11institutetext: Anna Yesypenko 22institutetext: Oden Institute, 22email:
<EMAIL_ADDRESS>33institutetext: Per-Gunnar Martinsson 44institutetext:
Oden Institute 44email<EMAIL_ADDRESS>
# Parallel Optimizations for the Hierarchical Poincaré-Steklov Scheme (HPS)

Anna Yesypenko and Per-Gunnar Martinsson

###### Abstract

Parallel optimizations for the 2D Hierarchical Poincaré-Steklov (HPS)
discretization scheme are described. HPS is a multi-domain spectral
collocation scheme that allows for combining very high order discretizations
with direct solvers, making the discretization powerful in resolving highly
oscillatory solutions to high accuracy. HPS can be viewed as a domain
decomposition scheme where the domains are connected directly through the use
of a sparse direct solver. This manuscript describes optimizations of HPS that
are simple to implement, and that leverage batched linear algebra on modern
hybrid architectures to improve the practical speed of the solver. In
particular, the manuscript demonstrates that the traditionally high cost of
performing local static condensation for discretizations involving very high
local order $p$ can be reduced dramatically.

##  1 Introduction

We describe methods for solving boundary value problems of the form

$\left\\{\begin{aligned} \mathcal{A}u(x)=&\ f(x),\qquad&x\in\Omega,\\\ u(x)=&\
g(x),\qquad&x\in\Gamma,\end{aligned}\right.$ (1)

where $\mathcal{A}$ is a second order elliptic differential operator, and
$\Omega$ is a rectangular domain in two dimensions with boundary $\Gamma$. For
the sake of concreteness, we will focus on the case where $\mathcal{A}$ is a
variable coefficient Helmholtz operator

$\mathcal{A}u(x)=-\Delta u(x)-\kappa^{2}b(x)u(x),$ (2)

where $\kappa$ is a reference wavenumber, and where $b(x)$ is a smooth non-
negative function that typically satisfies $0\leq b(x)\leq 1$. Upon
discretizing (1), one obtains a linear system

$\bm{\mathsf{A}}\bm{\mathsf{u}}=\bm{\mathsf{f}}$ (3)

involving a sparse coefficient matrix $\bm{\mathsf{A}}\in\mathbb{R}^{N\times
N}$. The focus of this work is on efficiently solving the sparse system (3)
for the Hierarchical Poincaré-Steklov (HPS) discretization. HPS is a multi-
domain spectral collocation scheme that allows for relatively high choices of
$p$, while interfacing well with sparse direct solvers. For (1) discretized
with HPS with local polynomial order $p$, the cost of factorizing
$\bm{\mathsf{A}}$ directly is

$T_{\rm build}=O\left(\underset{\rm leaf\ operations}{p^{4}N}+\underset{\rm
direct\ solver}{N^{3/2}}\right).$ (4)

After the leaf operations are complete, the cost to factorize the system
directly has no pre-factor dependence on $p$. The pre-factor cost of the leaf
operations, however, has long been viewed as prohibitively expensive. This
manuscript describes simple parallel optimizations using batched linear
algebra that substantially accelerate the leaf operations and shows compelling
results for $p$ up to 42. We also demonstrate that the choice of $p$ does not
have substantial effects on the build time for the direct factorization stage,
allowing $p$ to be chosen based on physical considerations instead of
practical concerns.

High order discretization is crucial in resolving variable-coefficient
scattering phenomena due to the well known “pollution effect” that generally
requires the number of points per wavelength to increase, the larger the
computational domain is. The pollution effect is very strong for low order
discretizations, but quickly gets less problematic as the discretization order
increases beriot2016efficient ; deraemaeker1999dispersion . HPS is less
sensitive to pollution because the scheme allows for high choices of local
polynomial order $p$ gillman2014direct ; martinsson2013direct . Combining HPS
discretization with efficient sparse direct solvers provides a powerful tool
for resolving challenging scattering phenomena to high accuracy, especially
for situations where no efficient preconditioners are known to exist (e.g.
trapped rays, multiple reflections, backscattering) ernst2012difficult .

##  2 HPS Discretization and Interfacing with Sparse Direct Solvers

We next discuss the HPS discretization and efficient methods to interface the
resulting sparse linear system with direct solvers. We introduce the HPS
briefly for the simple model problem (1), and refer the reader to
2013_martinsson_ItI ; 2019_martinsson_book ; hao2016direct  for details and
extensions. An important limitation of the discretization is that we assume
the solution is smooth and that the coefficients in the operator $\mathcal{A}$
of (1) are smooth as well.

The domain $\Omega$ is partitioned into non-overlapping subdomains. The
discretization is described by two parameters, $a$ and $p$, which are the
element size and local polynomial order, respectively. On each subdomain, we
place a $p\times p$ tensor product mesh of Chebyshev points. Internal to each
subdomain, the PDE is enforced locally via spectral differentiation and direct
collocation. On element boundaries, we enforce that the flux between adjacent
boundaries is continuous. On each subdomain of $p^{2}$ nodes, the spectral
differentiation operators lead to a dense matrix of interactions of size
$p^{2}\times p^{2}$. To improve efficiency of sparse direct solvers for HPS
discretizations, we “eliminate” the dense interactions of nodes interior to
each subdomain. This process is referred to as “static condensation”
cockburn2016static ; guyan1965reduction . The remaining active nodes are on
the boundaries between subdomains. As a result of the leaf elimination, we
produce a smaller system $\tilde{\bm{\mathsf{A}}}$ of size $\approx N/p$ with
equivalent body load $\tilde{\bm{\mathsf{f}}}$ on the active nodes

$\tilde{\bm{\mathsf{A}}}\tilde{\bm{\mathsf{u}}}=\tilde{\bm{\mathsf{f}}}.$ (5)

See Figure 1 for the full grid of points and the remaining active nodes after
static condensation.

$\Rightarrow$ Figure 1: Prior to interfacing with sparse direct solvers, we do
static condensation to produce an equivalent system (5) to solve on the
remaining active nodes. The original grid has $N$ points, and remaining grid
has $\approx N/p$ points.

The leaf elimination requires independent dense linear algebraic operations on
$N/{p^{2}}$ systems each of size $p^{2}\times p^{2}$, resulting in overall
cost $O(p^{4}N)$ to produce the equivalent system (5). For $p$ up to 42, these
operations can be efficiently parallelized with batched linear algebra (BLAS).
For larger $p$, methods that produce an sparser equivalent system are more
fitting brubeck2021scalable ; 2021_fortunato_ultraspherical . Traditionally,
it may be challenging to achieve high arithmetic intensity for many small
parallel tasks due to overhead costs. Batched BLAS is a highly optimized
software for parallel operations on matrices that are small enough to fit in
the top levels of the memory hierarchy (i.e. smaller than the L2 cache size)
dongarra2017design . The framework operates by grouping small inputs into
larger “batches” and can achieve particularly good performance on high-
throughput architectures, such as GPUs.

We combined the fast leaf factorization procedure with two methods for solving
the reduced system (5). The first option for solving (5) uses a black-box
sparse direct solver with the nested dissection (ND) ordering. ND is a based
on a multi-level graph partitioning of nodes and produces a sparse
factorization with minimal fill-in 2006_davis_directsolverbook ;
2016_acta_sparse_direct_survey . In 2D, sparse factorization using the ND
ordering requires $O\left(N^{3/2}\right)$ time to build and $O\left(N\log
N\right)$ time to apply.

As a second option for solving (5), we used a scheme we refer to as SlabLU,
which is a simplified two-level scheme (as opposed to standard hierarchical
schemes) that is designed for ease of parallelization yesypenko2022slablu . To
be precise, SlabLU uses a decomposition of the domain into elongated “slab”
subdomains. The nodes internal to each slab are eliminated in parallel, and
the resulting block-tridiagonal system on slab interfaces is solved directly.
The resulting two-level scheme has complexity $O(N^{5/3})$ to factorize
$\tilde{\bm{\mathsf{A}}}$ directly and $O\left(N^{7/6}\right)$ complexity to
apply the computed factors to solve (5). SlabLU is simple scheme that
leverages high concurrency and batched BLAS to achieve high performance on
modern hybrid architectures. Despite the asymptotically higher costs, SlabLU
performs favorably compared to multi-level nested dissection schemes in its
build time and memory footprint, as we demonstrate in Section 3.

###### Remark 1

Since the leaf computations are very efficient, we saved memory and reduced
communication by not explicitly storing the factorizations of the local
spectral differentiation matrices. Instead, these are reformed and refactored
after each solve involving the reduced system (5).

##  3 Numerical Experiments

We next demonstrate the effectiveness of HPS, combined with efficient sparse
direct solvers, in solving high frequency Helmholtz equations. The experiments
are run on a desktop with a 16-core Intel i9-12900k CPU with 128GB of memory
and a NVIDIA RTX 3090 GPU with 24GB of memory.

Figure 2: Leaf Operations for HPS require $O(p^{4}N)$ operations, though the
practical scaling for parallel operations has a small constant prefactor for
$p$ up to 42. Parallel HPS leaf operations are further accelerated on GPUs,
with a speed-up of at least 4x.

We demonstrate that simple parallel optimizations allow us to efficiently do
leaf operations for various local polynomial order $p$; see Figure 2. After
the leaf operations, we directly factorize the reduced system (5) using
efficient sparse direct solvers and demonstrate that the choice of $p$ does
not have much effect on the time to factorize $\tilde{\bm{\mathsf{A}}}$.
Having freedom over the choice of $p$ allows the user to resolve highly
oscillatory PDEs to high order accuracy, without concern for how the choice
may affect the cost of solving (3) directly.

To demonstrate the effectiveness of the HPS discretization resolving
oscillatory solutions to high accuracy, we report results for a PDE with a
known analytic solution

$\left\\{\begin{aligned} -\Delta u(x)-\kappa^{2}u(x)=&\
0,\qquad&x\in\Omega,\\\ u(x)=&\
u_{\text{true}}(x),\qquad&x\in\Gamma,\end{aligned}\right.$ (6)

where the true solution $u_{\text{true}}$ is given by
$u_{\text{true}}=J_{0}\left(\kappa|x-(-0.1,0.5)|\right)$, where $x\mapsto
J_{0}(\kappa|x|)$ is the free-space fundamental solution to the Helmholtz
equation. In the experiments, we discretize (6) using HPS for various choices
of $p$ and set the wavenumber $\kappa$ to increase with $N$ to maintain 10
points per wavelength with increasing problem size.

After applying a direct solver to solve (5) on the reduced HPS grid, we re-
factorize the linear systems on interior leaf nodes to calculate the solution
$\bm{\mathsf{u}}_{\rm calc}$ on the full HPS grid. The leaf solve requires
time $O(p^{4}N)$, but is particularly efficient using the parallel
optimizations described. The reported build times and solve times include the
leaf operations, which are done with batched BLAS on a GPU. We report relative
error with respect to the residual of the discretized system (3). When a true
solution is known, we also report relative error with respect to the true
solution $\bm{\mathsf{u}}_{\rm true}$ evaluated on the collocation points of
the full HPS grid

$\text{relerr}_{\text{res}}=\frac{{\|\bm{\mathsf{A}}\bm{\mathsf{u}}_{\text{calc}}-\bm{\mathsf{f}}\|}_{2}}{{\|\bm{\mathsf{f}}\|}_{2}},\qquad\text{relerr}_{\rm
true}=\frac{{\|\bm{\mathsf{u}}_{\rm calc}-\bm{\mathsf{u}}_{\rm
true}\|}_{2}}{{\|\bm{\mathsf{u}}_{\rm true}\|}_{2}}.$ (7)

###  3.1 Comparison of Sparse Direct Solvers

The system (5) is solved using two different sparse direct solvers, SuperLU
and SlabLU. SuperLU is a black-box solver that finds an appropriate ordering
of the system that minimizes fill-in while increasing concurrency by grouping
nodes into “super-nodes” 2011_li_supernodal . SuperLU is accessed through the
Scipy interface (version 1.8.1) and called with the COLAMD ordering. This
version of Scipy uses the CPU only. Not many GPU-aware sparse direct solvers
are widely available, though this is an active area of research. SuperLU also
uses a pivoting scheme that can exchange rows between supernodes to attain
almost machine precision accuracy in the residual of the computed solutions.
SlabLU uses an ordering based on a decomposition of the domain into “slabs”
that has a limited pivoting scheme. Despite this limitation, SlabLU can
achieve 10 digits of accuracy in the residual, which gives also gives high
order true relative accuracy, depending on the choice of $p$.

SlabLU is a simple two-level framework that achieves large speedups over
SuperLU by leveraging batched BLAS and GPU optimizations. See Figure 3 for a
comparison between SuperLU and SlabLU in factorizing $\tilde{\bm{\mathsf{A}}}$
to solve (5). See Figure 4 for a comparison of accuracies in the computed
solutions for various $p$. For PDEs on rectangular domains, SlabLU, compared
to SuperLU, is able to solve larger sparse systems (5) with a smaller memory
footprint. We demonstrate the ability of HPS for various $p$, combined with
SlabLU, to solve Helmholtz problems of size up to $900\lambda\times
900\lambda$ (for which $N$=81M) to high order accuracy. See Figure 5 which
reports build and solve times for various choices of $p$. See Figure 6 which
reports how the choice of $p$ affects the accuracy of the computed solutions.

Figure 3: Build time and memory footprint comparison between SuperLU and
SlabLU for (6) discretized with HPS ($p$=22), where $\kappa$ is increased to
maintain 10 points per wavelength. For $N$=5.06M, the SlabLU build time is
faster by a factor of 16x and the memory footprint is less by a factor of 14x.

Figure 4: Accuracy comparison between SuperLU and SlabLU for (6) discretized
with HPS ($p$=22,42), where $\kappa$ is increased to maintain 10 points per
wavelength. SuperLU uses a more sophisticated pivoting scheme to achieve high
accuracy in the residual error. For $p$=22, SlabLU and SuperLU both resolve
the solution to 6 digits in the true relative accuracy. For $p$=42, SuperLU is
able to achieve 10 digits in the true relative accuracy, while SlabLU achieves
8 digits.

Figure 5: Build time and Solve time for HPS with various $p$ for (6) where
$\kappa$ is increased with $N$ to maintain 10 points per wavelength. The
choice of $p$ does not substantially affect the time needed to factorize the
sparse linear system with SlabLU. As $p$ increases, the memory footprint
required to store the factorization decreases and the solve time increases.
Figure 6: Solution accuracy for HPS with various $p$ for (6) where $\kappa$ is
increased with $N$ to maintain 10 points per wavelength. Regardless of the
choice of $p$, slabLU resolves the solution to at least 10 digits of relative
accuracy in the residual. With increasing $p$, one can calculate solutions
with higher relative accuracy, compared to the true solution of the PDE.

###  3.2 Variable-Coefficient Scattering Problems

We next demonstrate the ability of HPS, combined with SlabLU as a sparse
direct solver, to solve complicated scattering phenomena. To be precise, we
solve BVP (1) with the variable-coefficient Helmholtz operator (2) and
discretize using HPS with $p$=32. The problems physically correspond to an
antenna releasing an electromagnetic signal in a closed room. Due to the
boundary conditions, multiple reflections occur because the wave cannot escape
the closed domain. The Dirichlet data $u_{\rm dir}$ for the experiments in
this section is given by a Gaussian pulse on the left boundary of the domain

$u_{\rm dir}(x)=\begin{cases}\
\text{exp}\left(-2000(x_{1}-0.5)^{2}\right),&\text{when}\ x_{0}=0\\\
0,&\text{otherwise}.\end{cases}$ (8)

See Figure 7 for the computed solution with smooth variable coefficient media
$b$, which mimics the presence of a crystal that blocks the propagating pulse.

Figure 7: Solutions for Dirichlet data $u_{\rm dir}$ with smooth scattering
field $b$ discretized using HPS with $p$=32 on a grid of $N$=1.2M points for
various wavenumbers $\kappa$.

##  4 Conclusions

HPS is a high-order convergent discretization scheme that interfaces well with
sparse direct solvers. This manuscript describes parallel optimizations of the
scheme that allow for the rapid and memory efficient direct solution of (3) of
the resulting linear systems. First, we perform the leaf operations in
parallel using batched BLAS, then we factorize a smaller system (5) of size
$\approx N/p$ using sparse direct solvers, where $p$ denotes the local order
of convergence (which we show can be chosen as high as $40$). The numerical
results feature comparisons between sparse direct solvers and demonstrate that
SlabLU can factorize systems corresponding to domains of size up to
$900\lambda\times 900\lambda$ (for which $N$=81M) in less than 20 minutes. The
approach is effective in resolving challenging scattering problems to high
accuracy.

Acknowledgments The work reported was supported by the Office of Naval
Research (N00014-18-1-2354), by the National Science Foundation (DMS-1952735
and DMS-2012606), and by the Department of Energy ASCR (DE-SC0022251).

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# Counting geodesic loops on surfaces of genus at least $2$ without conjugate
points

Mark Pollicott and Khadim War

###### Abstract.

In this paper we prove asymptotic estimates for closed geodesic loops on
compact surfaces with no conjugate points. These generalize the classical
counting results of Huber and Margulis and sector theorems  for surfaces of
strictly negative curvature. We will also prove more general sector theorems,
generalizing results of Nicholls and Sharp for special case of surfaces of
strictly negative curvature.

###### 2020 Mathematics Subject Classification:

37C35, 37D40, 53C22

##  1\. Introduction

For a closed surface $M$ of negative curvature there are classical results
which count the number of geodesic arcs starting and ending at a given
reference point $p\in M$ and whose length at most $t$, say. For constant
curvature surfaces these were proved by Huber in 1959, and for variable
curvature surfaces these were proved by Margulis in 1969, In particular, they
give simple asymptotic estimates for this counting function as $t\to+\infty$.
In this brief note we will extend these results in Corollary 1.6 to the more
general setting of surfaces without conjugate points.

There are refinements of the original counting results of Huber and Margulis
whereby the geodesics are restricted to lie in a sector. These are due to
shown for constant curvature surfaces by Nicholls in 1983, and for variable
curvature surfaces by Sharp in $2001$. We will describe generalizations of
these results to surfaces without conjugate points in Corollaries 1.5 and 1.6.
These will follow from a more general statement (Theorem 1.3) which appears
below.

We begin with some general notation. Let $(M,g)$ be a closed Riemannian
manifold, $SM$ the unit tangent bundle of $M$ and let $\pi:SM\to M$ be the
natural projection to the footpoint.

$\theta$$\theta$$v_{0}$$J(v_{0},\theta)$$J(v_{0}^{\prime},\theta^{\prime})$$v_{0}^{\prime}$$\theta^{\prime}$$\theta^{\prime}$$c$$\dot{c}(0)$$\dot{c}(\tau)$$p=c(0)=c(\tau)$
Figure 1. A geodesic loop $c$ which starts within an angle $\theta$ of $v_{0}$
and ends within an angle $\theta^{\prime}$ of $v_{0}^{\prime}$

Let $t,\theta,\theta^{\prime}>0$ and $v_{0},v_{0}^{\prime}\in SM$ with $\pi
v_{0}=\pi v_{0}^{\prime}=p$, say. We want to count geodesic loops
$c:[0,\tau]\to M$ which:

  1. (1)

start and finish at $p$ (i.e., $c(0)=c(\tau)=p$);

  2. (2)

have length $\tau$ less than $t$;

  3. (3)

leaves the fibre $S_{p}M$ at an angle at most $\theta$ to $v_{0}$; and

  4. (4)

enters the fibre $S_{p}M$ at an angle at most $\theta^{\prime}$ to
$v_{0}^{\prime}$.

(see Figure 1).

######  Definition 1.1.

Given an angle $0<\theta\leq\pi$ and a unit tangent vector $v_{0}\in SM$, we
define the following arc in the fibre $S_{\pi v_{0}}M$:

$J(v_{0},\theta):=\\{w\in S_{p}M:\measuredangle_{p}(v_{0},w)\leq\theta\\},$

i.e., the unit tangent vectors $w$ in the same fibre as $v_{0}$ at an angle at
most $\theta$.

This allows us to introduce convenient notation for the collection of geodesic
arcs satisfying properties (1)–(4).

######  Definition 1.2.

We let $\mathcal{C}(t,J(v_{0},\theta),J(v_{0}^{\prime},\theta^{\prime}))$
denote the set of geodesic loops $c:[0,\tau]\to M$ based at $c(0)=c(\tau)=p\in
M$ of length $\tau\leq t$ and satisfying $c^{\prime}(0)\in J(v_{0},\theta)$
and $c^{\prime}(\tau)\in J(v_{0}^{\prime},\theta^{\prime})$.

We will now consider the problem of counting geodesic the number

$\\#\mathcal{C}(t,J(v,\theta),J(v^{\prime},\theta^{\prime}))$

of such geodesic arcs.

We will work in the general setting of closed surfaces $M$ of genus at least
$2$ that have no conjugate points, i.e., for any two points $p,q\in M$ there
is no geodesic from $p$ to $q$ along which there is a non-trivial Jacobi field
vanishing at $p$ and $q$. 111Recall that a Jacobi vector field is a vector
field $J(t)$ on the geodesic $c(t)$ satisfying
$\frac{D^{2}}{dt^{2}}J(t)=R(J(t)),\dot{\gamma}(t),\dot{\gamma}(t)=0$, where
$D$ denotes the covariant derivative with respect to the Levi-Civita
connection, $R$ the Riemann curvature tensor $\dot{\gamma}(t)=d\gamma(t)/dt$
the tangent vector field. By the Cartan-Hadamard theorem, an equivalent
formulation is that there is a unique geodesic arc joining distinct points in
the universal cover $\widetilde{M}$. Examples include the special case that
$M$ has non-positive curvature. We refer to [1] for another well known
example.

Finally, using the following notation

$S^{2}M:=\\{(v,v^{\prime})\in SM\times SM:\pi v=\pi v^{\prime}\\}$

we can formulate our main result.

######  Theorem 1.3.

Let $M$ be a closed connected surface of genus at least $2$ without conjugate
points. Then there exists $\theta_{0}>0$, $h>0$ and a measurable positive
function $a:S^{2}M\times(0,\theta_{0})^{2}\to\mathbb{R}_{>0}$ such that

(1) $\\#\mathcal{C}(t,J(v,\theta),J(v^{\prime},\theta^{\prime}))\sim
a(v,v^{\prime},\theta,\theta^{\prime})e^{ht},\hbox{ as }t\to+\infty$

i.e.,
$\lim_{t\to+\infty}\frac{\\#\mathcal{C}(t,J(v,\theta),J(v^{\prime},\theta^{\prime}))}{a(v,v^{\prime},\theta,\theta^{\prime})e^{ht}}=1$.

Moreover if the geodesic flow is expansive 222A flow $\phi_{t}:SM\to SM$ is
_expansive_ if for all $\delta>0$ there exists $\epsilon>0$ such that if
$d(\phi_{t}(x),\phi_{s(t)}(y))<\delta$ for all $t\in\mathbb{R}$ for $x,y\in
SM$ and a continuous map $s:\mathbb{R}\to\mathbb{R}$ then $y=\phi_{t}(x)$
where $|t|<\epsilon$.  then the function $a(\cdot,\cdot,\cdot,\cdot)$ is
continuous.

In the statement of the theorem the value $h$ is the topological entropy of
the geodesic flow on the unit tangent bundle $SM$.

######  Remark 1.4.

In the special case that $M$ has constant curvature then $a(\cdot)$ is a
constant function, and when $M$ has variable negative curvature it is known
that $a(\cdot)$ is a continuous function (not least because it is expansive).

Theorem 1.3 has corollaries which extend several classical results from the
context of negative curvature. In particular, this leads to generalizations of
classical counting and sector theorems. For example, when we set
$\theta^{\prime}=\pi$ then this gives the following.

######  Corollary 1.5 (Sector Theorem).

Given $0<\theta\leq\pi$ there exists $a=a(p,\theta)>0$ such that the number of
geodesic arcs which: start at $p\in M$ and finish at $q\in M$; leave $S_{p}M$
at an angle at most $\theta$ to $v_{0}$; and have length at most $t$, is
asymptotic to $ae^{ht}$ as $t\to+\infty$.

This generalizes results from [8], [9], [10].

Furthermore, when $\theta=\theta^{\prime}=\pi$ then this further reduces to
the original counting result:

######  Corollary 1.6 (Arc counting).

There exists $a=a(p)>0$ such that the number of geodesic arcs which start at
$p\in M$, finish at $q\in M$ and have length at most $t$ is asymptotic to
$ae^{ht}$ as $t\to+\infty$.

This generalizes results from [5], [6], [7].

Finally, we can describe equidistribution result of a slightly different
flavour. Let $\widehat{M}$ be a finite cover for $M$. We can associate to any
geodesic arc $c$ on $M$ which starts and ends at $p\in M$ (and has length
$L_{c}$) a lift $\widehat{c}$ to $\widehat{M}$. The following corollary
estimates the proportion of geodesic arcs such that $\widehat{c}$ on
$\widehat{M}$ with $\widehat{c}(0)=\widehat{c}(L_{c})$

######  Corollary 1.7 (Equidistribution in finite covers).

The proportion of geodesic arcs $c$ which start and end at $p\in M$, have
lifts $\widehat{c}$ which start and end at the same point in $\widehat{M}$,
and have length at most $t$ is asymptotic to

$\frac{\hbox{\rm Area}(M)}{\hbox{\rm Area}(\widehat{M})}ae^{ht}\hbox{ as
$t\to+\infty$ }$

This corollary can be used to prove a corollary related to the first Homology
group $H_{1}(M,\mathbb{Z})$. Each closed loop $c$ based at $p$ gives rise
naturally to an element $\langle c\rangle\in H_{1}(M,\mathbb{Z})$. Let us
consider a finite index subgroup $G<H_{1}(M,\mathbb{Z})$ then for a geodesic
arc $c$ we can associate the coset $\langle c\rangle G\in
H_{1}(M,\mathbb{Z})G$.

######  Corollary 1.8 (Homological Equidistribution).

For a fixed coset $\alpha\in H_{1}(M,\mathbb{Z})/G$. The proportion of
geodesic arcs $c$ which start and finish at $p\in M$, satisfy $\langle
c\rangle\Gamma=\alpha$ and have length at most $t$ is asymptotic to

$\\#(H_{1}(M,\mathbb{Z})/\Gamma)ae^{ht}\hbox{ as }t\to+\infty$

######  Remark 1.9.

The theorem and each of the corollaries has a natural equivalent formulation
in terms of the action $\Gamma\times X\to X$ of the covering group
$\Gamma=\pi_{1}(M)$ on the universal cover $X$. For example, Corollary 1.6
gives an asymptotic estimate for $\\#\\{g\in\Gamma\hbox{ :
}d_{X}(\overline{p},g\overline{p})\leq t\\}$ where $\overline{p}\in X$ and
$d_{X}$ is the lifted Riemannian metric to $X$.

##  2\. Closed arcs and isometries

The structure of the proof of Theorem 1.3 follows the lines of Margulis’
original proof. However, it requires modifications using a number of recent
techniques from [2], [3]. A key ingredient is the construction of the measure
of maximal entropy for the geodesic flow $\phi_{t}:SM\to SM$.

###  2.1. Some Notation

Let $X$ be the universal cover of $M$ with the lifted metric. The covering
group $\Gamma\cong\pi_{1}(M)$ satisfies that $M=X/\Gamma$.

Let $SX$ denote the unit tangent bundle for $X$ and let $\overline{\pi}:SX\to
X$ denote the canonical projection of a unit tangent vector in $SX$ to its
footpoint in $X$. Let $\overline{p}\in X$ be a lift of $p\in M$ and let
$\underline{B}(\overline{p},R)\subset X$ denote a ball of radius $R>0$ about
$\overline{p}$. We can use this to give a convenient definition of topological
entropy [4].

######  Definition 2.1.

The _topological entropy_ $h=h(\phi)$ is given by

$h=\lim_{R\to+\infty}\frac{\log\hbox{\rm
Vol}(\underline{B}(\overline{p},R))}{R}.$

Given $\overline{v}\in SX$, let $c=c_{\overline{v}}:\mathbb{R}\to X$ denote
the unique geodesic such that
$c_{\overline{v}}(0)=\overline{\pi}(\overline{v})$ and
$c^{\prime}_{\overline{v}}(0)=\overline{v}$.

######  Definition 2.2.

Let $\partial X$ denote the ideal boundary of $X$ consisting of equivalence
classes $[c]$ of geodesics $c:\mathbb{R}\to X$ which stay a bounded distance
apart.

(See [2, Section 2] for a detailed description of the construction and
properties of $\partial X$). In particular, every geodesic
$c_{\overline{v}}:\mathbb{R}\to X$ defines two points $c(\pm\infty)\in\partial
X$, which it is convenient to denote $\overline{v}^{-}:=c(\\-\infty)$ and
$\overline{v}^{+}:=c(+\infty)$. The natural acton $\Gamma\times X\to X$
extends to an action of $\Gamma$ on $\partial X$ given by $g[c]=[gc]$, where
$g\in\Gamma$.

######  Definition 2.3.

Given $\overline{p}\in X$, the _Busemann function_
$b_{\overline{p}}(\cdot,\cdot):X\times\partial X\to\mathbb{R}$ is defined by

$b_{\overline{p}}(\overline{q},\xi)=\lim_{t\to+\infty}d(\overline{q},c_{v}(t))-t$

for $\overline{v}\in S_{\overline{p}}X$ satisfying
$\xi=c_{\overline{v}}(+\infty)$ [2, Definition 2.16].

We next recall the characterization of Patterson-Sullivan measures on the
boundary $\partial X$ constructed in [2, Proposition 5.1].

######  Definition 2.4.

The _Patterson-Sullivan measures_ on $\partial X$ are a family of measures
$\\{\mu_{\overline{p}}\hbox{ : }\overline{p}\in X\\}$ which transform under
the action of $\Gamma$ on $\partial X$ by

$\frac{d\mu_{{\overline{p}}}\gamma}{d\mu_{\overline{p}}}(\xi)=e^{-hb_{\overline{p}}(\gamma{\overline{p}},\xi)}$

for $\gamma\in\Gamma$ and $\xi\in\partial X$

The Busemann function is also used in defining horocycles.

######  Definition 2.5.

The _stable horocycle_ is defined by

$H_{\xi}({\overline{p}})=\\{\overline{q}\in X\hbox{ :
}b_{\overline{p}}(\overline{q},\xi)=0\\}$

and the _unstable horocycle_ is defined by

$H^{-}_{\xi}({\overline{p}})=\\{q\in X\hbox{ :
}b_{\overline{p}}(\overline{q},-\xi)=0\\}$

where $-\xi$ is the antipodal vector to $\xi$.

Finally, we define a class of tangent vectors which will serve us well in the
proof.

######  Definition 2.6.

We denote by $\mathcal{E}\subset SX$ the set of _expansive vectors_ consisting
of those unit tangent vectors whose stable and unstable horocycles intersect
at exactly one point.

$\xi$$q$$\overline{p}$$b_{\overline{p}}(\overline{q},\xi)$$H_{\xi}(\overline{p})$
$\xi$$\eta$$\overline{p}$$b_{\overline{p}}(\xi,\eta)$$H_{\xi}(\overline{p})$$H_{\eta}(\overline{p})$

Figure 2.  (i) The geometric interpretation of the Busemann function as the
signed distance of $\overline{q}$ from $H_{\xi}(\overline{p})$ corresponds to
$b_{p}(q,\xi)$; (ii) The distance between the horocycles
$H_{\xi}(\overline{p})$ and $H_{\eta}(\overline{p})$ corresponds to
$b_{\overline{p}}(\xi,\eta)$

###  2.2. The measure of maximal entropy

We begin with a correspondence which is useful in the construction of measures
of maximal entropy.

######  Definition 2.7.

The _Hopf map_ $H\colon SX\to\partial^{2}X\times\mathbb{R}$ is defined by

(2) $H({v}):=({\overline{v}}^{-},{\overline{v}}^{+},s({v}))\quad\text{ where
}\quad s({\overline{v}}):=b_{p}(\pi({\overline{v}},({\overline{v}}^{-}).$

In particular, following [2, Lemma 5.5] this family of measures defines a
$\Gamma$-invariant measure $\overline{\mu}$ on $\partial X\times\partial
X\setminus\text{diag}$ (where $\text{diag}\subset\partial X\times\partial X$
are the diagonal elements) characterized by

(3)
$d\overline{\mu}(\xi,\eta)=e^{h\beta_{\overline{p}}(\xi,\eta)}d\mu_{\overline{p}}(\xi)d\mu_{\overline{p}}(\eta),\hbox{
for }\xi,\eta\in\partial X,$

where $\beta_{\overline{p}}(\xi,\eta)$ is the distance in $X$ between the
horospheres $H_{\overline{p}}(\xi)$ and $H_{\overline{p}}(\eta)$, see Figure 2
(ii) (or [3, Figure 1]).

######  Definition 2.8.

The Hopf transform carries $d\overline{\mu}\times dt$ to a measure
$d\overline{m}:=H_{*}(d\overline{\mu}\times dt)$ on $SX$.

There is a natural projection from $SX$ to $SM$ (taking $v$ to $v\Gamma$). The
following result was proved in [2, Theorem 1.1].

######  Lemma 2.9.

The measure $\overline{m}$ on $SX$ projects (after normalization) to the
measure $\underline{m}$ maximal entropy for the geodesic flow on $SM$ (i.e.,
$\pi_{*}\overline{m}=\underline{m}$ and $h(\underline{m})=h$). Moreover,

  1. (1)

$\underline{m}$ is unique, strongly mixing 333$\underline{m}$ is even shown to
be Bernoulli and fully supported; and

  2. (2)

$\underline{m}(\mathcal{E})=1$ (cf. [2, Equation (2.10)]).

We now turn to the final ingredients in the proof.

###  2.3. Flow boxes

For the remainder of this section, we fix a choice of
$(v_{0},v_{0}^{\prime})\in S^{2}M\cap\mathcal{E}^{2}$. We can then associate
to the sets $J(v_{0},\theta),J(v_{0}^{\prime},\theta^{\prime})\subset SM$ in
Definition 1.2 a choice of lifts
$\overline{J}(v_{0},\theta),\overline{J}(v_{0}^{\prime},\theta^{\prime})\subset
SX$.

To proceed we want to consider the natural images of these sets in $\partial
X$:

######  Definition 2.10.

We can associate to $J(v_{0},\theta)$ and $J(v_{0}^{\prime},\theta^{\prime})$
their “future” and “past” subsets of $\partial X$ defined, respectively, by

$\mathbf{F}=\mathbf{F}_{\theta}:=\\{\bar{w}^{+}:\bar{w}\in\overline{J}(v_{0},\theta)\\}\text{
and
}\mathbf{P}=\mathbf{P}_{\theta}:=\\{\bar{w}^{-}:\bar{w}\in\overline{J}(v_{0},\theta)\\}$
$\mathbf{F}^{\prime}=\mathbf{F}_{\theta^{\prime}}:=\\{\bar{w}^{+}:\bar{w}\in\overline{J}(v_{0}^{\prime},\theta^{\prime})\\}\text{
and
}\mathbf{P}^{\prime}=\mathbf{P}_{\theta^{\prime}}:=\\{\bar{w}^{-}:\bar{w}\in\overline{J}(v_{0}^{\prime},\theta^{\prime})\\}.$

$v$$\mathbf{F}$$\mathbf{P}$$\theta$$\theta$$\widetilde{J}(v,\theta)$

Figure 3.  The sets $\mathbf{P}$ and $\mathbf{F}$ associated to
$\widetilde{J}(v,\theta)$

The sets
$\mathbf{F},\mathbf{P},\mathbf{F}^{\prime},\mathbf{P}^{\prime}\subset\partial
X$ will be used to construct flow boxes for the geodesic flow. Assume first
that $\epsilon>0$ is small (with respect to the injectivity radius of $M$) and
then choose $\theta_{1}>0$ such that for all $\theta<\theta_{1}$ we have

$\hbox{\rm diam}\left(\pi
H^{-1}(\mathbf{P}\times\mathbf{F}\times\\{0\\})\right)<\frac{\epsilon}{2}$

(see [3, Lemma 3.9]). For $\alpha\leq\frac{3}{2}\epsilon$ and
$\theta\in(0,\theta_{1})$ we define two different flow boxes 444for the
geodesic flow $\phi_{t}:SX\to SX$ on $SX$ ${B}_{\theta}^{\alpha}$ and
${B}_{\theta^{\prime}}^{\alpha^{\prime}}$ (of different “lengths” $\alpha$ and
$\epsilon^{2}$, respectively) in $SX$ by:

(4)
$\displaystyle\overline{B}_{\theta}^{\alpha}:=H^{-1}(\mathbf{P}\times\mathbf{F}\times[0,\alpha])\hbox{
and }$
$\displaystyle\overline{B}_{\theta^{\prime}}^{\epsilon^{2}}:=H^{-1}(\mathbf{P}^{\prime}\times\mathbf{F}^{\prime}\times[0,\epsilon^{2}]).$

(cf. [3, (3.11) and (3.12)]).

Let $\underline{{B}}_{\theta}^{\alpha}=\pi({B}_{\theta}^{\alpha})$ and
$\underline{{B}}_{\theta^{\prime}}^{\epsilon^{2}}=\pi({B}_{\theta^{\prime}}^{\epsilon^{2}})$
be their projections onto $SM$.

######  Remark 2.11.

Since the function
$\rho^{\prime}\to{m}(\underline{{B}}^{\epsilon^{2}}_{\rho^{\prime}})$ is
nondecreasing, and thus has countably many discontinuties (by Lebesgue’s
Theorem), we can suppose without loss of generality that
$\theta^{\prime}\in(0,\theta_{1})$ is a continuity point, and so, in
particular,

(5)
$\lim_{\rho^{\prime}\to\theta^{\prime}}{m}(\underline{{B}}^{\epsilon^{2}}_{\rho^{\prime}})={m}(\underline{{B}}^{\epsilon^{2}}_{\theta^{\prime}}).$

In order to give a dynamical approach to the counting problem the following
two definitions will prove useful. Let $\phi^{t}:SX\to SX$ denote the geodesic
flow on $SX$.

######  Definition 2.12.

For $t>0$ we can define two subsets of $\Gamma$ by:

(6) $\displaystyle\Gamma_{\theta,\theta^{\prime}}(t)$
$\displaystyle:=\\{\gamma\in\Gamma:\overline{B}_{\theta^{\prime}}^{\epsilon^{2}}\cap\phi^{-t}\gamma_{*}\overline{B}_{\theta}^{\alpha}\neq\emptyset\\}$
(7)
$\Gamma^{*}_{\theta,\theta^{\prime}}(t):=\\{\gamma\in\Gamma_{\theta,\theta^{\prime}}(t):\gamma\mathbf{F}\subset\mathbf{F}^{\prime}\text{
and }\gamma^{-1}\mathbf{P}\subset\mathbf{P}^{\prime}\\}.$

where the sets have an implicit dependence on
$\epsilon,\alpha,v_{0},v_{0}^{\prime}$. (cf. [3, (4.4) and (4.14)].)

By definition we have
$\Gamma_{\theta,\theta^{\prime}}^{*}(t)\subset\Gamma_{\theta,\theta^{\prime}}(t)$
and although we may not expect the reverse inclusion to be true, we have the
following slightly more modest result.

######  Lemma 2.13.

For every $\rho^{\prime}\in(0,\theta^{\prime})$ and $\rho\in(0,\theta)$, there
exists $t_{0}>0$ such that

$\Gamma_{\rho,\rho^{\prime}}(t)\subset\Gamma_{\theta,\theta^{\prime}}^{*}(t)\quad\text{
for all}\quad t\geq t_{0}.$

We postpone the proof of Lemma 2.13 until Appendix A.

The next lemma shows there is an inclusion of the set defined in Definition
1.2 into $\Gamma(t)$.

######  Lemma 2.14.

We have an injection

$\mathcal{C}(t,J(v_{0},\theta),J(v^{\prime}_{0},\theta^{\prime}))\hookrightarrow\Gamma(t)$

which associates to a geodesic $c$ the associated homotopy class
$[c]\in\pi_{1}(M)\cong\Gamma$.

We postpone the proof of Lemma 2.14 until Appendix A.

Although we may not expect the reverse inclusion in Lemma 2.14 to be true, we
at least have the following partial result.

######  Lemma 2.15.

For every $\rho^{\prime}\in(0,\theta^{\prime})$, there exists $t_{0}>0$ such
that there is an inclusion

$\Gamma_{\theta,\rho^{\prime}}(t)\hookrightarrow\mathcal{C}(t\pm
2\epsilon,J(v_{0},\theta),J(v^{\prime}_{0},\theta^{\prime}))\quad\forall
t>t_{0}.$

Again we postpone the proof of Lemma 2.15 until Appendix A.

##  3\. Proof of the counting results

In this section we will use results from the previous section to prove the
following proposition, which easily implies Theorem 1.3.

######  Proposition 3.1.

We have an asymptotic expression for the cardinality of $\Gamma(t)$ of the
form:

(8) $\\#\Gamma(t)\sim
e^{ht}\overline{m}({B})\frac{\mu_{\overline{p}}(\mathbf{F}^{\prime})}{\mu_{\overline{p}}(\mathbf{F})}\hbox{
as }t\to+\infty.$

Moreover, if the geodesic flow is expansive then the quantity
$m({B})\frac{\mu_{\overline{p}}(\mathbf{F}^{\prime})}{\mu_{\overline{p}}(\mathbf{F})}$
depends continuously on $v,v^{\prime},\theta,\theta^{\prime}$.

######  Remark 3.2.

The constant on the righthand side of (8) depends on $p$, but not then on the
choice of $\bar{p}\in\pi^{-1}(p)$.

We begin with a little more notation. Let

(9)
$S_{\theta}=H^{-1}\left(\mathbf{P}\times\mathbf{F}\times[0,\epsilon^{2}]\right)\subset
SX$

be another flow box and let

$\Gamma^{*}(t,\alpha):=\\{\gamma\in\Gamma^{*}\hbox{ :
}S_{\theta}\cap\gamma_{*}\phi^{-t}B_{\theta}^{\alpha}\neq\emptyset\\}.$

The proof of Proposition 3.1 now depends on the following two technical
lemmas.

######  Lemma 3.3.

For $\gamma\in\Gamma^{*}(t,\alpha)$, we have

${B}_{\theta^{\prime}}^{\epsilon^{2}}\cap\phi^{-(t+2\epsilon^{\frac{3}{2}})}\gamma_{*}{B}^{\alpha+4\epsilon^{\frac{3}{2}}}_{\theta}=H^{-1}(\mathbf{P}^{\prime}\times\gamma\mathbf{F}\times[0,\epsilon^{2}])=:{S}^{\gamma}.$

The next lemma describes the $\overline{m}$-measure of the set $S^{\gamma}$.

######  Lemma 3.4.

For each $\gamma\in\Gamma^{*}$, we have

$\overline{m}(\overline{S}^{\gamma})=\epsilon^{2}e^{\pm
4h\epsilon}e^{-ht}\mu_{p}(\mathbf{P}^{\prime})\mu_{p}(\mathbf{F}),$

and similarly with $\overline{m}$ and $\overline{S}^{\gamma}$ on $SX$ replaced
by the projections $m$ and $S^{\gamma}=\pi(\overline{S}^{\gamma})$ onto $SM$.

We postpone the proofs of both of these lemmas until Appendix B.

###### Proof of Proposition 3.1.

This follows the general lines of §5.2 in [3]. It follows from Lemmas 2.13 and
3.3 that given any ${\alpha}\in(0,\frac{3}{2}\epsilon]$ and
$\rho^{\prime}\in(0,\theta^{\prime}),\rho\in(0,\theta)$, for all sufficiently
large $t$ we have

$\underline{B}^{\epsilon^{2}}_{\rho^{\prime}}\cap\phi^{-t}\underline{B}_{\theta}^{\alpha}\subset\bigcup_{\gamma\in\Gamma^{*}(t,{\alpha})}\underline{S}^{\gamma}\subset\underline{B}^{\epsilon^{2}}_{\theta^{\prime}}\cap\phi^{-(t+2\epsilon^{2})}\underline{B}_{\theta}^{{\alpha}+4\epsilon^{2}}$

by proving the corresponding result on $SX$ and projecting to $SM$.

Using Lemma 3.4, for all $\gamma\in\Gamma^{*}(t)$, we have

$\displaystyle
e^{-4h\epsilon}\underline{m}(\underline{B}^{\epsilon^{2}}_{\rho^{\prime}}\cap\phi^{-t}\underline{B})$
$\displaystyle\leq\epsilon^{2}\\#\Gamma^{*}(t,{\alpha})e^{-ht}\mu_{p}(\mathbf{P}^{\prime})\mu_{p}(\mathbf{F})$
$\displaystyle\leq
e^{4h\epsilon}\underline{m}(\underline{B}^{\epsilon^{2}}_{\theta^{\prime}}\cap\phi^{-(t+2\epsilon^{2})}\underline{B}_{\theta}^{{\alpha}+4\epsilon^{2}}).$

Sending $t\to\infty$, using mixing, and dividing through by
$\underline{m}(\underline{B}^{\epsilon^{2}}_{\theta^{\prime}})\underline{m}(\underline{B}_{\theta}^{\alpha})=\overline{m}({B}_{\theta^{\prime}}^{\epsilon^{2}})\overline{m}({B}_{\theta}^{\alpha})$,
we get

$e^{-4h\epsilon}\frac{\overline{m}(B^{\epsilon^{2}}_{\rho^{\prime}})}{\overline{m}(B^{\epsilon^{2}}_{\theta^{\prime}})}\lesssim\frac{\epsilon^{2}\\#\Gamma^{*}(t,{\alpha})\mu_{p}(\mathbf{P}^{\prime})\mu_{p}(\mathbf{F})}{e^{ht}\overline{m}({B}_{\theta"}^{\epsilon^{2}})\overline{m}({B}_{\theta}^{\alpha})}\lesssim
e^{4h\epsilon}\frac{\overline{m}({B}_{\theta}^{{\alpha}+4\epsilon^{2}})}{\overline{m}({B}_{\theta}^{\alpha})}.$

By (5), assuming that $\theta^{\prime}$ is a point of continuity for
$\rho^{\prime}\mapsto{m}({B}^{\prime}_{\rho^{\prime}})$, so we can send
$\rho^{\prime}\nearrow\theta^{\prime}$ and obtain

(10)
$e^{-5h\epsilon}\lesssim\frac{\\#\Gamma^{*}(t,{\alpha})}{e^{ht}\overline{m}({B})}\frac{\mu_{p}(\mathbf{F})}{\mu_{p}(\mathbf{F}^{\prime})}\lesssim
e^{5h\epsilon}(1+4\epsilon^{2}/{\alpha}).$

Finally we need to replace $\\#\Gamma^{*}(t,\alpha)$ by $\\#\Gamma(t)$. (cf.
Compare with [3, (5.4)].)

This ends the proof of 8. Finally, if the geodesic flow is expansive then the
space of geodesics is in bijection with $\partial^{2}X$ then using that the
Busemann function $b_{p}(q,\xi)$ depends continuous on $(p,q,\xi)$ we have
$m({B})\frac{\mu_{\overline{p}}(\mathbf{F}^{\prime})}{\mu_{\overline{p}}(\mathbf{F})}$
depends continuously on $v,v^{\prime},\theta,\theta^{\prime}$. ∎

In order to allow for arbitrary $\theta$ and $\theta^{\prime}$ in the main
theorem we can break the arcs $J(\cdot,\cdot)$ into smaller pieces and apply
the proposition.

##  Appendix A Proofs of lemmas on isometries
and closed arcs

This section is devoted to the proof of Lemmas 2.13, 2.14 and 2.15. The proof
of Lemma 2.14 is relatively easy while Lemma 2.13 and 2.15 both uses a
geometric featture of surfaces without conjugate point that we first recall
here.

######  Definition A.1.

A simply connected Riemannian manifold $X$ without conjugate points is a
(uniform) visibility manifold if for every $\epsilon>0$ there exists $L>0$
such that whenever a geodesic $c:[a,b]\to X$ stays at a distance at least $L$
from some point $p\in X$, then the angle sustained by $c$ at $p$ is less than
$\epsilon$, that is

$\measuredangle_{p}(c)=\sup_{a\leq s,t\leq
b}\measuredangle_{p}((c(s),c(t))<\epsilon.$

###### Proof of Lemma 2.13.

The proof uses [3, Lemma 4.9] with the choices
$R=\mathbf{F}^{\prime}_{\rho^{\prime}}$,
$Q=\mathbf{P}^{\prime}_{\rho^{\prime}}$,
$V=int(\mathbf{F}^{\prime}_{\theta^{\prime}})$ and
$U=int(\mathbf{P}^{\prime}_{\theta^{\prime}})$. ∎

###### Proof of Lemma 2.14.

Let
$\underline{c}\in\mathcal{C}(t,J(v_{0},\theta),J(v^{\prime}_{0},\theta^{\prime}))$
and $c$ be the lift of $\underline{c}$ on $X$ with $\underline{c}(0)=p$. There
exists $\gamma\in\Gamma$ such that $c(t)=\gamma p=\gamma c(0)$. Let
$\operatorname{pr}_{*}:SX\to SM$ be the map associated to $\pi:X\to M$ then by
definition of
$\mathcal{C}(t,J(v_{0},\theta),J(v^{\prime}_{0},\theta^{\prime}))$, for
$w=c^{\prime}(t)$,
$\operatorname{pr}_{*}w\in{B}_{\theta^{\prime}}^{\epsilon^{2}}$ and
$\phi^{-t}w=c^{\prime}(0)\in{B}_{\theta}^{\alpha}$ implies that
$\bar{w}:=\gamma_{*}{w}\in B_{\theta^{\prime}}^{\epsilon^{2}}$ for some
$\gamma\in\Gamma$. Therefore $\bar{w}\in
B_{\theta^{\prime}}^{\epsilon^{2}}\cap\phi^{-t}\gamma_{*}B_{\theta}^{\alpha}$.
∎

555Compare with [2, (4.8)]

###### Proof of Lemma 2.15.

Let $\gamma\in\Gamma_{\theta,\rho^{\prime}}(t)$ and $w\in
B^{\epsilon^{2}}_{\rho^{\prime}}\cap\phi^{-t}\gamma_{*}B_{\theta}^{\alpha}$.
By the triangle inequaity

$\displaystyle d(p,\gamma p)$ $\displaystyle\leq d(p,\pi w)+d(\pi
w,\pi\phi^{t}w)+d(\pi\phi^{t}w,\gamma p).$

By [3, Lemma 3.10 ], we have
$d(p,\pi{w})\leq\operatorname{diam}(B^{\prime})\leq 2\epsilon$ and
$d(\pi\phi^{t}{w},\gamma p)\leq\operatorname{diam}({B})\leq 2\epsilon$.
Substituting these into the above display inequality gives

$d(p,\gamma p)\leq t+4\epsilon.$

We are left to prove that the geodesic $c:=c_{p,\gamma_{p}}$ connecting $p$ to
$\gamma p$ satisfies $c^{\prime}(0)\in J(v_{0},\theta)$ and
$c^{\prime}(d(p,\gamma p))\in J(v_{0}^{\prime},\theta^{\prime})$.

Let $v\in S_{p}X$ such that $v^{+}=w^{+}\in\mathbf{F}$, in particular, there
exists $R>0$ such that $d(c_{v}(t),c_{w}(t))\leq R$ and therefore the geodesic
connecting $\gamma p$ to $c_{v}(t)$ stays at distance at least $t-2R$. Then
using the uniform visabilty, there exists $t_{0}$ such that for all $t>t_{0}$
we have $\measuredangle_{p}(v,c^{\prime}(0))\leq\theta-\rho$ which implies
that $c^{\prime}(0)\in\mathbf{F}$. Therefore by the uniform visibility, we
have $\measuredangle_{p}(c_{p,\gamma
p}^{\prime}(0),c_{v}^{\prime}(0))\leq\theta-\rho$, in particular
$c^{\prime}_{p,\gamma p}(0)\in J(v_{0},\theta)$. Similarly we use the same
visibility condition for the point $\gamma p$ and the geodesic joining $p$ and
$c_{v}(-t)$ where $v\in S_{\gamma p}X$ with $v^{-}=w^{-}$. Thus the geodesic
$c_{p,\gamma p}$ belongs to $\mathcal{C}(t\pm
2\epsilon,J(v_{0},\theta),J(v^{\prime}_{0},\theta^{\prime}))$. ∎

##  Appendix B Counting

This section is devoted to the proof of Lemmas 3.3 and 3.4. The proof uses
some geometric quantities that we will define first.

######  Definition B.1.

For $\xi\in\partial X$ and $\gamma\in\Gamma$, we let
$b^{\gamma}_{\xi}:=b_{\xi}(\gamma p,p)$

######  Lemma B.2.

Given any $\gamma\in\Gamma^{*}=\\{\gamma\in\Gamma\hbox{ :
}\gamma\mathbf{F}\subset\mathbf{F}\hbox{ and
}\gamma^{-1}\mathbf{P}\subset\mathbf{P}\\}$ and any $t\in\mathbb{R}$, we have

${B}_{\theta^{\prime}}^{\epsilon^{2}}\cap\phi^{-t}\gamma_{*}{B}_{\theta}^{\alpha}=\\{{w}\in
E^{-1}(\mathbf{P}^{\prime}\times\gamma\mathbf{F}):s({w})\in[0,\epsilon^{2}]\cap(b_{{w}^{-}}^{\gamma}-t+[0,\alpha])\\}.$

666Compare this to the proof of [3, Lemma 4.13]

###### Proof of Lemma B.2.

To prove that
${B}_{\theta^{\prime}}^{\epsilon^{2}}\cap\phi^{-1}\gamma_{*}{B}_{\theta}^{\alpha}\subset
E^{-1}(\mathbf{P}^{\prime}\times\gamma\mathbf{F})$, we observe that if
$E({w})\notin\mathbf{P}^{\prime}\times\gamma\mathbf{F}$, then either
${w}^{-}\notin\mathbf{P}^{\prime}$, so
${w}\notin{B}_{\theta^{\prime}}^{\epsilon^{2}}$, or
${w}^{+}\notin\gamma\mathbf{F}$, so
${w}\notin\phi^{-t}\gamma_{*}{B}_{\theta}^{\alpha}$.

It remains to show that given ${w}\in
E^{-1}(\mathbf{P}^{\prime}\times\gamma\mathbf{F})$, we have

(11) $\displaystyle{w}\in{B}_{\theta^{\prime}}^{\epsilon^{2}}\ $
$\displaystyle\Leftrightarrow\ s({w})\in[0,\alpha],\text{ and}$ (12)
$\displaystyle{w}\in\phi^{-t}\gamma_{*}{B}_{\theta}^{\alpha}\ $
$\displaystyle\Leftrightarrow\ s({w})\in b_{{w}^{-}}^{\gamma}-t+[0,\alpha].$

The first of these is immediate from the definition of ${B}^{\prime}$. For the
second, we observe that
$s({v})=b_{{v}^{-}}(\pi{v},p)=b_{\gamma{v}^{-}}(\gamma\pi{v},\gamma p)$, and
thus

$\displaystyle\gamma_{*}{B}$ $\displaystyle=\\{\gamma_{*}{v}:{v}\in
E^{-1}(\mathbf{P}\times\mathbf{F})\text{ and
}b_{{v}^{-}}(\pi{v},p)\in[0,\alpha]\\}$ $\displaystyle=\\{{w}\in
E^{-1}(\gamma\mathbf{P}\times\gamma\mathbf{F}):b_{{w}^{-}}(\pi{w},\gamma
p)\in[0,\alpha]\\}$

By [3, Equation (3.1)] and [3, Equation (3.2)], we have

$b_{{w}^{-}}(\pi{w},\gamma p)=b_{{w}^{-}}(\pi{w},p)+b_{{w}^{-}}(p,\gamma
p)=s({w})-b_{{w}^{-}}^{\gamma};$

moreover, since $s(\phi^{t}{w})=s({w})+t$ by [3, Equation (3.8)], we see that
$\phi^{t}{w}\in\gamma_{*}{B}$ if and only if
$s({w})-b_{{w}^{-}}^{\gamma}+t\in[0,\alpha]$, which proves (12) and completes
the proof of the lemma. ∎

###### Proof of Lemma 3.3.

By Lemma B.2, the fact that
${B}_{\theta^{\prime}}^{\epsilon^{2}}\cap\phi^{-t}\gamma_{*}{B}_{\theta}^{\alpha}\neq\emptyset$
implies existence of $\eta\in\mathbf{P}^{\prime}$ such that

$(b_{\eta}^{\gamma}-t+[0,{\alpha}])\cap[0,\epsilon^{2}]\neq\emptyset$

from which we deduce that

$b_{\eta}^{\gamma}-t-\epsilon^{\frac{3}{2}}+[0,{\alpha}+2\epsilon^{\frac{3}{2}}]\supset[0,\epsilon^{2}]$

By [3, Lemma (4.11)], it follows that every $\xi\in\mathbf{P}^{\prime}$ has

$(b_{\xi}^{\gamma}-t-\epsilon^{\frac{3}{2}}+[0,\alpha+2\epsilon^{\frac{3}{2}}])\cap[0,\epsilon^{2}]\neq\emptyset$

which in turn implies that

$b_{\xi}^{\gamma}-t-2\epsilon^{\frac{3}{2}}+[0,{\alpha}+4\epsilon^{\frac{3}{2}}]\supset[0,\epsilon^{2}].$

By Lemma B.2, this completes the proof. ∎

###### Proof of Lemma 3.4.

By definition of $\underline{m}$, we have
$\underline{m}(\underline{S}^{\gamma})=\overline{m}(\overline{S}^{\gamma})=\epsilon^{2}\bar{\mu}(\mathbf{P}\times\gamma\mathbf{F})$.
Then we need to prove that $\bar{\mu}(\mathbf{P}\times\gamma\mathbf{F})=e^{\pm
4h\epsilon}e^{-ht}\mu_{p}(\mathbf{P}^{\prime})\mu_{p}(\mathbf{F})$

Given $(\xi,\eta)\in\mathbf{P}^{\prime}\times\gamma\mathbf{F}$, we can take
$q$ to lie on a geodesic connecting $\xi$ and $\eta$, with $b_{\xi}(q,p)=0$;
then we have

$|\beta_{p}(\xi,\eta)|:=|b_{\xi}(q,p)+b_{\eta}(q,p)|\leq d(q,p)<\epsilon/2,$

where the last inequality uses [3, Lemma 3.9]. Using this together with (3)
gives

$\bar{\mu}(\mathbf{P}^{\prime}\times\gamma\mathbf{F})=e^{\pm
h\epsilon/2}\mu_{p}(\mathbf{P}^{\prime})\mu_{p}(\gamma\mathbf{F}),$

Using [2, Proposition 5.1 (a)] gives

$\mu_{p}(\gamma\mathbf{F})=\mu_{\gamma^{-1}p}(\mathbf{F}),$

and [2, Proposition 5.1 (b)] gives

$\frac{d\mu_{\gamma^{-1}p}}{d\mu_{p}}(\eta)=e^{-hb_{\eta}(\gamma^{-1}p,p)}.$

When $\eta={c}(-\infty)$, where ${c}:=c_{p,\gamma^{-1}p}$. Using the
visibility condition as in the proof of Lemma 2.15, for $t$ large enough
$\eta\in\mathbf{F}^{\prime}_{\theta^{\prime}+\iota}$ for some $\iota>0$ very
small. Using Lemma 2.15, $b_{\eta}(p,\gamma p)=t\pm 4\epsilon$. By [3, Lemma
4.11], for $\xi\in\mathbf{F}^{\prime}$, $b_{\xi}(\gamma^{-1}p,p)$ varies by at
most $\epsilon^{2}$. We conclude that $\mu_{p}(\gamma\mathbf{F})=e^{\pm
5\epsilon}e^{-ht}\mu_{p}(\mathbf{F})$, and and this proves the lemma. ∎

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