The action differences
of orbit pairs of all cases are estimated at $\ln(1+u_{1}s_{1})(1+u_{2}s_{2})$
with explicit error bounds. This paper also provides a new symbolic dynamics
for orbit pairs via conjugacy classes.

The paper is organized as follows. In Section 2 we recall background and
materials, including Poincaré sections, the Anosov and closing lemmas,
conjugacy classes and rigorous definitions of encounters, partners in the
Hadamard-Gutwiller model. Section 3 considers periodic orbits with one single
2-antiparallel encounter. In the last section we consider periodic orbits with
two 2-antiparallel encounters serial, with two 2-parallel encounter
intertwined, and with one 2-parallel encounter and one 2-antiparallel
encounter intertwined. In each case, we prove the existence of partner orbits,
estimate the action differences as well as provide symbolic dynamics for orbit
pairs.

##  2 The Hadamard-Gutzwiller model

The Hadamard-Gutzwiller model is the geodesic flow on compact Riemann surfaces
of constant negative curvature. It is well-known that any compact orientable
surface with constant negative curvature is isometric to a factor
$\Gamma\backslash\mathbb{H}^{2}$, where
$\mathbb{H}^{2}=\\{z=x+iy\in\mathbb{C}:\,y>0\\}$ is the hyperbolic plane
endowed with the hyperbolic metric $ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}}$ and
$\Gamma$ is a discrete subgroup of the projective Lie group ${\rm
PSL}(2,\mathbb{R})={\rm SL}(2,\mathbb{R})/\\{\pm E_{2}\\}$. The hyperbolic
plane has constant Gaussian curvature $-1$. The group ${\rm
PSL}(2,\mathbb{R})$ acts transitively on $\mathbb{H}^{2}$ by Möbius
transformations $z\mapsto\frac{az+b}{cz+d}$. If the action has no fixed
points, then the factor $\Gamma\backslash\mathbb{H}^{2}$ has a Riemann surface
structure. Such a surface is a closed Riemann surface of genus at least $2$
and has the hyperbolic plane $\mathbb{H}^{2}$ as the universal covering; so
the natural projection
$\pi_{\Gamma}:\mathbb{H}^{2}\rightarrow\Gamma\backslash\mathbb{H}^{2},\pi_{\Gamma}(z)=\Gamma
z,\ z\in\mathbb{H}^{2}$ becomes a local isometry. This implies that
$\Gamma\backslash\mathbb{H}^{2}$ also has constant curvature $-1$. The
geodesic flow $(\varphi_{t}^{\mathcal{X}})_{t\in\mathbb{R}}$ on the unit
tangent bundle $\mathcal{X}=T^{1}(\Gamma\backslash\mathbb{H}^{2})$ goes along
the unit speed geodesics on $\Gamma\backslash\mathbb{H}^{2}$. On the other
hand, the unit tangent bundle $T^{1}(\Gamma\backslash\mathbb{H}^{2})$ is
isometric to the quotient space $\Gamma\backslash{\rm
PSL}(2,\mathbb{R})=\\{\Gamma g,g\in{\rm PSL}(2,\mathbb{R})\\}$, which is the
system of right co-sets of $\Gamma$ in ${\rm PSL}(2,\mathbb{R})$, by an
isometry $\Xi$. Then the geodesic flow
$(\varphi_{t}^{\mathcal{X}})_{t\in\mathbb{R}}$ can be equivalently expressed
as the natural “quotient flow” $\varphi_{t}(\Gamma g)=\Gamma ga_{t}$ on
$X=\Gamma\backslash{\rm PSL}(2,\mathbb{R})$ associated to the flow
$\varphi^{\mathsf{G}}_{t}(g)=ga_{t}$ on $\mathsf{G}:={\rm PSL}(2,\mathbb{R})$
by the conjugate relation

$\varphi_{t}^{\mathcal{X}}=\Xi^{-1}\circ\varphi_{t}\circ\Xi\quad\mbox{for
all}\quad t\in\mathbb{R}.$

Here $a_{t}\in{\rm PSL}(2,\mathbb{R})$ denotes the equivalence class obtained
from the matrix $A_{t}=\scriptsize\Big{(}\begin{array}[]{cc}e^{t/2}&0\\\
0&e^{-t/2}\end{array}\Big{)}\in{\rm SL}(2,\mathbb{R})$.

There are some more advantages to work on $X=\Gamma\backslash{\rm
PSL}(2,\mathbb{R})$ rather than on
$\mathcal{X}=T^{1}(\Gamma\backslash\mathbb{H}^{2})$. One can calculate
explicitly the stable and unstable manifolds at a point $x=\Gamma g\in X$ to
be

$W^{s}_{X}(x)=\\{\Gamma gb_{s},s\in\mathbb{R}\\}\quad\mbox{and}\quad
W^{u}_{X}(x)=\\{\Gamma gc_{u},u\in\mathbb{R}\\},$

where $b_{s}=\\{\pm B_{s}\\},c_{u}=\\{\pm C_{u}\\}\in{\rm PSL}(2,\mathbb{R})$
denote the equivalence classes obtained from
$B_{s}=\scriptsize\Big{(}\begin{array}[]{cc}1&s\\\ 0&1\end{array}\Big{)},\
C_{u}=\scriptsize\Big{(}\begin{array}[]{cc}1&0\\\ u&1\end{array}\Big{)}\in{\rm
SL}(2,\mathbb{R})$. If the space $X$ is compact, then the flow
$(\varphi_{t})_{t\in\mathbb{R}}$ is a hyperbolic flow.

There is a natural Riemannian metric on $\mathsf{G}={\rm PSL}(2,\mathbb{R})$
such that the induced metric function $d_{\mathsf{G}}$ is left-invariant under
$\mathsf{G}$. We define a metric function $d_{X}$ on $X=\Gamma\backslash{\rm
PSL}(2,\mathbb{R})$ by

$d_{X}(x_{1},x_{2})=\inf_{\gamma_{1},\gamma_{2}\in\Gamma}d_{\mathsf{G}}(\gamma_{1}g_{1},\gamma_{2}g_{2})=\inf_{\gamma\in\Gamma}d_{\mathsf{G}}(g_{1},\gamma
g_{2}),$

where $x_{1}=\Gamma g_{1}$, $x_{2}=\Gamma g_{2}$.

General references for this section are [1, 6], and these works may be
consulted for the proofs to all results which are stated above

###  2.1 Poincaré sections

It is well-known that the Riemann surface $\Gamma\backslash\mathbb{H}^{2}$ is
compact if and only if the quotient space $X=\Gamma\backslash{\rm
PSL}(2,\mathbb{R})$ is compact.

First we recall the definitions of Poincaré sections in [12, 13].

######  Definition 2.1.

Let $x\in X$ and $\varepsilon>0$. The Poincaré sections of radius
$\varepsilon$ at $x$ are defined by

${\mathcal{P}}_{\varepsilon}(x)=\\{\Gamma
gc_{u}b_{s}:|u|<\varepsilon,|s|<\varepsilon\\},$

and

${\mathcal{P}}^{\prime}_{\varepsilon}(x)=\\{\Gamma
gb_{s}c_{u}:|s|<\varepsilon,\ |u|<\varepsilon\\},$

where $g\in\mathsf{G}$ is such that $x=\Gamma g$.

If $z=\Gamma gc_{u}b_{s}\in{\mathcal{P}}_{\varepsilon}(x)$ (resp. $z=\Gamma
gb_{s}c_{u}\in{\mathcal{P}}^{\prime}_{\varepsilon}(x)$), we write
$z=(u,s)_{x}$ (resp. $z=(s,u)^{\prime}_{x}$). Note that the couple $(u,s)$ are
not unique. As we will see below, if $X$ is compact and $\varepsilon$ is
sufficiently small, then the uniqueness of couple $(u,s)$ is obtained.

######  Lemma 2.1.

If the space $X=\Gamma\backslash{\rm PSL}(2,\mathbb{R})$ is compact, then
there exists $\sigma_{0}>0$ such that

$d_{\mathsf{G}}(\gamma g,g)>\sigma_{0}\quad\mbox{for
all}\quad\gamma\in\Gamma\setminus\\{e\\}.$ (2.1)

See [20, Lemma 1, p. 237] for a similar result on
$\Gamma\backslash\mathbb{H}^{2}$.

######  Lemma 2.2 ([13]).

If the space $X=\Gamma\backslash{\rm PSL}(2,\mathbb{R})$ is compact and
$\varepsilon\in(0,\frac{\sigma_{0}}{4})$, then for each
$z\in{\mathcal{P}}_{\varepsilon}(x)$ there exist a unique couple
$(u_{z},s_{z})\in(0,\varepsilon)^{2}$ such that $z=\Gamma
gc_{u_{z}}b_{s_{z}}$, where $g\in{\rm PSL}(2,\mathbb{R})$ is such that
$x=\Gamma g$, and we call $(u_{z},s_{z})$ the coordinates of $z$.

###  2.2 Conjugacy classes

Let $\Gamma$ be a discrete subgroup of ${\rm PSL}(2,\mathbb{R})$.

######  Definition 2.2.

(a) An element $\gamma\in\Gamma$ is called primitive if $\gamma=\zeta^{m}$ for
some $\zeta\in\Gamma$ implies that $m=1$ or $m=-1$.

(b) The conjugacy class of $\gamma\in\Gamma$ is defined by

${\\{\gamma\\}}_{\Gamma}=\\{\sigma\gamma\sigma^{-1}:\sigma\in\Gamma\\}.$

The collection of all conjugacy classes of primitive elements in
$\Gamma\setminus\\{e\\}$ are denoted by ${\mathfrak{C}}_{\Gamma}$; here
$e=[E_{2}]$ denotes the unity of ${\rm PSL}(2,\mathbb{R})$.

For $g=[G]\in{\rm
PSL}(2,\mathbb{R}),G=\big{(}{\scriptsize\begin{array}[]{cc}a&b\\\
c&d\end{array}}\big{)}\in{\rm SL}(2,\mathbb{R})$, the trace of $g$ is defined
by ${\rm tr}(g)=|a+d|.$ If the action of $\Gamma$ on $\mathbb{H}^{2}$ is free
and the factor $\Gamma\backslash\mathbb{H}^{2}$ is compact then all elements
$g\in\Gamma\setminus\\{e\\}$ are hyperbolic [20, Theorem 6.6.6], i.e. ${\rm
tr}(g)>2$.

Denote by ${\mathcal{P}O}_{X}$ the set of all periodic orbits of the flow
$(\varphi_{t})_{t\in\mathbb{R}}$. We define a mapping

$\varsigma:{{\mathcal{P}O}}_{X}\to\mathfrak{C}_{\Gamma}$ (2.2)

as follows. Take a periodic orbit $c$ of the flow, any point $x$ on $c$, and
let $T>0$ be the prime period for $x$. Then $\varphi_{T}(x)=x$, and the
definition of the flow implies that there are $g\in{\rm PSL}(2,\mathbb{R})$
and $\gamma\in\Gamma$ such that $x=\Pi_{\Gamma}(g)$ and $\gamma=ga_{T}g^{-1}$,
due to $\Gamma ga_{T}=xa_{T}=\varphi_{T}(x)=x=\Gamma g$; note that $\gamma\neq
e$, since otherwise $a_{T}=e$ so that $T=0$. Then put
$\varsigma(c)={\\{\gamma\\}}_{\Gamma}$.

######  Lemma 2.3.

Suppose that all elements in $\Gamma\backslash\\{e\\}$ are hyperbolic. Then
the mapping $\varsigma$ defined by (2.2) is a bijection between the periodic
orbits ${\mathcal{P}O}_{X}$ of the flow $(\varphi_{t})_{t\in\mathbb{R}}$ and
the collection of all conjugacy classes of primitive elements
${\mathfrak{C}_{\Gamma}}$ in $\Gamma\setminus\\{e\\}$.

###  2.3 Anosov closing lemma, connecting lemma

The next two results are illustrated in Figure 1 (a). For proofs, see [12,
13].

######  Lemma 2.4 (Anosov closing lemma I).

Suppose that $\varepsilon\in(0,\frac{1}{4})$, $x\in X$, $T\geq 1$, and
$\varphi_{T}(x)\in{\mathcal{P}}_{\varepsilon}(x)$. If
$\varphi_{T}(x)={(u,s)}_{x}\in{\mathcal{P}}_{\varepsilon}(x)$, in the notation
from Definition 2.1, then there are
$x^{\prime}=(\sigma,\eta)_{x}\in{\mathcal{P}}_{2\varepsilon}(x)$ and
$T^{\prime}\in\mathbb{R}$ so that

$\varphi_{T^{\prime}}(x^{\prime})=x^{\prime}\quad\mbox{and}\quad
d_{X}(\varphi_{t}(x),\varphi_{t}(x^{\prime}))<2|u|+|\eta|<4\varepsilon\quad\mbox{for
all}\quad t\in[0,T].$

Furthermore,

$\Big{|}\frac{T^{\prime}-T}{2}-\ln(1+us)\Big{|}<5|us|e^{-T}$

and

$\displaystyle|\sigma|<2|u|e^{-T},\quad|\eta|<\frac{3|s|}{2}.$

######  Remark 2.1.

According to the proof of the Anosov closing lemma I in [12, Theorem 2.3],
$x=\Gamma g,g\in{\rm PSL}(2,\mathbb{R})$ and $\zeta\in\Gamma$ is such that
$ga_{T}=\zeta gc_{u}b_{s}$ then $gc_{\sigma}b_{\eta}a_{T^{\prime}}=\zeta
gc_{\sigma}b_{\eta}$. This yields that the periodic orbit of
$x^{\prime}=\Gamma gc_{\sigma}b_{\eta}$ corresponds to the conjugacy class
$\\{\zeta\\}_{\Gamma}$, provided that all elements in
$\Gamma\backslash\\{e\\}$ are hyperbolic.

Using the other version of Poincaré sections, we have a respective statement
for the Anosov closing lemma which will be also useful afterwards.

######  Lemma 2.5 (Anosov closing lemma II).

Suppose that $\varepsilon\in\,(0,\frac{1}{4})$, $x\in X$, $T\geq 1$, and
$\varphi_{T}(x)\in{\mathcal{P}}^{\prime}_{\varepsilon}(x)$. If
$\varphi_{T}(x)={(s,u)}^{\prime}_{x}\in{\mathcal{P}}^{\prime}_{\varepsilon}(x)$,
in the notation from Definition 2.1, then there are
$x^{\prime}=(\eta,\sigma)^{\prime}_{x}\in{\mathcal{P}}^{\prime}_{2\varepsilon}(x)$
and $T^{\prime}\in\mathbb{R}$ so that

$\displaystyle\varphi_{T^{\prime}}(x)=x\quad\mbox{and}\quad
d_{X}(\varphi_{t}(x),\varphi_{t}(x^{\prime}))\leq
2|u|+|\eta|<4\varepsilon\quad\mbox{for all}\quad t\in[0,T].$

Furthermore,

$\Big{|}\frac{T^{\prime}-T}{2}\Big{|}<4|us|e^{-T}$

and

$|\sigma|<2|u|e^{-T},\quad|\eta|\leq\frac{3|s|}{2}.$

######  Lemma 2.6 (Connecting lemma).

Let $x_{j}\in X$ be $T_{j}$-periodic point of the flow
$(\varphi_{t})_{t\in\mathbb{R}}$ for $j=1,2$ and $T_{1}+T_{2}\geq 1$ and let
$\varepsilon\in(0,\frac{1}{4})$. If
$x_{2}=(u_{1},s_{1})_{x_{1}}\in{\mathcal{P}}_{\varepsilon}(x_{1})$, then there
are $x\in X$ and $T>0$ such that $\varphi_{T}(x)=x$,

$\displaystyle d_{X}(\varphi_{t}(x),\varphi_{t}(x_{1}))$ $\displaystyle<$
$\displaystyle 5\varepsilon\quad\mbox{for all}\quad t\in[0,T_{1}],$ (2.3)
$\displaystyle d_{X}(\varphi_{t+T_{1}}(x),\varphi_{t}(x_{2}))$
$\displaystyle<$ $\displaystyle 5\varepsilon\quad\mbox{for all}\quad
t\in[0,T_{2}],$ (2.4)

and

$\Big{|}\frac{T-(T_{1}+T_{2})}{2}-\ln(1+us)\Big{|}<7|us|(e^{-T_{1}}+e^{-T_{2}}).$
(2.5)

Furthermore, if $x_{i}=\Gamma g_{i}$ for some $g_{i}\in{\rm
PSL}(2,\mathbb{R})$, then $x=\Gamma g_{1}c_{ue^{-T_{1}}+\sigma}b_{\eta}$,
where $\sigma,\eta\in\mathbb{R}$ satisfy

$|\sigma|<2|u|e^{-T_{1}-T_{2}},\ \ |\eta|<\frac{3|s|}{2}$ (2.6)

and the orbit of $x$ corresponds to the conjugacy class
$\\{\gamma_{1}\gamma_{2}\\}_{\Gamma}$, where $\gamma_{1},\gamma_{2}\in\Gamma$
such that $g_{1}a_{T_{1}}=\gamma_{1}g_{1},g_{2}a_{T_{2}}=\gamma_{2}g_{2}$,
provided that all elements in $\Gamma\backslash\\{e\\}$ are hyperbolic.

See Figure 1 (b) for an illustration.

###### Proof.

For the existence of a periodic point $x$ and (2.3)-(2.6), see [13, Theorem
2.6]. For the last assertion, we can choose $g_{1},g_{2}\in{\rm
PSL}(2,\mathbb{R})$ such that $x=\Gamma g_{1},y=\Gamma g_{2}$ and
$g_{2}=g_{1}c_{u}b_{s}$. According to the proof of the previous lemma in [13,
Theorem 2.6], if $\gamma_{1},\gamma_{2}\in\Gamma$ such that
$g_{1}a_{T_{1}}=\gamma_{1}g_{1},g_{2}a_{T_{2}}=\gamma_{2}g_{2}$ then
$g_{1}c_{ue^{-T_{1}}+\sigma}b_{\eta}a_{T}=\gamma
g_{1}c_{ue^{-T_{1}}+\sigma}b_{\eta}$ with

$\displaystyle\gamma$ $\displaystyle=$ $\displaystyle
g_{2}b_{-s}a_{-T_{1}}a_{T_{1}+T_{2}}b_{-s(1-e^{-T_{2}})}c_{-u(1-e^{-T_{1}})}a_{T_{1}}c_{-u}g_{1}^{-1}$
$\displaystyle=$ $\displaystyle
g_{2}a_{T_{2}}b_{-se^{-T_{2}}}b_{-s(1-e^{-T_{2}})}c_{-u(1-e^{-T_{1}})}c_{-ue^{-T_{1}}}a_{T_{1}}g_{1}^{-1}$
$\displaystyle=$ $\displaystyle
g_{2}a_{T_{2}}b_{-s}c_{-u}g_{1}^{-1}\gamma_{1}$ $\displaystyle=$
$\displaystyle\gamma_{2}g_{2}b_{-s}c_{-u}g_{1}^{-1}\gamma_{1}$
$\displaystyle=$ $\displaystyle\gamma_{2}\gamma_{1}.$

This implies that the orbit of $x$ corresponds to the conjugacy class

$\\{\gamma_{2}\gamma_{1}\\}_{\Gamma}=\\{\gamma_{1}\gamma_{2}\\}_{\Gamma}.$

∎

Figure 1: (a) Anosov closing lemma (b) Connecting lemma (c) Orbit with a self-
crossing

###  2.4 Self-crossings

Recall that ${\mathcal{X}}=T^{1}(\Gamma\backslash\mathbb{H}^{2})$ denotes the
unit tangent bundle of the factor $\Gamma\backslash\mathbb{H}^{2}$; see Figure
1 (c) for an illustration for the next result.

######  Lemma 2.7 (Self-crossings,[12]).

Suppose that all elements of $\Gamma\setminus\\{e\\}$ are hyperbolic and let
$\tau\in\mathbb{R},\,L>0,\,\theta\in\,(0,\pi)$, and
${\mathtt{x}}\in\mathcal{X}$ be given. The orbit of ${\mathtt{x}}$ under the
geodesic flow ${(\varphi^{\mathcal{X}}_{t})}_{t\in\mathbb{R}}$ crosses itself
in configuration space at the time $\tau$, at the angle $\theta$, and creates
a loop of length $L$ if and only if

$\mbox{either}\quad\Gamma ga_{\tau+L}=\Gamma
ga_{\tau}d_{\theta}\quad\mbox{or}\quad\Gamma g_{\tau+L}=\Gamma
ga_{\tau}d_{-\theta}$ (2.7)

holds for any $g\in{\rm PSL}(2,\mathbb{R}),\Gamma g=\Xi({\mathtt{x}})$.
Furthermore,

$e^{-L}<\cos^{2}\Big{(}\frac{\theta}{2}\Big{)}.$ (2.8)

###  2.5 Encounters and partner orbits

######  Definition 2.3 (Time reversal).

The time reversal map ${\mathcal{T}}:\mathcal{X}\rightarrow\mathcal{X}$ is
defined by

${\mathcal{T}}(p,\xi)=(-p,\xi)\quad\mbox{for}\quad(p,\xi)\in\mathcal{X}.$

The respective time reversal map on $X=\Gamma\backslash{\rm
PSL}(2,\mathbb{R})$ is determined by

${\mathcal{T}}(x)=\Gamma gd_{\pi}\quad\mbox{for}\quad x=\Gamma g\in X,$

where $d_{\pi}\in{\rm PSL}(2,\mathbb{R})$ is the equivalence class of the
matrix $D_{\pi}=\scriptsize\Big{(}\begin{array}[]{cc}0&1\\\
-1&0\end{array}\Big{)}\in{\rm SL(2,\mathbb{R})}$.

Using Lemma 2.9 below, we have

$\varphi_{t}({\mathcal{T}}(x))={\mathcal{T}}(\varphi_{-t}(x))\quad\mbox{for}\quad
x\in X\quad\mbox{and}\quad t\in\mathbb{R}.$ (2.9)

Next, we recall the notions of orbit pairs and partner orbits. Roughly
speaking, two periodic orbits are called an orbit pair if they are close
enough to each other in configuration space, not for the whole time, since
otherwise they would be identical, but they decompose to the same number of
parts and any part of one orbit is close to some part of the other. The
following is a rigorous definition of orbit pairs, which is recalled from
[13].

######  Definition 2.4 (Orbit pair/Partner orbit).

Let $\varepsilon>0$ be given. Two given $T$-periodic orbit $c$ and
$T^{\prime}$-periodic orbit $c^{\prime}$ of the flow
$(\varphi_{t})_{t\in\mathbb{R}}$ are called an $\varepsilon$-orbit pair if
there are $L\geq 2,L\in\mathbb{Z}$ and two decompositions of $[0,T]$ and
$[0,T^{\prime}]:$ $0=t_{0}<\cdots<t_{L}=T$ and
$0=t_{0}^{\prime}<\cdots<t_{L}^{\prime}=T^{\prime}$, and a permutation
$\sigma:\\{0,1,\dots,L-1\\}\rightarrow\\{0,1,\dots,L-1\\}$ such that for each
$j\in\\{0,\dots,L-1\\}$, either

$d_{X}(\varphi_{t+t_{j}}(x),\varphi_{t+t_{\sigma(j)}^{\prime}}(x^{\prime}))<\varepsilon\quad\mbox{for
all}\quad t\in[0,t_{j+1}-t_{j}]$

or

$d_{X}\Big{(}\varphi_{t+t_{j}}(x),\varphi_{t-t_{\sigma(j)+1}^{\prime}}({\mathcal{T}}(x^{\prime}))\big{)}<\varepsilon\quad\mbox{for
all}\quad t\in[0,t_{j+1}-t_{j}]$

holds for some $x\in c$ and $x^{\prime}\in c^{\prime}$. Then $c^{\prime}$ is
called an $\varepsilon$-partner orbit of $c$ and vice versa.

######  Definition 2.5 (Encounter).

Let $\varepsilon>0$ and $L\in\mathbb{Z},L\geq 2$ be given. We say that a
periodic orbit $c$ of the flow $(\varphi_{t})_{t\in\mathbb{R}}$ has an
$(L,\varepsilon)$-encounter if there are $x_{1},\dots,x_{L}\in c$ such that
for each $j\in\\{2,\dots,L\\}$,

$\mbox{either}\quad
x_{j}\in{\mathcal{P}}_{\varepsilon}(x_{1})\quad\mbox{or}\quad{\mathcal{T}}(x_{j})\in{\mathcal{P}}_{\varepsilon}(x_{1}).$

The point $x_{2},\dots,x_{L}$ are called piercing points. If either
$x_{j}\in{\mathcal{P}}_{\varepsilon}(x_{1})$ holds for all $2=1,\dots,L$ or
${\mathcal{T}}(x_{j})\in{\mathcal{P}}_{\varepsilon}(x)$ holds for all
$j=2,\dots,L$ then the encounter is called parallel encounter; otherwise it is
called antiparallel encounter.

###  2.6 Auxiliary results

The next result is a decomposition of ${\rm PSL}(2,\mathbb{R})$.

######  Lemma 2.8 ([12]).

Let $g=[G]\in{\rm PSL}(2,\mathbb{R})$ for
$G=\scriptsize\Big{(}\begin{array}[]{cc}a&b\\\ c&d\end{array}\Big{)}\in{\rm
SL}(2,\mathbb{R})$.

  * (a)

If $a\neq 0$, then $g=c_{u}b_{s}a_{t}$ for

$t=2\ln|a|,\quad s=ab,\quad u=\frac{c}{a}.$

  * (b)

If $d\neq 0$, then $g=b_{s}c_{u}a_{t}$ for

$t=-2\ln|d|,\quad s=\frac{b}{d},\quad u=cd.$

######  Lemma 2.9.

The following relations hold for $t\in\mathbb{R}$:

$a_{t}d_{\pi}=d_{\pi}a_{-t},\quad b_{t}d_{\pi}=d_{\pi}c_{-t},\quad
c_{t}d_{\pi}=d_{\pi}b_{-t}.$ (2.10)

###### Proof.

In ${\rm SL}(2,\mathbb{R})$ we calculate

$\displaystyle A_{t}D_{\pi}$ $\displaystyle=$
$\displaystyle\Bigg{(}\begin{array}[]{cc}e^{t/2}&0\\\
0&e^{-t/2}\end{array}\Bigg{)}\Bigg{(}\begin{array}[]{cc}0&1\\\
-1&0\end{array}\Bigg{)}=\Bigg{(}\begin{array}[]{cc}0&e^{t/2}\\\
-e^{-t/2}&0\end{array}\Bigg{)}$ $\displaystyle=$
$\displaystyle\Bigg{(}\begin{array}[]{cc}0&1\\\
-1&0\end{array}\Bigg{)}\Bigg{(}\begin{array}[]{cc}e^{-t/2}&0\\\
0&e^{t/2}\end{array}\Bigg{)}=D_{\pi}A_{-t}$

which upon projection yields the first one. The argument is analogous for the
others.

∎

Owing to the hyperbolicity, the flow $(\varphi_{t})_{t\in\mathbb{R}}$ is
expansive, i.e., two orbits cannot stay too close together without being
identical; see [3, Lemma 1.5]. For periodic orbits, we have the following
property; see [12, Theorem 3.14] for a proof.

######  Lemma 2.10.

Let $X=\Gamma\backslash{\rm PSL}(2,\mathbb{R})$ be compact. Then there is
$\varepsilon_{\ast}>0$ with the following property. If
$\varepsilon\in(0,\varepsilon_{\ast})$ and if $x_{1},x_{2}\in X$ are periodic
points of ${(\varphi_{t}^{X})}_{t\in\mathbb{R}}$ having the periods
$T_{1},T_{2}>0$ such that $|T_{1}-T_{2}|\leq\sqrt{2}\varepsilon$ and

$d_{X}(\varphi_{t}^{X}(x_{1}),\varphi_{t}^{X}(x_{2}))<\varepsilon\quad\mbox{for
all}\quad t\in[0,\min\\{T_{1},T_{2}\\}],$

then $T_{1}=T_{2}$ and the orbits of $x_{1}$ and $x_{2}$ under
${(\varphi_{t}^{X})}_{t\in\mathbb{R}}$ are identical.

##  3 Periodic orbits with one single 2-antiparallel encounter

Let us first recall from [14] periodic orbits with one single 2-antiparallel
encounter. It was shown that a given periodic orbit including one single
2-antiparallel encounter has a partner orbit. The action difference between
the orbit pair is estimated with an exponentially small error bound. Periodic
orbits having small-angle self-crossing are special cases of this phenomenon.
The results in this section will be applied for the main results in Subsection
4.1.

######  Theorem 3.1.

Suppose that $X=\Gamma\backslash{\rm PSL}(2,\mathbb{R})$ is compact and let
$\varepsilon\in(0,\frac{\sigma_{0}}{8})$. If a periodic orbit $c$ of the flow
$(\varphi_{t})_{t\in\mathbb{R}}$ on $X$ with period $T>1$ has a
$(2,\varepsilon)$-antiparallel encounter, then it has a partner. Furthermore,
let $x,y\in c$, ${\mathcal{T}}(y)=(u,s)_{x}\in{\mathcal{P}}_{\varepsilon}(x)$
and $\varphi_{T_{1}}(x)=y,0<T_{1}<T$. Then the partner is
$\varepsilon^{\prime}$-partner with
$\varepsilon^{\prime}=\varepsilon+2(|u-se^{-T_{1}}|+|s-ue^{T_{1}-T}|)<8\varepsilon$
and the action difference between the orbit pair satisfies

$\Big{|}\frac{T^{\prime}-T}{2}-\ln(1+us)|<12\varepsilon^{2}(e^{-T_{1}}+e^{T_{1}-T}),$
(3.1)

where $T^{\prime}$ is the period of the partner. If
$\varepsilon\in(0,\frac{\varepsilon_{*}}{18})$, then the partner orbit is
unique.

###### Proof.

For the existence of a $T^{\prime}$-periodic point $v$, whose orbit under the
flow $(\varphi_{t})_{t\in\mathbb{R}}$ is a $\varepsilon^{\prime}$-partner
orbit, see [14, Theorem 9]. It was shown that the action difference satisfies

$\Big{|}\frac{T^{\prime}-T}{2}-\ln\big{(}1+(u-se^{-T_{1}})(s-ue^{T_{1}-T})\big{)}\Big{|}\leq|(u-se^{-T_{1}})(s-ue^{T_{1}-T})|e^{-T},$

which implies (3.1). For the last assertion, it follows from (3.1) that
$|T-T^{\prime}|<30\varepsilon^{2}$. Suppose that there is another partner
orbit which has the same property, i.e., it is also $8\varepsilon$-close to
the original one and its period called $T^{\prime\prime}$ satisfies
$|T^{\prime\prime}-T|<30\varepsilon^{2}$. Then these two partner orbits are
$\varepsilon_{*}$-close to each other for the whole time and their periods
satisfy

$|T^{\prime\prime}-T^{\prime}|\leq|T^{\prime\prime}-T|+|T^{\prime}-T|\leq
60\varepsilon^{2}<\sqrt{2}\varepsilon_{*}.$

By Lemma 2.10, the partner orbits must coincide. ∎

Figure 2: Periodic orbit with a single 2-antiparallel encounter has a partner
orbit.

######  Remark 3.1.

According to the proofs of [14, Theorem 9] and the Anosov closing lemma I, we
have $v=\Gamma ga_{-T_{2}}c_{ue^{-T_{2}}+\sigma}b_{\eta}$, where
$T_{2}=T_{1}-T,\sigma,\eta\in\mathbb{R}$ satisfy

$|\eta-\tilde{s}|<2\tilde{s}^{2}|\tilde{u}|+2|\tilde{s}|e^{-T}\quad\mbox{and}\quad|\sigma|<2|\tilde{u}|e^{-T}$
(3.2)

with $\tilde{s}=u-se^{-T_{1}}$ and $\tilde{u}=s-ue^{-T_{2}}$. $\diamond$

The next result is a new view of symbolic dynamics.

######  Theorem 3.2 (Symbolic dynamics).

In the setting of Theorem 3.1, let $g,h\in{\rm PSL}(2,\mathbb{R})$ be such
that $x=\Gamma g,y=\Gamma h$ and $hd_{\pi}=h^{\prime}=gc_{u}b_{s}$. If
$\gamma_{1},\gamma_{2}\in\Gamma$ such that $\gamma_{1}=ga_{T_{1}}h^{-1}$ and
$\gamma_{2}=ha_{T_{2}}g^{-1}$, where $T_{2}=T-T_{1}$, then the original orbit
corresponds to the conjugacy class $\\{\gamma_{1}\gamma_{2}\\}_{\Gamma}$ and
the partner orbit corresponds to the conjugacy class
$\\{\gamma_{1}^{-1}\gamma_{2}\\}_{\Gamma}$.

###### Proof.

Note that due to $\varphi_{T_{1}}(x)=y$ and $\varphi_{T_{2}}(y)=x$, there are
$\gamma_{1},\gamma_{2}\in\Gamma$ such that $\gamma_{1}=ga_{T_{1}}h^{-1}$ and
$\gamma_{2}=ha_{T_{2}}g^{-1}$ as assumption. Since $x=\Gamma g$ is a
$T$-periodic orbit, we have

$\gamma=ga_{T}g^{-1}$

for some $\gamma\in\Gamma$, and by Lemma 2.3, the orbit through $x$ (called
$c$) corresponds to the conjugacy class ${\\{\gamma\\}}_{\Gamma}$:

$\gamma_{1}\gamma_{2}=(ga_{T_{1}}h^{-1})(ha_{T_{2}}g^{-1})=ga_{T}g^{-1}=\gamma.$

This means that the orbit $c$ corresponds to the conjugacy class
${\\{\gamma\\}}_{\Gamma}={\\{\gamma_{1}\gamma_{2}\\}}_{\Gamma}$. Next, let
$\zeta\in\Gamma$ be such that

$\zeta=\bar{g}a_{T}b_{-s^{\prime}}c_{-u^{\prime}}(\bar{g})^{-1}$

for $\bar{g}=gc_{u}a_{-T_{2}}=h^{\prime}b_{-s}a_{T_{2}}$. According to the
proof of Theorem 3.1 and Remark 2.1, the partner orbit corresponds to the
conjugacy class $\\{\zeta\\}_{\Gamma}$. Now,

$\displaystyle\zeta$ $\displaystyle=$ $\displaystyle
h^{\prime}b_{-s}a_{-T_{2}}a_{T}b_{-(u-s^{-T_{1}})}c_{-(s-ue^{-T_{2}})}a_{T_{2}}c_{-u}g^{-1}$
$\displaystyle=$ $\displaystyle
h^{\prime}a_{T_{1}}b_{-se^{-T_{1}}}b_{-(u-s^{-T_{1}})}c_{-(s-ue^{-T_{2}})}c_{-ue^{-T_{2}}}a_{T_{2}}g^{-1}$
$\displaystyle=$ $\displaystyle
h^{\prime}a_{T_{1}}b_{-u}c_{-s}a_{T_{2}}g^{-1}$ $\displaystyle=$
$\displaystyle\gamma_{1}^{-1}\gamma_{2},$

noting $h^{\prime}a_{T_{1}}=\gamma_{1}^{-1}g^{\prime}$ owing to
$ga_{T_{1}}=\gamma_{1}h$. Therefore, the partner orbit $c^{\prime}$
corresponds to the conjugacy class
$\\{\zeta\\}_{\Gamma}=\\{\gamma_{1}^{-1}\gamma_{2}\\}_{\Gamma}$, completing
the proof. ∎

##  4 Periodic orbits including 2 encounters responsible for the third order
term

Let us first review orbit pairs responsible for the cubic contribution to
$K(\tau)$. Note that a sufficiently long periodic orbit has a huge number of
self-encounters which may involve arbitrarily many orbit stretches. Heusler,
Müller et al. [11, 19] show that only orbit pairs differing in two
2-encounters or in one single 3-encounter are responsible for the third order.
There are two ways of connections of orbit stretches forming two 2-encounters,
namely serial and intertwined, whereas the two stretches of each encounter may
be either close in phase space (depicted by nearly parallel arrows
$\mathrel{\,\lower
0.6458pt\hbox{\begin{picture}(4.0,3.0)\put(0.0,0.8){\vector(1,0){4.0}}\put(0.0,2.0){\vector(1,0){4.0}}
\end{picture}}\,}$ or $\mathrel{\,\lower
0.6458pt\hbox{\begin{picture}(4.0,3.0)\put(0.0,2.4){\vector(2,-1){4.0}}\put(0.0,0.4){\vector(2,1){4.0}}
\end{picture}}\,}$), which is called parallel encounter, or almost mutually
time-reversed (like in $\mathrel{\,\lower
0.6458pt\hbox{\begin{picture}(4.0,3.0)\put(4.0,0.8){\vector(-1,0){4.0}}\put(0.0,2.0){\vector(1,0){4.0}}
\end{picture}}\,}$ or $\mathrel{\,\lower
0.6458pt\hbox{\begin{picture}(4.0,3.0)\put(4.0,0.4){\vector(-2,1){4.0}}\put(0.0,0.4){\vector(2,1){4.0}}
\end{picture}}\,}$), which is called antiparallel encounter. In each way,
therefore, there are three possibilities: both 2-encounters are parallel-
encounters , one 2-parallel encounter and one 2-antiparallel encounter, and
two 2-antiparallel encounters, i.e. there are totally six cases. Only three of
them lead to (genuine) periodic orbits: two 2-antiparallel encounters serial,
one parallel-encounter and one antiparallel encounters intertwined and two
anti-parallel encounters intertwined, which are responsible in the cubic order
to the form factor. The others form pseudo-periodic orbits and do not
contribute to the spectral form factor.

In this section we only consider periodic orbits including two 2-antiparallel
encounters contributing to the third term of the spectral form factor. The
case of one 3-parallel encounter is rigorously done in [13, Section 3.3].
Orbits with a single 3-antiparallel encounter can be done analogously.

Throughout this section, we assume that the space $\Gamma\backslash{\rm
PSL}(2,\mathbb{R})$ is compact.

###  4.1 Periodic orbits including two 2-antiparallel encounters serial

In this subsection we consider periodic orbits having two 2-antiparallel
encounters serial, which is so called antiparallel-antiparallel serial (aas
for short) in [19]. Periodic orbits with either two small-angle self-crossings
in configuration space ($\mathrel{\,\lower
0.6458pt\hbox{\begin{picture}(4.0,3.0)\put(4.0,0.4){\vector(-2,1){4.0}}\put(0.0,0.4){\vector(2,1){4.0}}
\end{picture}}\,}$ and $\mathrel{\,\lower
0.6458pt\hbox{\begin{picture}(4.0,3.0)\put(4.0,0.4){\vector(-2,1){4.0}}\put(0.0,0.4){\vector(2,1){4.0}}
\end{picture}}\,}$) or one small-angle self-crossing and one anti-parallel
avoided self-crossing ($\mathrel{\,\lower
0.6458pt\hbox{\begin{picture}(4.0,3.0)\put(4.0,0.4){\vector(-2,1){4.0}}\put(0.0,0.4){\vector(2,1){4.0}}
\end{picture}}\,}$ and $\mathrel{\,\lower
0.6458pt\hbox{\begin{picture}(4.0,3.0)\put(4.0,0.8){\vector(-1,0){4.0}}\put(0.0,2.0){\vector(1,0){4.0}}
\end{picture}}\,}$), or two anti-parallel avoided self-crossings
($\mathrel{\,\lower
0.6458pt\hbox{\begin{picture}(4.0,3.0)\put(4.0,0.8){\vector(-1,0){4.0}}\put(0.0,2.0){\vector(1,0){4.0}}
\end{picture}}\,}$ and $\mathrel{\,\lower
0.6458pt\hbox{\begin{picture}(4.0,3.0)\put(4.0,0.8){\vector(-1,0){4.0}}\put(0.0,2.0){\vector(1,0){4.0}}
\end{picture}}\,}$) in configuration space are special cases of aas.

######  Theorem 4.1.

Let $\varepsilon\in(0,\frac{\varepsilon_{0}}{24})$ and let $c$ be a
$T$-periodic orbit of the flow $(\varphi_{t})_{t\in\mathbb{R}}$ on $X$ having
two $(2,\varepsilon)$-antiparallel encounters serial. More precisely, let
$x,y,z,w\in c$ and $T_{1},T_{2},T_{3},T_{4}>0$ be such that
$T_{1}+T_{2}+T_{3}+T_{4}=T$,
$\varphi_{T_{1}}(x)=y,\varphi_{T_{2}}(y)=z,\varphi_{T_{3}}(z)=w$ and
$\varphi_{T_{4}}(w)=x$, and
${\mathcal{T}}(z)=(u_{1},s_{1})_{y}\in{\mathcal{P}}_{\varepsilon}(y),{\mathcal{T}}(w)=(u_{2},s_{2})_{x}\in{\mathcal{P}}_{\varepsilon}(x)$
satisfying

$\displaystyle|u_{1}|>6\varepsilon
e^{-T_{2}},\quad|s_{1}|>30\varepsilon^{3}+13\varepsilon
e^{-T_{1}}+5\varepsilon e^{-T_{2}}+3\varepsilon e^{-T_{3}}.$ (4.1)

Then $c$ has a $20\varepsilon$-partner orbit which differs in both encounters
of the original orbit and the action difference satisfies

$\Big{|}\frac{T^{\prime}-T}{2}-\ln(1+u_{1}s_{1})(1+u_{2}s_{2})\Big{|}<\varepsilon^{2}(21e^{-T_{1}}+30e^{-T_{2}}+12e^{-T_{3}}+19e^{-T_{4}}).$
(4.2)

In addition, if $\varepsilon\in(0,\frac{\varepsilon_{*}}{40})$, then the
partner is unique.

###### Proof.

The sketch of the proof is as follows; see Figure 3 for an illustration.
First, we apply Theorem 3.1 for the left encounter to have one partner orbit,
which is depicted by the dashed line in Figure 3 (b). Next, we show that the
new orbit admits one 2-antiparallel encounter; see Figure 3 (c). Finally, we
apply Theorem 3.1 again to get a partner orbit depicted by the dashed line in
Figure 3 (d).

Figure 3: Construction of partner orbit for a given periodic orbit with two
2-antiparallel encounters serial.

Let $x=\Gamma g,y=\Gamma h,z=\Gamma k,w=\Gamma l$ for some $g,h,k,l\in{\rm
PSL}(2,\mathbb{R})$ and set
$g^{\prime}=gd_{\pi},h^{\prime}=hd_{\pi},k^{\prime}=kd_{\pi},l^{\prime}=ld_{\pi}$.
By hypothesis, $\varphi_{T_{123}}(x)=w$ and
${\mathcal{T}}(w)=(s_{2},u_{2})_{x}\in{\mathcal{P}}_{\varepsilon}(x)$, where
$T_{123}=T_{1}+T_{2}+T_{3}$. Due to Theorem 3.1, there are $\tilde{x}=\Gamma
l^{\prime}b_{-s_{2}}a_{-T_{4}}c_{\sigma_{2}}b_{\eta_{2}}=\Gamma
gc_{u_{2}}a_{-T_{4}}c_{\sigma_{2}}b_{\eta_{2}}\in X$ and $\widetilde{T}>0$
such that

$\Big{|}\frac{\widetilde{T}-T}{2}-\ln(1+u_{2}s_{2})\Big{|}\leq
12\varepsilon^{2}(e^{-T_{4}}+e^{-T_{123}}).$ (4.3)

Furthermore,

$\displaystyle
d_{X}(\varphi_{t}(\tilde{x}),\varphi_{t}(w))<7\varepsilon\quad\mbox{for}\quad
t\in[0,T_{4}]$ (4.4)

and

$d_{X}(\varphi_{t}(\tilde{x}),\varphi_{t}(x^{\prime}))<7\varepsilon\quad\mbox{for}\quad
t\in[T_{4},T].$ (4.5)

Next we show that the orbit of $\tilde{x}$ possesses one 2-antiparallel
encounter. We write

$\displaystyle{\mathcal{T}}(\tilde{z}):=\varphi_{T_{4}+T_{3}}(\tilde{x})$
$\displaystyle=$ $\displaystyle\Gamma
l^{\prime}b_{-s_{2}}a_{-T_{4}}c_{\sigma_{2}}b_{\eta_{2}}a_{T_{4}+T_{3}}$
$\displaystyle=$ $\displaystyle\Gamma
l^{\prime}b_{-s_{2}}a_{T_{3}}c_{\sigma_{2}e^{T_{3}+T_{4}}}b_{\eta_{2}e^{-T_{3}-T_{4}}}$
$\displaystyle=$ $\displaystyle\Gamma
l^{\prime}a_{T_{3}}b_{-s_{2}e^{-T_{3}}}c_{\sigma_{2}e^{T_{3}+T_{4}}}b_{\eta_{2}e^{-T_{3}-T_{4}}}$
$\displaystyle=$ $\displaystyle\Gamma
k^{\prime}b_{-s_{2}e^{-T_{3}}}c_{\sigma_{2}e^{T_{3}+T_{4}}}b_{\eta_{2}e^{-T_{3}-T_{4}}},$

using $\varphi_{T_{3}}({\mathcal{T}}(w))={\mathcal{T}}(z)$ and
$b_{s}a_{t}=a_{t}b_{se^{-t}}$, $c_{u}a_{t}=a_{t}c_{ue^{t}}$ for all
$u,s,t\in\mathbb{R}$. Then

$\tilde{z}=\Gamma
kc_{s_{2}e^{-T_{3}}}b_{-\sigma_{2}e^{T_{3}+T_{4}}}c_{-\eta_{2}e^{-T_{3}-T_{4}}};$
(4.6)

recall (2.9). Now,

$\displaystyle{\mathcal{T}}(\hat{y})$ $\displaystyle:=$
$\displaystyle\varphi_{T_{4}+T_{3}+T_{2}}(\tilde{x})=\Gamma
h^{\prime}b_{-s_{2}e^{-T_{2}-T_{3}}}c_{\sigma_{2}e^{T_{2}+T_{3}+T_{4}}}b_{\eta_{2}e^{-T_{2}-T_{3}-T_{4}}}$
$\displaystyle=$ $\displaystyle(\Gamma
kc_{s_{1}}b_{u_{1}})b_{-s_{2}e^{-T_{2}-T_{3}}}c_{\sigma_{2}e^{T_{2}+T_{3}+T_{4}}}b_{\eta_{2}e^{-T_{2}-T_{3}-T_{4}}}$
$\displaystyle=$ $\displaystyle(\Gamma
kc_{s_{2}e^{-T_{3}}}b_{-\sigma_{2}e^{T_{3}+T_{4}}}c_{-\eta_{2}e^{-T_{3}-T_{4}}})$
$\displaystyle
c_{\eta_{2}e^{-T_{3}-T_{4}}}b_{\sigma_{2}e^{T_{3}+T_{4}}}c_{-s_{2}e^{-T_{3}}+s_{1}}b_{u_{1}-s_{2}e^{-T_{2}-T_{3}}}c_{\sigma_{2}e^{T_{2}+T_{3}+T_{4}}}b_{\eta_{2}e^{-T_{2}-T_{3}-T_{4}}}$
$\displaystyle=$ $\displaystyle(\Gamma
kc_{s_{2}e^{-T_{3}}}b_{-\sigma_{2}e^{T_{3}+T_{4}}}c_{-\eta_{2}e^{-T_{3}-T_{4}}})c_{\tilde{u}_{1}}b_{\tilde{s}_{1}}a_{\tilde{\tau}_{1}},$

where

$\displaystyle\tilde{u}_{1}$ $\displaystyle=$ $\displaystyle
s_{1}+\eta_{2}e^{-T_{3}-T_{4}}-s_{2}e^{-T_{3}}+\sigma_{2}e^{T_{2}+T_{3}+T_{4}}+\frac{1}{1+\tilde{\rho}_{1}}\times$
(4.7) $\displaystyle\quad\
\times\,\big{(}(s_{1}-s_{2}e^{-T_{3}})\sigma_{2}e^{T_{2}+T_{3}+T_{4}}(u_{1}-s_{2}e^{-T_{2}-T_{3}})$
$\displaystyle\quad\
-(s_{1}-s_{2}e^{-T_{3}}+\sigma_{2}e^{T_{2}+T_{3}+T_{4}})\tilde{\rho}_{1}\big{)},$
$\displaystyle\tilde{s}_{1}$ $\displaystyle=$ $\displaystyle
u_{1}+\sigma_{2}e^{T_{3}+T_{4}}-s_{2}e^{-T_{2}-T_{3}}+\eta_{2}e^{-T_{2}-T_{3}-T_{4}}$
(4.8)
$\displaystyle\quad\,+\,\tilde{\rho}_{1}\big{(}(2+\tilde{\rho}_{1})\eta_{2}e^{-T_{2}-T_{3}-T_{4}}+\sigma_{2}e^{T_{3}+T_{4}}+u_{1}-s_{2}e^{-T_{2}-T_{3}}\big{)}$
$\displaystyle\quad\,+\,(s_{1}-s_{2}e^{-T_{3}})\sigma_{2}e^{T_{3}+T_{4}}(u_{1}-s_{2}e^{-T_{2}-T_{3}})(1+\tilde{\rho}_{1}),$
$\displaystyle\tilde{\tau}_{1}$ $\displaystyle=$ $\displaystyle
2\ln(1+\tilde{\rho}_{1}),$ (4.9)

with

$\tilde{\rho}_{1}=\sigma_{2}e^{T_{2}+T_{3}+T_{4}}(\sigma_{2}e^{T_{3}+T_{4}}+u_{1}-s_{2}e^{-T_{2}-T_{3}})+(s_{1}-s_{2}e^{-T_{3}})\sigma_{2}e^{T_{3}+T_{4}}\big{(}1+\sigma_{2}e^{T_{2}+T_{3}+T_{4}}(u_{1}-s_{2}e^{-T_{2}-T_{3}})\big{)},$

owing to Lemma 2.8 (a). A short calculation shows that
$|\tilde{u}_{1}|<2\varepsilon$ and $|\tilde{s}_{1}|<2\varepsilon$. This means
that

${\mathcal{T}}(\tilde{y})=\varphi_{-\tilde{\tau}_{1}}({\mathcal{T}}(\hat{y}))=(\tilde{u}_{1},\tilde{s}_{1})_{\tilde{z}}\in{\mathcal{P}}_{2\varepsilon}(\tilde{z}),$

where $\tilde{y}:=\varphi_{\tilde{\tau}_{1}}(\hat{y})$. Apply Theorem 3.1 to
obtain
$v=\tilde{z}a_{-\widetilde{T}_{2}}c_{\tilde{u}_{1}e^{-\widetilde{T}_{2}}+\sigma_{1}}b_{\eta_{1}}\in
X$ and $T^{\prime}\in\mathbb{R}$ such that

$\Big{|}\frac{T^{\prime}-\widetilde{T}}{2}-\ln(1+\tilde{u}_{1}\tilde{s}_{1})\Big{|}\leq
28\varepsilon^{2}(e^{-T_{1}-T_{3}-T_{4}+\tilde{\tau}_{1}}+e^{T_{1}+T_{3}+T_{4}-\tilde{\tau}_{1}-\widetilde{T}})<7\varepsilon^{2}e^{-T_{4}}+30\varepsilon^{2}e^{-T_{2}}$
(4.10)

and

$\displaystyle|\eta_{1}-\hat{s}_{1}|$ $\displaystyle<$ $\displaystyle
2\hat{s}_{1}^{2}|\hat{u}_{1}|+2|\hat{s}_{1}|e^{-\widetilde{T}}<30\varepsilon^{3}+10\varepsilon
e^{-\widetilde{T}},$ (4.11) $\displaystyle|\sigma_{1}|$ $\displaystyle<$
$\displaystyle 6\varepsilon e^{-\widetilde{T}},$ (4.12)

where
$\hat{u}_{1}=\tilde{s}_{1}-\tilde{u}_{1}e^{\widetilde{T}_{2}-\widetilde{T}},\hat{s}_{1}=\tilde{u}_{1}-\tilde{s}_{1}e^{-\widetilde{T}_{2}}$.
Furthermore,

$\displaystyle
d_{X}(\varphi_{t}(v),\varphi_{t}(\tilde{y}))<13\varepsilon\quad\mbox{for}\quad
t\in[0,T_{2}]$

and

$d_{X}(\varphi_{t}(v),\varphi_{t}({\mathcal{T}}(\tilde{z})))<13\varepsilon\quad\mbox{for}\quad
t\in[T_{2},\widetilde{T}].$

This means that the orbit of $v$ is $13\varepsilon$-close to the orbit of
$\tilde{x}$. Recalling from (4.4) and (4.5) that the orbit of $\tilde{x}$ is
$7\varepsilon$-close to the orbit of $x$, we deduce that the orbit of $v$ is
$20\varepsilon$-close to the original one.

Next, in order to establish an estimate for the action difference, observe
that by (4.7), (4.8),

$|\tilde{u}_{1}-s_{1}|<7\varepsilon e^{-T_{1}}+2\varepsilon
e^{-T_{3}},\quad|\tilde{s}_{1}-u_{1}|<\varepsilon e^{-T_{3}}$ (4.13)

after a short calculation. This yields

$|\ln(1+\tilde{u}_{1}\tilde{s}_{1})-\ln(1+u_{1}s_{1})|\leq
21\varepsilon^{2}e^{-T_{1}}+11\varepsilon^{2}e^{-T_{3}}$

and hence

$\displaystyle\Big{|}\frac{T^{\prime}-\widetilde{T}}{2}-\ln(1+u_{1}s_{1})\Big{|}\leq
21\varepsilon^{2}e^{-T_{1}}+30\varepsilon^{2}e^{-T_{2}}+11\varepsilon^{2}e^{-T_{3}}+7\varepsilon^{2}e^{-T_{4}},$
(4.14)

using (4.10). The estimate (4.2) follows from (4.3) and (4.14).

Next we are going to show that the partner orbit is different from the
original one. For, we find a point in the partner orbit which lies in the
Poincaré section of $y$ and is different from $y$ and ${\cal T}(z)$. Letting
$\widetilde{T}_{2}=T_{2}-\tilde{\tau}_{1}$, we have
$\varphi_{\widetilde{T}-\widetilde{T}_{2}}(\tilde{z})=\tilde{y}$ and
$\varphi_{\widetilde{T}_{2}}(\tilde{y})=\tilde{z}$. Recall
$v=\tilde{z}a_{-\widetilde{T}_{2}}c_{\tilde{u}_{1}e^{-\widetilde{T}_{2}}+\sigma_{1}}b_{\eta_{1}}$.
Using (4.6) and Lemma 2.8 (a), we write

$\displaystyle v$ $\displaystyle=$
$\displaystyle\tilde{z}a_{-\widetilde{T}_{2}}c_{\tilde{u}_{1}e^{-\widetilde{T}_{2}}+\sigma_{1}}b_{\eta_{1}}$
$\displaystyle=$ $\displaystyle\Gamma
kc_{s_{2}e^{-T_{3}}}b_{-\sigma_{2}e^{T_{3}+T_{4}}}c_{-\eta_{2}e^{-T_{3}-T_{4}}}a_{-T_{2}+\tilde{\tau}_{1}}c_{\tilde{u}_{1}e^{-\widetilde{T}_{2}}+\sigma_{1}}b_{\eta_{1}}$
$\displaystyle=$ $\displaystyle(\Gamma
ka_{-T_{2}})c_{s_{2}e^{-T_{2}-T_{3}}}b_{-\sigma_{2}e^{T_{2}+T_{3}+T_{4}}}c_{-\eta_{2}e^{-T_{2}-T_{3}-T_{4}}+\tilde{u}_{1}e^{-T_{2}}+\sigma_{1}e^{-\tilde{\tau}_{1}}}b_{\eta_{1}e^{\tilde{\tau}_{1}}}a_{\tilde{\tau}_{1}}$
$\displaystyle=$ $\displaystyle\Gamma hc_{u_{v}}b_{s_{v}}a_{\tau_{v}},$

where

$\displaystyle u_{v}$ $\displaystyle=$ $\displaystyle
s_{2}e^{-T_{2}-T_{3}}+\frac{-\eta_{2}e^{-T_{2}-T_{3}-T_{4}}+\tilde{u}_{1}e^{-T_{2}}+\sigma_{1}e^{-\tilde{\tau}_{1}}}{1+\rho_{v}},$
$\displaystyle s_{v}$ $\displaystyle=$
$\displaystyle(-\sigma_{2}e^{T_{2}+T_{3}+T_{4}}+\eta_{1}e^{\tilde{\tau}_{1}}+\eta_{1}e^{\tilde{\tau}_{1}}\rho_{v})(1+\rho_{v}),$
$\displaystyle\tau_{v}$ $\displaystyle=$ $\displaystyle 2\ln(1+\rho_{v}),$

here

$\rho_{v}=(-\sigma_{2}e^{T_{2}+T_{3}+T_{4}})(-\eta_{2}e^{-T_{2}-T_{3}-T_{4}}+\tilde{u}_{2}e^{-T_{2}}+\sigma_{1}e^{-\tilde{\tau}_{1}});$

recall $\tilde{\tau}_{1}$ from (4.9). A short calculation shows that

$|u_{v}|<6\varepsilon e^{-T_{2}},|s_{v}-\eta_{1}|<6\varepsilon e^{-T_{1}},$
(4.15)

which imply that
$\tilde{v}=\varphi_{\tau_{v}}(z)=(u_{v},s_{v})_{y}\in{\mathcal{P}}_{6\varepsilon}(y)$.
By assumption, $\varepsilon<\frac{\sigma_{0}}{24}$ and
${\mathcal{T}}(z)=(u_{1},s_{1})_{y}$ with

$|u_{1}|\geq 6\varepsilon e^{-T_{2}}>|u_{v}|.$

As a consequence of Lemma 2.2, we get $\tilde{v}\neq{\mathcal{T}}(z)$. It
remains to check that $\tilde{v}\neq y$. Recalling that
$\hat{s}_{1}=\tilde{u}_{1}-\tilde{s}_{1}e^{-\widetilde{T}_{2}}$, it follows
from (4.11) and (4.13) that

$\displaystyle|\eta_{1}-s_{1}|$ $\displaystyle\leq$
$\displaystyle|\eta_{1}-\hat{s}_{1}|+|\hat{s}_{1}-\tilde{u}_{1}|+|\tilde{u}_{1}-s_{1}|$
(4.16) $\displaystyle<$ $\displaystyle 30\varepsilon^{3}+7\varepsilon
e^{-T_{1}}+5\varepsilon e^{-T_{2}}+3\varepsilon e^{-T_{3}}.$

Using (4.15), (4.16), together with the assumption (4.1) we deduce

$|s_{v}|\geq|s_{1}|-|s_{v}-\eta_{1}|-|\eta_{1}-s_{1}|>0,$

which shows $\tilde{v}\neq y$. Consequently, the orbit of $v$ is different
from the obit of $x$ and its time reverse.

For the last assertion, suppose that there is another $20\varepsilon$-partner
orbit which differs in both encounters of the original orbit. Then the two
partner orbits are $40\varepsilon$-close to each other for the whole time and
the period difference $|T^{\prime\prime}-T^{\prime}|<50\varepsilon^{2}$, where
$T^{\prime\prime}$ is the period of the new partner. Due to
$\varepsilon<\frac{\varepsilon_{*}}{40}$, the two partners are
$\varepsilon_{*}$-close to each other for the whole time and
$|T^{\prime\prime}-T^{\prime}|<\varepsilon_{*}$; so they must be identical by
Lemma 2.10. The proof is complete. ∎

######  Remark 4.1.

(a) The orbit of $\tilde{x}$ is also a partner orbit of the original one,
which is depicted by the dotted line in Figure 3 (b). However, this partner
orbit differs only in one encounter and this orbit pair only contribute to the
second order term of the spectral form factor as a Sieber-Richter pair.

(b) If we first apply Theorem 3.1 for the other encounter, then we have
another partner orbit for the original one. This orbit pair also contribute to
the second order term of $K(\tau)$. Then, using the same argument of the
previous proof, we get the same partner orbit.

The next result provides a new view of symbolic dynamics of orbit pair in the
preceding theorem.

######  Theorem 4.2.

In the setting of Theorem 4.1, let
$\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\in\Gamma$ be such that
$ga_{T_{1}}=\gamma_{1}h,ha_{T_{2}}=\gamma_{2}k,ka_{T_{3}}=\gamma_{3}l,la_{T_{4}}=\gamma_{4}g$.
Then the original orbit and the partner orbit correspond to the conjugacy
classes $\\{\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\\}_{\Gamma}$ and
$\\{\gamma_{1}^{-1}\gamma_{4}\gamma_{3}^{-1}\gamma_{2}\\}_{\Gamma}$,
respectively.

###### Proof.

First observe that
$a_{T_{1}}=g^{-1}\gamma_{1}h,a_{T_{2}}=h^{-1}\gamma_{2}k,a_{T_{3}}=k^{-1}\gamma_{3}l,a_{T_{4}}=l^{-1}\gamma_{4}g$
leads to

$ga_{T}=ga_{T_{1}}a_{T_{2}}a_{T_{3}}a_{T_{4}}=g(g^{-1}\gamma_{1}h)(h^{-1}\gamma_{2}k)(k^{-1}\gamma_{3}l)(l^{-1}\gamma_{4}g)=\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}g.$

This yields the orbit of $x=\Gamma g$ under the flow
$(\varphi_{t})_{t\in\mathbb{R}}$, which is the original orbit, corresponds to
the conjugacy class $\\{\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\\}_{\Gamma}$
by the definition of the mapping $\varsigma$ in (2.2).

Next, due to $ga_{T_{1}+T_{2}+T_{3}}=\gamma_{1}\gamma_{2}\gamma_{3}l$ and
$la_{T_{4}}=\gamma_{4}g$, the orbit of $\hat{x}$ corresponds to the conjugacy
class

$\displaystyle\\{\zeta\\}_{\Gamma}=\\{(\gamma_{1}\gamma_{2}\gamma_{3})^{-1}\gamma_{4}\\}_{\Gamma}=\\{\gamma_{3}^{-1}\gamma_{2}^{-1}\gamma_{1}^{-1}\gamma_{4}\\}_{\Gamma}=\\{\gamma_{2}^{-1}\gamma_{1}^{-1}\gamma_{4}\gamma_{3}^{-1}\\}_{\Gamma}.$

by Theorem 3.2. Similarly, the orbit of $v$ corresponds to the conjugacy class

$\displaystyle\\{\xi\\}_{\Gamma}=\\{({\gamma_{2}^{-1}})^{-1}\gamma_{1}^{-1}\gamma_{4}\gamma_{3}^{-1}\\}_{\Gamma}=\\{\gamma_{2}\gamma_{1}^{-1}\gamma_{4}\gamma_{3}^{-1}\\}_{\Gamma}=\\{\gamma_{1}^{-1}\gamma_{4}\gamma_{3}^{-1}\gamma_{2}\\}_{\Gamma},$

as was to be shown. ∎

The following corollary considers periodic orbits of the geodesic flow having
two small-angle self-crossings.

######  Corollary 4.1.

Suppose that all elements in $\Gamma\setminus\\{e\\}$ are hyperbolic. Suppose
that $T$-periodic orbit of the geodesic flow
${(\varphi^{\mathcal{X}}_{t})}_{t\in\mathbb{R}}$ on
$\mathcal{X}=T^{1}(\Gamma\backslash\mathbb{H}^{2})$ crosses itself in
configuration space at a time $T_{1}$, at an angle $\theta_{1}$, creates a
loop of length $T_{2}$ and then crosses itself again after a time $T_{3}$ at
an angle $\theta_{2}$, and creates another loop of length $T_{4}$ with
$T_{1}+T_{2}+T_{3}+T_{4}=T$. If $0<\phi<\frac{1}{3}$ for
$\phi=\max\\{\pi-\theta_{1},\pi-\theta_{2}\\}$, then it has a
$36|\sin(\phi/2)|$-partner orbit; the period of the partner orbit denoted by
$T^{\prime}$ satisfies

$\displaystyle\Big{|}\frac{T^{\prime}-T}{2}-$
$\displaystyle\ln\left(1+\sin^{2}(\phi_{1}/2)\right)\left(1+\sin^{2}(\phi_{2}/2)\right)\Big{|}$
$\displaystyle<\sin^{2}(\phi/2)(21e^{-T_{1}}+31e^{-T_{2}}+13e^{-T_{3}}+19e^{-T_{4}}).$
(4.17)

Furthermore, if $\Gamma\backslash\mathbb{H}^{2}$ is compact and
$\phi<\frac{\varepsilon_{*}}{20}$ then the partner is unique.

###### Proof.

Let ${\mathtt{x}}\in{\mathcal{X}}=T^{1}(\Gamma\backslash\mathbb{H}^{2})$ be a
$T$-periodic point of the flow $(\varphi_{t}^{\mathcal{X}})_{t\in\mathbb{R}}$
and let

$\varphi_{T_{1}}^{{\mathcal{X}}}({\mathtt{x}})={\mathtt{y}},\quad\varphi_{T_{2}}^{{\mathcal{X}}}({\mathtt{y}})={\mathtt{z}},\quad\varphi_{T_{3}}^{{\mathcal{X}}}({\mathtt{z}})={\mathtt{w}},\quad\varphi_{T_{4}}^{{\mathcal{X}}}({\mathtt{w}})={\mathtt{x}}.$
(4.18)

Let $\phi_{1}=\pi-\theta_{1},\phi_{2}=\pi-\theta_{2}$,
$\phi=\max\\{\phi_{1},\phi_{2}\\}$ and assume that $|\phi|<\frac{1}{3}$. Then
in particular

$\Big{|}\sin\Big{(}\frac{\phi_{i}}{2}\Big{)}\Big{|}\leq\frac{|\phi_{i}|}{2}<\frac{1}{6}$
(4.19)

holds for $i=1,2$. Set
$x=\Xi({\mathtt{x}}),y=\Xi({\mathtt{y}}),z=\Xi({\mathtt{z}})$, and
$w=\Xi({\mathtt{w}})$; recall the isometry
$\Xi:T^{1}(\Gamma\backslash\mathbb{H}^{2})\to\Gamma\backslash{\rm
PSL}(2,\mathbb{R})$. It follows from Theorem 2.7 that

$\mbox{either}\quad\Gamma h=\Gamma kd_{\theta_{1}}\quad\mbox{or}\quad\Gamma
h=\Gamma kd_{-\theta_{1}}$

and

$\mbox{either}\quad\Gamma l=\Gamma gd_{\theta_{2}}\quad\mbox{or}\quad\Gamma
l=\Gamma gd_{-\theta_{2}}$

with some $g,h,k,l\in{\rm PSL}(2,\mathbb{R})$ such that $\Gamma g=x,\Gamma
h=y,\Gamma k=z$ and $\Gamma l=w$. We only consider $\Gamma h=\Gamma
kd_{\theta_{1}}$ and $\Gamma l=\Gamma gd_{\theta_{2}}$, the other cases are
similar. Define $z^{\prime}={\mathcal{T}}(z)$ and
$w^{\prime}={\mathcal{T}}(w)$; recall the notation ${\mathcal{T}}$ from
Definition 2.3. Then

$z^{\prime}=\Gamma hd_{\pi-\theta_{1}}=\Gamma
hd_{\phi_{1}}\quad\mbox{and}\quad w^{\prime}=\Gamma gd_{\pi-\theta_{2}}=\Gamma
gd_{\phi_{2}}.$

For $i=1,2$, apply Lemma 2.8 (a) to write

$d_{\phi_{i}}=c_{u_{i}}b_{s_{i}}a_{\tau_{i}},$ (4.20)

where

$\tau_{i}=2\ln(\cos(\phi_{i}/2)),\quad u_{i}=\tan(\phi/2),\quad
s_{i}=-\sin(\phi_{i}/2)\cos(\phi_{i}/2).$ (4.21)

Owing to (4.19), we have

$\cos\Big{(}\frac{\phi_{i}}{2}\Big{)}>\frac{5}{6}.$ (4.22)

Define $\varepsilon=\frac{6}{5}|\sin(\phi/2)$. Observe that

$|u_{i}|=|\tan(\phi_{i}/2)|\leq\frac{6}{5}|\sin(\phi_{i}/2)|\leq\varepsilon,\
|s_{i}|=|\sin(\phi_{i}/2)\cos(\phi_{i}/2)|\leq|\sin(\phi_{i}/2)|<\varepsilon,$
(4.23)

and

$|\tau_{i}|=|\ln(1-\sin^{2}(\phi_{i}/2))|\leq
2\sin^{2}(\phi_{i}/2)\leq\frac{1}{2}\,\varepsilon^{2},$

due to $|\ln(1+z)|\leq 2|z|$ for $|z|\leq 1/2$. Denote
$\tilde{z}=\varphi_{\tau_{1}}(z)$ and $\tilde{w}=\varphi_{\tau_{2}}(w)$. This
leads to

$\displaystyle{\mathcal{T}}(\tilde{z})$
$\displaystyle=\varphi_{-\tau_{1}}(z^{\prime})=\Gamma
k^{\prime}a_{-\tau_{1}}=\Gamma
hc_{u_{1}}b_{s_{1}}=(u_{1},s_{1})_{y}\in{\mathcal{P}}_{\varepsilon}(y),$
$\displaystyle{\mathcal{T}}(\tilde{w})$
$\displaystyle=\varphi_{-\tau_{2}}(w^{\prime})=\Gamma
l^{\prime}a_{-\tau_{2}}=\Gamma
gc_{u_{2}}b_{s_{2}}=(u_{2},s_{2})_{x}\in{\mathcal{P}}_{\varepsilon}(x),$

using (2.9). Define
$\widetilde{T}_{2}=T_{2}+\tau_{1},\tilde{T}_{3}=T_{3}-\tau_{1}+\tau_{2},\widetilde{T}_{4}=T_{4}-\tau_{2},\widetilde{T}_{1}=T_{1}$.
Then $T_{1}+\widetilde{T}_{2}+\widetilde{T}_{3}+\widetilde{T}_{4}=T$,
$\varphi_{T_{1}}(x)=y,\varphi_{\widetilde{T}_{2}}(y)=\tilde{z},\varphi_{\widetilde{T}_{3}}(z)=\tilde{w}$,
and $\varphi_{\widetilde{T}_{4}}(\tilde{w})=x$. We now apply Theorem 3.1 with
$\varepsilon=\frac{6}{5}|\sin(\phi/2)|$ to have a partner orbit, which is
$33|\sin(\phi/2)|$-close to the original one and has period $T^{\prime}$
satisfying

$\displaystyle\Big{|}\frac{T^{\prime}-T}{2}-\ln(1+\sin^{2}(\phi_{1}/2))(1+\sin^{2}(\phi_{2}/2))\Big{|}$
$\displaystyle<\varepsilon^{2}(21e^{-T_{1}}+30e^{-\widetilde{T}_{2}}+12e^{-\widetilde{T}_{3}}+19e^{-\widetilde{T}_{4}})$
$\displaystyle<\varepsilon^{2}(21e^{-T_{1}}+31e^{-T_{2}}+13e^{-T_{3}}+19e^{-T_{4}}),$

which is (4.17).

For the last assertion, if $\phi<\frac{\varepsilon_{*}}{12}$, then
$\varepsilon=\frac{6}{5}|\sin(\phi/2)|<\frac{\varepsilon_{*}}{20}$ and whence
the partner orbit is unique according to Theorem 4.1; see Figure 4 for an
illustration. ∎

Figure 4: A periodic orbit with 2 small-angle-self-crossings has a partner
orbit which has 2 avoided crossings.

######  Remark 4.2.

The existence of the partner orbit in the preceding theorem does not need the
condition that the space is compact. In fact, according [12, Theorem 3.11] and
the proof of Theorem 4.1, the partner orbit has a smaller period
$T^{\prime}<T$.

###  4.2 Periodic orbits including two 2-parallel encounters intertwined

In this subsection we consider periodic orbits with two 2-parallel encounters
intertwined, which is so called parallel-parallel intertwined (ppi for short)
in [19].

######  Theorem 4.3.

Let $\varepsilon\in(0,\frac{\sigma_{0}}{20})$. Suppose that a $T$-periodic
orbit $c$ of the flow $(\varphi_{t})_{t\in\mathbb{R}}$ on $X$ with period $T$
has two $(2,\varepsilon)$-parallel encounters intertwined. For instance,
suppose that there are $x,y,z,w\in c$ and $T_{1},T_{2},T_{3},T_{4}>0$ such
that $T_{1}+T_{2}+T_{3}+T_{4}=T$,
$\varphi_{T_{1}}(x)=y,\varphi_{T_{2}}(y)=z,\varphi_{T_{3}}(z)=w$ and
$\varphi_{T_{4}}(w)=x$ with
$z=(u_{1},s_{1})_{x}\in{\mathcal{P}}_{\varepsilon}(x),w=(u_{2},s_{2})_{y}\in{\mathcal{P}}_{\varepsilon}(y)$
satisfying

$\displaystyle|u_{2}|>9\varepsilon e^{-T_{4}},\
|s_{2}|>72\varepsilon^{3}+5\varepsilon e^{-T_{1}}+2\varepsilon e^{-T_{3}}.$
(4.24)

Then $c$ has a $19\varepsilon$-partner orbit of period $T^{\prime}$ which
differs in both encounters and the action difference satisfies

$\Big{|}\frac{T^{\prime}-T}{2}-\ln(1+s_{1}u_{1})(1+s_{2}u_{2})|\leq
54\varepsilon^{4}+25\varepsilon^{2}(e^{-T_{1}}+e^{-T_{2}}+e^{-T_{3}}+e^{-T_{4}}).$
(4.25)

If $\varepsilon\in(0,\frac{\varepsilon_{*}}{38})$, then the partner orbit is
unique.

###### Proof.

The construction of a partner orbit is summarized as follows. We first apply
the Anosov closing lemmas for the first encounter to obtain two shorter
periodic orbits, which are expressed by dashed line and dotted line in Figure
5 (b). We next show that the obtained orbits create a pseudo-orbit (see Figure
5 (c)). Finally we use the connecting lemma to get a new periodic orbit, which
is illustrated by the dashed line in Figure 5 (d).

Figure 5: Reconnection to give the partner orbit for orbit with two parallel
encounters intertwined.

Denote $T_{12}=T_{1}+T_{2}$ and $T_{34}=T_{3}+T_{4}$. By assumption,
$\varphi_{T_{12}}(x)=z=(u_{1},s_{1})_{x}\in{\mathcal{P}}_{\varepsilon}(x)$.
According to the Anosov closing lemma I, there exist $\tilde{x}=\Gamma
hb_{\eta_{x}}c_{\sigma_{x}}\in X$ and $T_{12}^{\prime}\in\mathbb{R}$ such that
$\varphi_{T^{\prime}_{12}}(\tilde{x})=\tilde{x}$,

$\Big{|}\frac{T_{12}^{\prime}-T_{12}}{2}-\ln(1+u_{1}s_{1})\Big{|}\leq
5\varepsilon^{2}e^{-T_{1}-T_{2}}$ (4.26)

and

$d_{X}(\varphi_{t}(\tilde{x}),\varphi_{t}(x))<4\varepsilon,\quad\mbox{for
all}\quad t\in[0,T_{1}+T_{2}].$ (4.27)

The orbit of $\tilde{x}$ is depicted by the dashed line in Figure 5 (b).
Observe that
$\varphi_{T_{34}}(z)=x=(-s_{1},-u_{1})_{z}^{\prime}\in{\mathcal{P}}^{\prime}_{\varepsilon}(z)$.
Apply the Anosov closing lemma II, there exist $\tilde{z}=\Gamma
lb_{\eta_{z}}c_{\sigma_{z}}\in X$ and $T_{34}^{\prime}\in\mathbb{R}$ such that
$\varphi_{T_{34}^{\prime}}(\tilde{z})=\tilde{z}$,

$\Big{|}\frac{T_{34}^{\prime}-T_{34}}{2}\Big{|}\leq
4\varepsilon^{2}e^{-T_{3}-T_{4}}$ (4.28)

and

$d_{X}(\varphi_{t}(\tilde{z}),\varphi_{t}(z))<4\varepsilon,\quad\mbox{for
all}\quad t\in[0,T_{3}+T_{4}].$ (4.29)

The orbit of $\tilde{z}$ is depicted by the dotted line in Figure 5 (b).

Next we show that two orbits of $\tilde{x}$ and $\tilde{z}$ form a pseudo-
orbit, and hence it is possible to connect them to obtain a longer periodic
orbit. Using Lemma 2.8 (b), we write

$\displaystyle\varphi_{T_{3}}(\tilde{z})=\Gamma
lb_{\eta_{z}}c_{\sigma_{z}}a_{T_{3}}$ $\displaystyle=$ $\displaystyle\Gamma
la_{T_{3}}b_{\eta_{z}e^{-T_{3}}}c_{\sigma_{z}e^{T_{3}}}=\Gamma
kb_{\eta_{z}e^{-T_{3}}}c_{\sigma_{z}e^{T_{3}}}$ $\displaystyle=$
$\displaystyle\Gamma
hc_{u_{2}}b_{s_{2}}b_{\eta_{z}e^{-T_{3}}}c_{\sigma_{z}e^{T_{3}}}$
$\displaystyle=$ $\displaystyle(\Gamma
hc_{\sigma_{x}e^{T_{1}}}b_{\eta_{x}e^{-T_{1}}})b_{-\eta_{x}e^{-T_{1}}}c_{u_{2}-\sigma_{x}e^{T_{1}}}b_{s_{2}+\eta_{z}e^{-T_{3}}}c_{\sigma_{z}e^{T_{3}}}$
$\displaystyle=$ $\displaystyle(\Gamma
hc_{\sigma_{x}e^{T_{1}}}b_{\eta_{x}e^{-T_{1}}})b_{s}c_{u}a_{\tau},$

where

$\displaystyle s$ $\displaystyle=$
$\displaystyle\eta_{z}e^{-T_{1}}+\frac{s_{2}+\eta_{z}e^{-T_{3}}}{1+(u_{2}-\sigma_{x}e^{T_{1}})(s_{2}+\eta_{z}e^{-T_{3}})},$
$\displaystyle u$ $\displaystyle=$
$\displaystyle\left(u_{2}-\sigma_{x}e^{T_{1}}+\sigma_{z}e^{T_{3}}+(u_{2}-\sigma_{x}e^{T_{1}})(s_{2}+\eta_{z}e^{-T_{3}})\sigma_{z}e^{T_{3}}\right)\left(1+(u_{2}-\sigma_{x}e^{T_{1}})(s_{2}+\eta_{z}e^{-T_{3}})\right),$
$\displaystyle\tau$ $\displaystyle=$
$\displaystyle-2\ln\left(1+(u_{2}-\sigma_{x}e^{T_{1}})(s_{2}+\eta_{z}e^{-T_{3}})\right).$
(4.30)

A short calculation shows that $|u|<3\varepsilon,|s|<3\varepsilon$. If we set

$\displaystyle\tilde{y}:=\varphi_{T_{1}}(\tilde{x})=\Gamma
gc_{\sigma_{x}}b_{\eta_{x}}a_{T_{1}}=\Gamma
ga_{T_{1}}c_{\sigma_{x}e^{T_{1}}}b_{\eta_{x}e^{-T_{1}}}=\Gamma
hc_{\sigma_{x}e^{T_{1}}}b_{\eta_{x}e^{-T_{1}}}$

and $\tilde{w}:=\varphi_{T_{3}-\tau}(\tilde{z})$, then
$\tilde{y}=(-u,-s)_{\tilde{w}}\in{\mathcal{P}}_{3\varepsilon}(\tilde{w})$.
Apply the connecting lemma I to obtain a $T^{\prime}$-periodic point
$v=\tilde{w}c_{-ue^{-T_{12}^{\prime}}+\sigma}b_{\eta}$ satisfying

$\displaystyle\Big{|}\frac{T^{\prime}-(T^{\prime}_{12}+T^{\prime}_{34})}{2}-\ln(1+us)\Big{|}<63\varepsilon^{2}(e^{-T^{\prime}_{12}}+e^{-T^{\prime}_{34}})$
(4.31)

and

$|\eta+s|<2s^{2}|u|+2|s|e^{-T_{1}-T_{2}}<54\varepsilon^{3}+6\varepsilon
e^{-T_{1}-T_{2}}.$ (4.32)

Furthermore,

$d_{X}(\varphi_{t}(v),\varphi_{t}(\tilde{z}))<15\varepsilon\quad\mbox{for}\quad
t\in[0,T^{\prime}_{3,4}]$

and

$d_{X}(\varphi_{t+T^{\prime}_{34}}(v),\varphi_{t}(\tilde{w}))<15\varepsilon\quad\mbox{for}\quad
t\in[0,T^{\prime}_{1,2}].$

This yields that the orbit of $v$ is $15\varepsilon$-close to the orbits of
$\tilde{x}$ and $\tilde{z}$. Together with (4.27) and (4.29), this implies
that the orbit of $v$ is $19\varepsilon$-close to the orbit of $x$.

Next, we estimate the action difference. According to the Anosov closing
lemmas,

$|\eta_{x}|<\frac{3\varepsilon}{2},|\eta_{z}|<\frac{3\varepsilon}{2},|\sigma_{x}|<2\varepsilon
e^{-T_{1}-T_{2}},|\sigma_{z}|<2\varepsilon e^{-T_{3}-T_{4}}.$

Observe that

$|u-u_{2}|<9\varepsilon^{3}+2\varepsilon e^{-T_{2}}+2\varepsilon
e^{-T_{4}}\quad\mbox{and}\quad|s-s_{2}|<3\varepsilon^{3}+2\varepsilon
e^{-T_{1}}+2\varepsilon e^{-T_{3}}$ (4.33)

imply

$|\ln(1+us)-\ln(1+u_{2}s_{2})|<54\varepsilon^{4}+18\varepsilon^{2}e^{-T_{1}}+6\varepsilon^{2}e^{-T_{2}}+18\varepsilon^{2}e^{-T_{3}}+6\varepsilon^{2}e^{-T_{4}},$

and hence

$\displaystyle\Big{|}\frac{T^{\prime}-T^{\prime}_{12}-T^{\prime}_{34}}{2}$
$\displaystyle-\ln(1+u_{2}s_{2})\Big{|}$ $\displaystyle\leq
54\varepsilon^{4}+24\varepsilon^{2}e^{-T_{1}}+24\varepsilon^{2}e^{-T_{2}}+24\varepsilon^{2}e^{-T_{3}}+24\varepsilon^{2}e^{-T_{4}},$
(4.34)

owing to (4.31). The estimate (4.25) follows from (4.26), (4.28) and (4.34).

Next, we are going to show that the orbit of $v$ is different from the orbit
of $x$. Analogously to the proof of Theorem 4.1, we write

$\displaystyle v$ $\displaystyle=$
$\displaystyle\tilde{w}c_{ue^{-T_{34}^{\prime}}+\sigma}b_{\eta}$
$\displaystyle=$ $\displaystyle(\Gamma
hc_{\sigma_{x}e^{T_{1}}}b_{\eta_{x}e^{-T_{1}}})b_{s}c_{u}c_{ue^{-T_{34}^{\prime}}+\sigma}b_{\eta}$
$\displaystyle=$ $\displaystyle\Gamma
hc_{\sigma_{x}e^{T_{1}}}b_{\eta_{x}e^{-T_{1}}+s}c_{u+ue^{-T_{34}^{\prime}}+\sigma}b_{\eta}$
$\displaystyle=$ $\displaystyle\Gamma hc_{u_{v}}b_{s_{v}}a_{\tau_{v}}$
$\displaystyle=$ $\displaystyle\Gamma
hc_{u_{2}}b_{s_{2}}b_{\eta_{z}e^{-T_{3}}}c_{\sigma_{z}e^{T_{3}}}a_{-\tau}c_{ue^{-T_{34}^{\prime}}+\sigma}b_{\eta}$
$\displaystyle=$ $\displaystyle\Gamma
hc_{u_{2}}b_{s_{2}+\eta_{z}e^{-T_{3}}}c_{\sigma_{z}e^{T_{3}}+ue^{-T_{34}^{\prime}+\tau}+\sigma}b_{\eta
e^{-\tau}}a_{-\tau}$ $\displaystyle=$ $\displaystyle\Gamma
hc_{u_{v}}b_{s_{v}}a_{\tau_{v}},$

where

$\displaystyle u_{v}$ $\displaystyle=$ $\displaystyle
u_{2}+\frac{\sigma_{z}e^{T_{3}}+ue^{-T_{34}^{\prime}+\tau}+\sigma}{1+(s_{2}+\eta_{z}e^{-T_{3}})(\sigma_{z}e^{T_{3}}+ue^{-T_{34}^{\prime}+\tau}+\sigma)},$
$\displaystyle s_{v}$ $\displaystyle=$
$\displaystyle(s_{2}+\eta_{z}e^{-T_{3}}+\eta e^{-\tau}+\eta
e^{-\tau}\rho_{v})(1+\rho_{v}),$ $\displaystyle\tau_{v}$ $\displaystyle=$
$\displaystyle 2\ln\big{(}1+\rho_{v}\big{)}-\tau,$

recalling $\tau$ from (4.30); here

$\rho_{v}=(s_{2}+\eta_{z}e^{-T_{3}})(\sigma_{z}e^{T_{3}}+ue^{-T_{34}^{\prime}+\tau}+\sigma).$

A short calculation shows that $|s_{v}|<5\varepsilon$ and
$|u_{v}|<5\varepsilon$. Consequently,
$\tilde{v}:=\varphi_{-\tau_{v}}(v)=(u_{v},s_{v})_{y}\in{\mathcal{P}}_{5\varepsilon}(y)$.
We need to check that $\tilde{v}\neq y$ and $\tilde{v}\neq w$. For, observe
that

$\displaystyle|u_{v}-u_{2}|<9\varepsilon
e^{-T_{4}},\quad|s_{v}-s_{2}-\eta|<15\varepsilon^{3}.$

Furthermore, it follows from (4.32) and (4.33) that

$|\eta+s_{2}|\leq|\eta+s|+|s-s_{2}|<57\varepsilon^{3}+5\varepsilon
e^{-T_{1}}+2\varepsilon e^{-T_{3}}.$

Using assumption (4.24), we derive

$\displaystyle|u_{v}|$ $\displaystyle>$
$\displaystyle|u_{2}|-|u_{v}-u_{2}|>|u_{2}|-9\varepsilon e^{-T_{4}}>0$
$\displaystyle|s_{v}-s_{2}|$ $\displaystyle>$
$\displaystyle|s_{2}|-|s_{v}-s_{2}-\eta|-|\eta+s_{2}|>0.$

This means $u_{v}\neq 0$ and $s_{v}\neq s_{2}$. Owing to
$\varepsilon<\frac{\sigma_{0}}{20}$, we get $\tilde{v}\neq y,\tilde{v}\neq w$
and hence the orbit of $v$ is different from the orbit of $x$.

For the last assertion, recall that the orbit of $v$ is 19$\varepsilon$-close
to the original orbit. The uniqueness of partner orbit can be done analogously
to the proof of Theorem 4.1. ∎

######  Remark 4.3.

(a) The period of the partner orbit can be explicitly computed according to
the proof of Anosov closing lemmas and the connecting lemma. The term
$33\varepsilon^{4}$ appears when we change the coordinates $(u,s)$ to
$(u_{2},s_{2})$ (see (4.33)) and it cannot avoid.

(b) Condition (4.24) can be replaced by

$\displaystyle|u_{1}|>9\varepsilon e^{-T_{3}},\
|s_{1}|>72\varepsilon^{3}+5\varepsilon e^{-T_{4}}+2\varepsilon e^{-T_{2}}.$
(4.35)

######  Theorem 4.4.

In the setting of Theorem 4.3, suppose that $x=\Gamma g,y=\Gamma h,z=\Gamma
k,w=\Gamma l$ for some $g,h,k,l\in{\rm PSL}(2,\mathbb{R})$. If
$\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\in\Gamma$ satisfy
$ga_{T_{1}}=\gamma_{1}h,ha_{T_{2}}=\gamma_{2}k,ka_{T_{3}}=\gamma_{3}l,la_{T_{4}}=\gamma_{4}g$,
then the original orbit corresponds to the conjugacy class
$\\{\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\\}$ and the partner orbit
corresponds to the conjugacy class
$\\{\gamma_{2}\gamma_{1}\gamma_{4}\gamma_{3}\\}$.

###### Proof.

That the original orbit corresponds to the conjugacy class
$\\{\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\\}$ can be done analogously to
Theorem 4.2. For the last assertion, we can choose $g,h,k,l\in{\rm
PSL}(2,\mathbb{R})$ such that $k=gc_{u_{1}}b_{s_{1}}$ and
$l=hc_{u_{2}}b_{s_{2}}$. Then $\varphi_{T_{1}+T_{2}}(\Gamma g)=\Gamma
gc_{u_{1}}b_{s_{1}}$ and
$ga_{T_{1}+T_{2}}=\gamma_{1}ha_{T_{2}}=\gamma_{1}\gamma_{2}k=\gamma_{1}\gamma_{2}gc_{u_{1}}b_{s_{1}}$.
Similarly, $\varphi_{T_{3}+T_{4}}(\Gamma k)=\Gamma kb_{-s_{1}}c_{-u_{1}}$ and
$ka_{T_{3}+T_{4}}=\gamma_{3}la_{T_{4}}=\gamma_{3}\gamma_{4}g=\gamma_{3}\gamma_{4}kb_{-s_{1}}c_{-u_{1}}$.
By the Remark 2.1, the orbit of $\tilde{x}$ corresponds to the conjugacy class
$\\{\gamma_{1}\gamma_{2}\\}_{\Gamma}=\\{\gamma_{2}\gamma_{1}\\}_{\Gamma}$ and
the orbit of $\tilde{z}$ corresponds to the class
$\\{\gamma_{3}\gamma_{4}\\}_{\Gamma}=\\{\gamma_{4}\gamma_{3}\\}_{\Gamma}$. The
partner orbit corresponds to the conjugacy class

$\\{\gamma_{2}\gamma_{1}\gamma_{4}\gamma_{3}\\}_{\Gamma};$

recall the last assertion of Lemma 2.6. ∎

###  4.3 Periodic orbits including one 2-antiparallel encounter and one
2-parallel encounter intertwined

This subsection deals with periodic orbits with one 2-antiparallel encounter
and one 2-parallel encounter intertwined, which is so called antiparallel-
parallel intertwined (api for short) in [19].

Figure 6: Reconnecting encounter stretches to form a partner orbit in api
case.

######  Theorem 4.5.

Let $\varepsilon\in(0,\frac{\sigma_{0}}{20})$. Suppose that a periodic orbit
$c$ of the flow $(\varphi_{t})_{t\in\mathbb{R}}$ on $X$ with period $T>1$ has
one $(2,\varepsilon)$-parallel encounter and one
$(2,\varepsilon)$-antiparallel encounter intertwined. More precisely, suppose
that there are $x,y,z,w\in c$ and $T_{1},T_{2},T_{3},T_{4}>0$ be such that
$T_{1}+T_{2}+T_{3}+T_{4}=T$,
$\varphi_{T_{1}}(x)=y,\varphi_{T_{2}}(y)=z,\varphi_{T_{3}}(z)=w$ and
$\varphi_{T_{4}}(w)=x$ and
$z=(u_{1},s_{1})_{x}\in{\mathcal{P}}_{\varepsilon}(x),{\mathcal{T}}(w)=(u_{2},s_{2})_{y}\in{\mathcal{P}}_{\varepsilon}(y)$
and

$\displaystyle|u_{1}|>9\varepsilon
e^{-T_{3}},\quad|s_{1}|>72\varepsilon^{3}+5\varepsilon e^{-T_{4}}+2\varepsilon
e^{-T_{2}}.$ (4.36)

Then $c$ has a $19\varepsilon$-partner orbit of period $T^{\prime}$ which
differs in both encounters and the action difference satisfies

$\Big{|}\frac{T^{\prime}-T}{2}-\ln(1+s_{1}u_{1})(1+s_{2}u_{2})|\leq
54\varepsilon^{4}+25\varepsilon^{2}(e^{-T_{1}}+e^{-T_{3}}+e^{-T_{4}}).$

If $\varepsilon\in(0,\frac{\varepsilon_{*}}{38})$, then the partner orbit is
unique.

###### Proof.

The proof of this theorem is similar to that of Theorem 4.3. Note that after
obtaining two shorter periodic orbits, the dashed and dotted lines, we
consider the time reversal of the former one (Figure 6 (c)), and apply the
connecting lemma to have a partner depicted by dashed line in Figure 6 (d). ∎

######  Remark 4.4.

Analogously to Theorem 4.4, in the setting of the previous theorem, if the
original orbit corresponds to the conjugacy class
$\\{\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}\\}_{\Gamma}$, then the partner
orbit corresponds to the conjugacy class

$\\{(\gamma_{2}\gamma_{1})^{-1}\gamma_{4}\gamma_{3}\\}_{\Gamma}=\\{\gamma_{1}^{-1}\gamma_{2}^{-1}\gamma_{4}\gamma_{3}\\}_{\Gamma}.$

######  Remark 4.5.

(a) The assumption of compactness is unnecessary for the existence of partner
orbits in all cases above. However, we need it for the uniqueness.

(b) The conditions expressed by the coordinates of the piercing points (4.1),
(4.24), (4.36) guarantee that the encounter stretches are separated by non-
vanishing loops and whence the partner orbit and the original orbit do not
coincide. A similar condition is needed in physics literature; see [16, 19].

(c) The approach in the present paper can be applied to consider periodic
orbits responsible for all order in $\tau$ to the spectral form factor
$K(\tau)$.

## References

  * [1] T. Bedford, M. Keane, and C. Series (Eds.),  Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford University Press, Oxford 1991.
  * [2] M.V. Berry, “Semiclassical theory of spectral rigidity”, Proc. Roy. Soc. London Ser. A 400, 229-251 (1985).
  * [3] R. Bowen, “Symbolic dynamics for hyperbolic flows”, Amer. J. Math. 95(2) (1973), 429-460.
  * [4] P. Braun, S. Heusler, S. Müller, and F. Haake, “Statistics of self-crossings and avoided crossings of periodic orbits in the Hadamard-Gutzwiller model”, Eur. Phys. J. B 30, 189-206 (2002).
  * [5] K. Efetov, Supersymmetry in Disorder and Chaos, Cambridge University Press, Cambridge 1997.
  * [6] M. Einsiedler and T. Ward, Ergodic Theory With a View Towards Number Theory, Springer, Berlin-New York 2011
  * [7] B. Gutkin and V.A. Osipov, “Clustering of periodic orbits in chaotic systems”, Nonlinearity 2, 177-199 (2012).
  * [8] F. Haake, Quantum Signatures of Chaos, 3rd edition, Springer, Berlin-New York 2010.
  * [9] J.H. Hannay and A.M. Ozorio de Almeida, “Periodic orbits and a correlation function for the semiclassical density of state”, J. Phys. A: Math. Gen. 17, 3429-3440 (1984).
  * [10] H. Heusler, S. Müller, A. Altland, P. Braun, and F. Haake, “Periodic orbit theory of level correlations”, Phys. Rev. Lett. 98, 044103 (2007).
  * [11] H. Heusler, S. Müller, P. Braun, and F. Haake, “Universal spectral form factor for chaotic dynamics”, J. Phys. A 37, L31-L37 (2004).
  * [12] H.M. Huynh and M. Kunze, “Partner orbits and action differences on compact factors of the hyperbolic plane. I: Sieber-Richter pairs”, Nonlinearity 28, 593-623 (2015).
  * [13] H.M. Huynh, “Partner orbits and action differences on compact factors of the hyperbolic plane. II: Higher-order encounters”, Physica D. 314, 35-53 (2016).
  * [14] H.M. Huynh, “On 2-antiparallel encounters on factors of the hyperbolic plane”, Quy Nhon J. Sci. 11(1), 5-15 (2017).
  * [15] J.P. Keating and J.M. Robbins, “Discrete symmetries and spectral statistics”, J. Phys. A: Math. Gen. 30, L177-L181 (1997)
  * [16] S. Müller, Periodic-Orbit Approach to Universality in Quantum Chaos, PhD thesis, Universität Duisburg-Essen 2005.
  * [17] S. Müller, S. Heusler, A. Altland, P. Braun, and F. Haake, “Periodic-orbit theory of universal level correlations in quantum chaos”, New J. Phys. 11, 103025 (2009).
  * [18] S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, “Semiclassical foundation of universality in quantum chaos”, Phys. Rev. Lett. 93, 014103 (2004).
  * [19] S. Müller, S. Heusler, P. Braun, F. Haake, and A. Altland, “Periodic-orbit theory of universality in quantum chaos”, Phys. Rev. E 72, 046207 (2005).
  * [20] J. Ratcliff, Foundations of Hyperbolic Manifolds, 2nd edition, Springer, Berlin-Heidelberg-New York, 2006.
  * [21] M. Sieber, “Semiclassical approach to spectral correlation functions”, in Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology, Eds. Bolte J. & Steiner F., London Math. Soc. LNS 397, Cambridge University Press, Cambridge-New York 2012, pp. 121-142.
  * [22] M. Sieber and K. Richter, “Correlations between periodic orbits and their rôle in spectral statistics”, Physica Scripta T90, 128-133 (2001).
  * [23] M. Turek, D. Spehner, S. Müller, and K. Richter, “Semiclassical form factor for spectral and matrix element fluctuation of multidimensional chaotic systems”, Phys. Rev. E 71, 1-25 (2005).