2.1 Limit set

Let $\partial _\infty (\mathbb {H}^{n})$ denote the boundary at infinity and $\overline {\mathbb {H}^{n}} = \mathbb {H}^{n} \cup \partial _\infty (\mathbb {H}^{n})$ denote the compactification of $\mathbb {H}^{n}$.

We assume that $\Gamma$ is convex cocompact in the entire paper.

2.2 Patterson–Sullivan density

Let $\{\mu ^{\mathrm {PS}}_x: x \in \mathbb {H}^{n}\}$ denote the Patterson–Sullivan density of $\Gamma$ [Reference PattersonPat76, Reference SullivanSul79], that is, the set of finite Borel measures on $\partial _\infty (\mathbb {H}^{n})$ supported on $\Lambda (\Gamma )$ such that

1. (1) $g_*\mu ^{\mathrm {PS}}_x = \mu ^{\mathrm {PS}}_{gx}$ for all $g \in \Gamma$ and $x \in \mathbb {H}^{n}$,

2. (2) $({d\mu ^{\mathrm {PS}}_x}/{d\mu ^{\mathrm {PS}}_y})(\xi ) = e^{\delta _\Gamma \beta _{\xi }(y, x)}$ for all $\xi \in \partial _\infty (\mathbb {H}^{n})$ and $x, y \in \mathbb {H}^{n}$

where $\beta _{\xi }$ denotes the Busemann function at $\xi \in \partial _\infty (\mathbb {H}^{n})$ defined by $\beta _{\xi }(y, x) = \lim _{t \to \infty } (d(\xi (t), y) - d(\xi (t), x))$, and $\xi :\mathbb {R} \to \mathbb {H}^{n}$ is any geodesic such that $\lim _{t \to \infty } \xi (t) = \xi$. We allow tangent vector arguments for the Busemann function as well, in which case we will use their basepoints in the definition. Since $\Gamma$ is convex cocompact, for all $x \in \mathbb {H}^{n}$, the measure $\mu ^{\mathrm {PS}}_x$ is the $\delta _\Gamma$-dimensional Hausdorff measure on $\partial _\infty (\mathbb {H}^{n})$ supported on $\Lambda (\Gamma )$ corresponding to the spherical metric on $\partial _\infty (\mathbb {H}^{n})$ with respect to $x$, up to scalar multiples.

2.3 Bowen–Margulis–Sullivan measure

For all $u \in \operatorname {T}^{1}(\mathbb {H}^{n})$, let $u^{+}$ and $u^{-}$ denote its forward and backward limit points. Using the Hopf parametrization via the homeomorphism $G/M \cong \operatorname {T}^{1}(\mathbb {H}^{n}) \to \{(u^{+}, u^{-}) \in \partial _\infty (\mathbb {H}^{n}) \times \partial _\infty (\mathbb {H}^{n}): u^{+} \neq u^{-}\} \times \mathbb {R}$ defined by $u \mapsto (u^{+}, u^{-}, t = \beta _{u^{-}}(o, u))$, we define the Bowen–Margulis–Sullivan (BMS) measure $\mathsf {m}$ on $G/M$ [Reference MargulisMar04, Reference BowenBow71, Reference KaimanovichKai90] by

\[ d\mathsf{m}(u) = e^{\delta_\Gamma \beta_{u^{+}}(o, u)} e^{\delta_\Gamma \beta_{u^{-}}(o, u)} \, d\mu^{\mathrm{PS}}_o(u^{+}) \, d\mu^{\mathrm{PS}}_o(u^{-}) \, dt. \]

Note that this definition depends only on $\Gamma$ and not on the choice of reference point $o \in \mathbb {H}^{n}$. Moreover, $\mathsf {m}$ is left $\Gamma$-invariant. We now define induced measures on other spaces, all of which we call the BMS measures and denote by $\mathsf {m}$ by abuse of notation. By left $\Gamma$-invariance, $\mathsf {m}$ descends to a measure on $\Gamma \backslash G/M$. We normalize it to a probability measure so that $\mathsf {m}(\Gamma \backslash G/M) = 1$. Since $M$ is compact, we can then use the Haar probability measure on $M$ to lift $\mathsf {m}$ to a right $M$-invariant measure on $\Gamma \backslash G$. It can be checked that the BMS measures are invariant with respect to the geodesic flow or the frame flow as appropriate, that is, they are right $A$-invariant. We denote the right $A$-invariant subset $\Omega = \operatorname {supp}(\mathsf {m}) \subset \Gamma \backslash G/M$ which is compact since $\Gamma$ is convex cocompact.

3.1 Markov sections

We will use a Markov section on $\Omega \subset \operatorname {T}^{1}(X) \cong \Gamma \backslash G/M$, as developed by Bowen and Ratner [Reference BowenBow70, Reference RatnerRat73], to obtain a symbolic coding of the dynamical system at hand. Recall that the geodesic flow on $\operatorname {T}^{1}(X)$ is Anosov. Let $W^{\mathrm {su}}(w) \subset \operatorname {T}^{1}(X)$ and $W^{\mathrm {ss}}(w) \subset \operatorname {T}^{1}(X)$ denote the leaves through $w \in \operatorname {T}^{1}(X)$ of the strong unstable and strong stable foliations, and $W_{\epsilon }^{\mathrm {su}}(w) \subset W^{\mathrm {su}}(w)$ and $W_{\epsilon }^{\mathrm {ss}}(w) \subset W^{\mathrm {ss}}(w)$ denote the open balls of radius $\epsilon > 0$ with respect to the induced distance functions $d_{\mathrm {su}}$ and $d_{\mathrm {ss}}$, respectively. We use similar notations for the weak unstable and weak stable foliations by replacing ‘su’ with ‘wu’ and ‘ss’ with ‘ws’, respectively. The Anosov property provides a constant $C_{\mathrm {Ano}} > 0$ such that for all $w \in \operatorname {T}^{1}(X)$, we have

\[ d_{\mathrm{su}}(ua_{{-}t}, va_{{-}t}) \leq C_{\mathrm{Ano}}e^{{-}t}d_{\mathrm{su}}(u, v),\quad d_{\mathrm{ss}}(ua_t, va_t) \leq C_{\mathrm{Ano}}e^{{-}t}d_{\mathrm{ss}}(u, v) \]

for all $t \geq 0$, for all $u, v \in W^{\mathrm {su}}(w)$ or for all $u, v \in W^{\mathrm {ss}}(w)$, respectively. We recall that there exist $\epsilon _0, \epsilon _0' > 0$ such that for all $w \in \operatorname {T}^{1}(X)$, $u \in W_{\epsilon _0}^{\mathrm {wu}}(w)$, and $s \in W_{\epsilon _0}^{\mathrm {ss}}(w)$, there exists a unique intersection denoted by

(1)\begin{equation} [u, s] = W_{\epsilon_0'}^{\mathrm{ss}}(u) \cap W_{\epsilon_0'}^{\mathrm{wu}}(s), \end{equation}

and moreover, $[\cdot , \cdot ]$ defines a homeomorphism from $W_{\epsilon _0}^{\mathrm {wu}}(w) \times W_{\epsilon _0}^{\mathrm {ss}}(w)$ onto its image [Reference RatnerRat73]. Subsets $U \subset W_{\epsilon _0}^{\mathrm {su}}(w) \cap \Omega$ and $S \subset W_{\epsilon _0}^{\mathrm {ss}}(w) \cap \Omega$ for some $w \in \Omega$ are called proper if $U = \overline {\operatorname {int}(U)}$ and $S = \overline {\operatorname {int}(S)}$, where the interiors and closures are taken in the topology of $W^{\mathrm {su}}(w) \cap \Omega$ and $W^{\mathrm {ss}}(w) \cap \Omega$, respectively. For any $\hat {\delta } > 0$ and proper sets $U \subset W_{\epsilon _0}^{\mathrm {su}}(w) \cap \Omega$ and $S \subset W_{\epsilon _0}^{\mathrm {ss}}(w) \cap \Omega$ containing some $w \in \Omega$, we call

\[ R = [U, S] = \{[u, s] \in \Omega: u \in U, s \in S\} \subset \Omega \]

a rectangle of size $\hat {\delta }$ if $\operatorname {diam}_{d_{\mathrm {su}}}(U), \operatorname {diam}_{d_{\mathrm {ss}}}(S) \leq \hat {\delta }$, and we call $w$ the center of $R$. For any rectangle $R = [U, S]$, we generalize the notation and define $[v_1, v_2] = [u_1, s_2]$ for all $v_1 = [u_1, s_1] \in R$ and $v_2 = [u_2, s_2] \in R$.

Definition 3.1 (Complete set of rectangles)

Let $\hat {\delta } > 0$ and $N \in \mathbb {N}$. A set

\[ \mathcal {R} = \{R_1, R_2, \dotsc , R_N\} = \{[U_1, S_1], [U_2, S_2], \dotsc , [U_N, S_N]\} \]

consisting of rectangles in $\Omega$ is called a complete set of rectangles of size $\hat {\delta }$ if:

1. (1) $R_j \cap R_k = \varnothing$ for all $1 \leq j, k \leq N$ with $j \neq k$;

2. (2) $\operatorname {diam}_{d_{\mathrm {su}}}(U_j), \operatorname {diam}_{d_{\mathrm {ss}}}(S_j) \leq \hat {\delta }$ for all $1 \leq j \leq N$;

3. (3) $\Omega = \bigcup _{j = 1}^{N} \bigcup _{t \in [0, \hat {\delta }]} R_j a_t$.

Henceforth, we fix

(2)\begin{equation} 0 < \hat{\delta} < \min(1, \epsilon_0, \epsilon_0'), \end{equation}

where $\epsilon _0$ and $\epsilon _0'$ are from (1). We also fix

\[ \mathcal{R} = \{R_1, R_2, \dotsc, R_N\} = \{[U_1, S_1], [U_2, S_2], \dotsc, [U_N, S_N]\} \]

to be a complete set of rectangles of size $\hat {\delta }$ in $\Omega$. We denote

\[ R = \bigsqcup_{j = 1}^{N} R_j,\quad U= \bigsqcup_{j = 1}^{N} U_j. \]

We introduce the distance function $d$ on $U$ defined by

\[ d(u, v) = \begin{cases} d_{\mathrm{su}}(u, v), & u, v \in U_j \text{ for some } 1 \leq j \leq N, \\ 1, & \text{otherwise,} \end{cases} \quad \text{for all } u, v \in U. \]

We will use $d_{\mathrm {su}}$ whenever further clarity is required. Denote by $\tau : R \to \mathbb {R}$ the first return time map defined by

\[ \tau(u) = \inf\{t > 0: ua_t \in R\} \quad \text{for all }u \in R. \]

Note that $\tau$ is constant on $[u, S_j]$ for all $u \in U_j$ and $1 \leq j \leq N$. Denote by $\mathcal {P}: R \to R$ the Poincaré first return map defined by

\[ \mathcal{P}(u) = ua_{\tau(u)} \quad \text{for all}\ u \in R. \]

Let $\sigma = (\operatorname {proj}_U \circ \mathcal {P})|_U: U \to U$ be its projection, where $\operatorname {proj}_U: R \to U$ is the projection defined by $\operatorname {proj}_U([u, s]) = u$ for all $[u, s] \in R$. Define the cores

\begin{align*} \hat{R} &= \{u \in R: \mathcal{P}^{k}(u) \in \operatorname{int}(R) \text{ for all } k \in \mathbb{Z}\},\nonumber \\ \hat{U} &= \{u \in U: \sigma^{k}(u) \in \operatorname{int}(U) \text{ for all } k \in \mathbb{Z}_{{\geq} 0}\}, \end{align*}

which are both residual subsets (complements of meager sets) of $R$ and $U$, respectively.

Definition 3.2 (Markov section)

Let $\hat {\delta } > 0$ and $N \in \mathbb {N}$. We call a complete set of rectangles $\mathcal {R}$ of size $\hat {\delta }$ a Markov section if in addition to properties (1)–(3), the following property, called the Markov property, is satisfied:

1. (4) $[\operatorname {int}(U_k), \mathcal {P}(u)] \subset \mathcal {P}([\operatorname {int}(U_j), u])$ and $\mathcal {P}([u, \operatorname {int}(S_j)]) \subset [\mathcal {P}(u), \operatorname {int}(S_k)]$ for all $u \in R$ such that $u \in \operatorname {int}(R_j) \cap \mathcal {P}^{-1}(\operatorname {int}(R_k)) \neq \varnothing$, for all $1 \leq j, k \leq N$.

This can be understood pictorially in Figure 1.

Figure 1.

The Markov property.

The existence of Markov sections of arbitrarily small size for Anosov flows was proved by Bowen and Ratner [Reference BowenBow70, Reference RatnerRat73]. Thus, we assume henceforth that $\mathcal {R}$ is a Markov section.

3.2 Symbolic dynamics

Let $\mathcal {A} = \{1, 2, \dotsc , N\}$ be the alphabet for the coding corresponding to the Markov section. Define the $N \times N$ transition matrix $T$ by

\[ T_{j, k} = \begin{cases} 1, & \operatorname{int}(R_j) \cap \mathcal{P}^{{-}1}(\operatorname{int}(R_k)) \neq \varnothing, \\ 0, & \text{otherwise}, \end{cases} \quad \text{for all }1 \leq j, k \leq N. \]

The transition matrix $T$ is topologically mixing [Reference RatnerRat73, Theorem 4.3], that is, there exists $N_T \in \mathbb {N}$ such that all the entries of $T^{N_T}$ are positive. This definition is equivalent to the one in [Reference RatnerRat73] in the setting of Markov sections. Define the spaces of bi-infinite and infinite admissible sequences

\begin{align*} \Sigma &= \{(\dotsc, x_{{-}1}, x_0, x_1, \dotsc) \in \mathcal{A}^{\mathbb{Z}}: T_{x_j, x_{j + 1}} = 1 \text{ for all } j \in \mathbb{Z}\},\\ \Sigma^{+} &= \{(x_0, x_1, \dotsc) \in \mathcal{A}^{\mathbb{Z}_{{\geq} 0}}: T_{x_j, x_{j + 1}} = 1 \text{ for all } j \in \mathbb{Z}_{{\geq} 0}\}, \end{align*}

respectively. We will also use the term admissible sequences for finite sequences in the natural way. For any $\theta \in (0, 1)$, we can endow $\Sigma$ with the distance function $d_\theta$ defined by $d_\theta (x, y) = \theta ^{\inf \{|j| \in \mathbb {Z}_{\geq 0}: x_j \neq y_j\}}$ for all $x, y \in \Sigma$. We can similarly endow $\Sigma ^{+}$ with a distance function which we also denote by $d_\theta$.

Definition 3.3 (Cylinder)

For all $k \in \mathbb {Z}_{\geq 0}$ and for all admissible sequences $x = (x_0, x_1, \dotsc , x_k)$, we define the corresponding cylinder to be

\[ \mathtt{C}[x] = \{u \in U: \sigma^{j}(u) \in \operatorname{int}(U_{x_j}) \text{ for all } 0 \leq j \leq k\} \]

with length $\operatorname {len}(\mathtt{C}[x]) := \operatorname {len}(x) := k$. We will denote cylinders simply by $\mathtt {C}$ (or other typewriter-style letters) when we do not need to specify the corresponding admissible sequence.

Although $\sigma$ and $\tau$ are not even continuous, we note that for all admissible pairs $(j, k)$, the restricted maps $\sigma |_{\mathtt {C}[j, k]}: \mathtt {C}[j, k] \to \operatorname {int}(U_k)$, $(\sigma |_{\mathtt {C}[j, k]})^{-1}: \operatorname {int}(U_k) \to \mathtt {C}[j, k]$, and $\tau |_{\mathtt {C}[j, k]}: \mathtt {C}[j, k] \to \mathbb {R}$ are Lipschitz in our setting.

By a slight abuse of notation, let $\sigma$ also denote the shift map on $\Sigma$ or $\Sigma ^{+}$. There exist natural continuous surjections $\zeta : \Sigma \to R$ and $\zeta ^{+}: \Sigma ^{+} \to U$ defined by $\zeta (x) = \bigcap _{j = -\infty }^{\infty } \overline {\mathcal {P}^{-j}(\operatorname {int}(R_{x_j}))}$ for all $x \in \Sigma$ and $\zeta ^{+}(x) = \bigcap _{j = 0}^{\infty } \overline {\sigma ^{-j}(\operatorname {int}(U_{x_j}))}$ for all $x \in \Sigma ^{+}$. Define $\hat {\Sigma } = \zeta ^{-1}(\hat {R})$ and $\hat {\Sigma }^{+} = (\zeta ^{+})^{-1}(\hat {U})$. Then the restrictions $\zeta |_{\hat {\Sigma }}: \hat {\Sigma } \to \hat {R}$ and $\zeta ^{+}|_{\hat {\Sigma }^{+}}: \hat {\Sigma }^{+} \to \hat {U}$ are bijective and satisfy $\zeta |_{\hat {\Sigma }} \circ \sigma |_{\hat {\Sigma }} = \mathcal {P}|_{\hat {R}} \circ \zeta |_{\hat {\Sigma }}$ and $\zeta ^{+}|_{\hat {\Sigma }^{+}} \circ \sigma |_{\hat {\Sigma }^{+}} = \sigma |_{\hat {U}} \circ \zeta ^{+}|_{\hat {\Sigma }^{+}}$.

For $\theta \in (0, 1)$ sufficiently close to $1$, the maps $\zeta$ and $\zeta ^{+}$ are Lipschitz [Reference BowenBow73, Lemma 2.2] with some Lipschitz constant $C_\theta > 0$. We now fix $\theta$ to be any such constant. Let $C^{\operatorname {Lip}(d_\theta )}(\Sigma , \mathbb {R})$ denote the space of Lipschitz functions $f: \Sigma \to \mathbb {R}$. We use similar notation for other domains, codomains, and metrics.

Since $(\tau \circ \zeta )|_{\hat {\Sigma }}$ and $(\tau \circ \zeta ^{+})|_{\hat {\Sigma }^{+}}$ are Lipschitz, there exist unique Lipschitz extensions $\tau _\Sigma : \Sigma \to \mathbb {R}$ and $\tau _{\Sigma ^{+}}: \Sigma ^{+} \to \mathbb {R}$, respectively. Note that the resulting maps are distinct from $\tau \circ \zeta$ and $\tau \circ \zeta ^{+}$ because they may differ precisely on $x \in \Sigma$ for which $\operatorname {proj}_U(\zeta (x)) \in \partial (\mathtt {C})$, and $x \in \Sigma ^{+}$ for which $\zeta ^{+}(x) \in \partial (\mathtt {C})$ respectively, for some cylinder $\mathtt {C} \subset U$ with $\operatorname {len}(\mathtt {C}) = 1$. Then the previous properties extend to $\zeta (\sigma (x)) = \zeta (x)a_{\tau _\Sigma (x)}$ for all $x \in \Sigma$ and $\zeta ^{+}(\sigma (x)) = \operatorname {proj}_U(\zeta ^{+}(x)a_{\tau _{\Sigma ^{+}}(x)})$ for all $x \in \Sigma ^{+}$.

3.3 Thermodynamics

Definition 3.4 (Pressure)

For all potential $f \in C^{\operatorname {Lip}(d_\theta )}(\Sigma , \mathbb {R})$, the pressure is defined by

\[ \operatorname{Pr}_\sigma(f) = \sup_{\nu \in \mathcal{M}^{1}_\sigma(\Sigma)}\biggl\{\int_\Sigma f \, d\nu + h_\nu(\sigma)\biggr\}, \]

where $\mathcal {M}^{1}_\sigma (\Sigma )$ is the set of $\sigma$-invariant Borel probability measures on $\Sigma$ and $h_\nu (\sigma )$ is the measure theoretic entropy of $\sigma$ with respect to $\nu$.

For all $f \in C^{\operatorname {Lip}(d_\theta )}(\Sigma , \mathbb {R})$, there is in fact a unique $\sigma$-invariant Borel probability measure $\nu _f$ on $\Sigma$ which attains the supremum in Definition 3.4 called the $f$-equilibrium state [Reference BowenBow08, Theorems 2.17 and 2.20] and it satisfies $\nu _f(\hat {\Sigma }) = 1$ [Reference ChernovChe02, Corollary 3.2]. In particular, we will consider the probability measure $\nu _{-\delta _\Gamma \tau _\Sigma }$ on $\Sigma$ which we will denote simply by $\nu _\Sigma$ and which has corresponding pressure $\operatorname {Pr}_\sigma (-\delta _\Gamma \tau _\Sigma ) = 0$. According to the foregoing, $\nu _\Sigma (\hat {\Sigma }) = 1$. Define the corresponding probability measure $\nu _R = \zeta _*(\nu _\Sigma )$ on $R$ and note that $\nu _R(\hat {R}) = 1$. Now consider the suspension space $R^{\tau } = (R \times \mathbb {R}_{\geq 0})/\mathord {\sim }$ where $\sim$ is the equivalence relation on $R \times \mathbb {R}_{\geq 0}$ defined by $(u, t + \tau (u)) \sim (\mathcal {P}(u), t)$. Then we have a bijection $R^{\tau } \to \Omega$ defined by $(u, t) \mapsto ua_t$. We can define the measure $\nu ^{\tau }$ on $R^{\tau }$ as the product measure $\nu _R \times m^{\mathrm {Leb}}$ on $\{(u, t) \in R \times \mathbb {R}_{\geq 0}: 0 \leq t < \tau (u)\}$. Then, using the aforementioned bijection, we have the pushforward measure which, by abuse of notation, we also denote by $\nu ^{\tau }$ on $\operatorname {T}^{1}(X)$ supported on $\Omega$. By [Reference SullivanSul84] and [Reference ChernovChe02, Theorem 4.4], we have $\mathsf {m} = \nu ^{\tau }/\nu _R(\tau )$ because they are the unique measure of maximal entropy for the geodesic flow on $\operatorname {T}^{1}(X)$. Finally, we define the probability measure $\nu _U = (\operatorname {proj}_U)_*(\nu _R)$ and note that $\nu _U(\hat {U}) = 1$ and $\nu _U(\tau ) = \nu _R(\tau )$.

5.1 Modified constructions using the smooth structure on $G$

We need to use the smooth structure on $G$ to apply Lie theoretic arguments to derive the LNIC later in § 6.1. However, the smooth structure is not readily available on $U$ since it is fractal in nature. Thus, we need an appropriately enlarged open set $\tilde {U}$ of the strong unstable foliation containing $U$. Since the strong unstable foliation is smooth, $\tilde {U} \subset \operatorname {T}^{1}(X)$ would then be a smooth submanifold and provide a smooth structure at our disposal. At the same time, we would like to extend $\sigma$ to a map on $\tilde {U}$, but this is difficult due to the expanding nature. Conveniently, we can avoid this problem altogether by extending the local inverses in the following sense. Let $w_j$ be the center of $R_j$ for all $j \in \mathcal {A}$. Using arguments of [Reference RuelleRue89, Lemma 1.2] with a sufficiently small $\delta > 0$, and increasing $\hat {\delta }$ if necessary while ensuring that (2) still holds, there exist open sets $U_j \subset \tilde {U}_j$ such that $\overline {\tilde {U}_j} \subset W_{\epsilon _0}^{\mathrm {su}}(w_j)$ with $\operatorname {diam}_{d_{\mathrm {su}}}(\tilde {U}_j) \leq \hat {\delta }$ for all $j \in \mathcal {A}$ such that for all admissible pairs $(j, k)$, we can naturally extend the inverse $(\sigma |_{\mathtt {C}[j, k]})^{-1}: \operatorname {int}(U_k) \to \mathtt {C}[j, k]$ to a smooth injective map $\sigma ^{-(j, k)}: \tilde {U}_k \to \tilde {U}_j$. More specifically, assuming that $\epsilon _0$ and $\delta$ are sufficiently small without loss of generality, taking any $u_0 \in U_j$ such that $\sigma (u_0) \in U_k$, we can define $\sigma ^{-(j, k)}(u)$ to be the unique intersection

\[ \sigma^{-(j, k)}(u) = \biggl(\bigcup_{t \in (-\tau(u_0) - \inf(\tau), -\tau(u_0) + \inf(\tau))} W_{\epsilon_0}^{\mathrm{ss}}(u)a_t\biggr) \cap W_{\epsilon_0}^{\mathrm{su}}(w_j) \]

for all $u \in \tilde {U}_k$. We define $\tilde {U} = \bigsqcup _{j = 1}^{N} \tilde {U}_j$. Also define the measure $\nu _{\tilde {U}}$ on $\tilde {U}$ simply by $\nu _{\tilde {U}}(B) = \nu _U(B \cap U)$ for all Borel subsets $B \subset \tilde {U}$. Let $j \in \mathbb {Z}_{\geq 0}$ and $\alpha = (\alpha _0, \alpha _1, \dotsc , \alpha _j)$ be an admissible sequence. Define $\sigma ^{-\alpha } = \sigma ^{-(\alpha _0, \alpha _1)} \circ \sigma ^{-(\alpha _1, \alpha _2)} \circ \cdots \circ \sigma ^{-(\alpha _{j - 1}, \alpha _j)}: \tilde {U}_{\alpha _j} \to \tilde {U}_{\alpha _0}$ if $j > 0$ and $\sigma ^{-\alpha } = \operatorname {Id}_{\tilde {U}_{\alpha _0}}$ if $j = 0$. Define the cylinder $\tilde {\mathtt {C}}[\alpha ] = \sigma ^{-\alpha }(\tilde {U}_{\alpha _j}) \supset \mathtt {C}[\alpha ]$. Define the smooth maps $\sigma ^{\alpha } = (\sigma ^{-\alpha })^{-1}: \tilde {\mathtt {C}}[\alpha ] \to \tilde {U}_{\alpha _j}$. These maps are sufficient for our purposes in defining transfer operators. For convenience we define $\tilde {R}_j = [\tilde {U}_j, S_j]$ for all $j \in \mathcal {A}$.

We define more extended maps. Let $(j, k)$ be an admissible pair. The maps $\tau |_{\mathtt {C}[j, k]}$ and $\vartheta |_{\mathtt {C}[j, k]}$ naturally extend to smooth maps $\tau _{(j, k)}: \tilde {\mathtt {C}}[j, k] \to \mathbb {R}$ and $\vartheta ^{(j, k)}: \tilde {\mathtt {C}}[j, k] \to M$ as follows. In light of the above definition of $\sigma ^{-(j, k)}$, using the same notation and writing $u' = \sigma ^{(j, k)}(u)$, we define $\tau _{(j, k)}(u) \in (\tau (u_0) - \inf (\tau ), \tau (u_0) + \inf (\tau ))$ uniquely such that $W_{\epsilon _0}^{\mathrm {ss}}(u')a_{-\tau _{(j, k)}(u)} \cap W_{\epsilon _0}^{\mathrm {su}}(w_k) \neq \varnothing$ for all $u \in \tilde {\mathtt {C}}[j, k]$. Similar to before, $\vartheta ^{(j, k)}(u)$ is such that $F(u)a_{\tau _{(j, k)}(u)} = F\bigl (ua_{\tau _{(j, k)}(u)}\bigr )\vartheta ^{(j, k)}(u)^{-1}$ for all $u \in \tilde {\mathtt {C}}[j, k]$. Now for all $k \in \mathbb {N}$ and admissible sequences $\alpha = (\alpha _0, \alpha _1, \dotsc , \alpha _k)$, we define the smooth maps $\tau _\alpha : \tilde {\mathtt {C}}[\alpha ] \to \mathbb {R}$, $\vartheta ^{\alpha }: \tilde {\mathtt {C}}[\alpha ] \to M$, and $\Phi ^{\alpha }: \tilde {\mathtt {C}}[\alpha ] \to AM$ by

\begin{align*} \tau_\alpha(u) &= \sum_{j = 0}^{k - 1} \tau_{(\alpha_j, \alpha_{j + 1})}(\sigma^{(\alpha_0, \alpha_1, \dotsc, \alpha_j)}(u)),\\ \vartheta^{\alpha}(u) &= \prod_{j = 0}^{k - 1} \vartheta^{(\alpha_j, \alpha_{j + 1})}(\sigma^{(\alpha_0, \alpha_1, \dotsc, \alpha_j)}(u)),\\ \Phi^{\alpha}(u) &= a_{\tau_\alpha(u)}\vartheta^{\alpha}(u) = \prod_{j = 0}^{k - 1} \Phi^{(\alpha_j, \alpha_{j + 1})}(\sigma^{(\alpha_0, \alpha_1, \dotsc, \alpha_j)}(u)) \end{align*}

for all $u \in \tilde {\mathtt {C}}[\alpha ]$, where the terms of the products are to be in ascending order from left to right. For all admissible sequences $\alpha$ with $\operatorname {len}(\alpha ) = 0$, we define $\tau _\alpha (u) = 0$ and $\vartheta ^{\alpha }(u) = \Phi ^{\alpha }(u) = e \in AM$ for all $u \in \tilde {\mathtt {C}}[\alpha ]$. For all $u \in U$, there is a corresponding unique admissible sequence in $\Sigma ^{+}$ and hence we can instead use the notation $\tau _k(u)$, $\vartheta ^{k}(u)$, and $\Phi ^{k}(u)$ for all $k \in \mathbb {Z}_{\geq 0}$.

5.2 Transfer operators

We will use the notation $\xi = a + ib \in \mathbb {C}$ for the complex parameter for the transfer operators and use the convention that sums over sequences are actually sums over admissible sequences, throughout the paper.

Definition 5.1 (Transfer operator with holonomy)

For all $\xi \in \mathbb {C}$ and $\rho \in \hat {M}$, the transfer operator with holonomy $\tilde {\mathcal {M}}_{\xi \tau , \rho }: C(\tilde {U}, V_\rho ^{\oplus \dim (\rho )}) \to C(\tilde {U}, V_\rho ^{\oplus \dim (\rho )})$ is defined by

\[ \tilde{\mathcal{M}}_{\xi\tau, \rho}(H)(u) = \sum_{\substack{(j, k)\\ u' = \sigma^{-(j, k)}(u)}} e^{\xi\tau_{(j, k)}(u')} \rho(\vartheta^{(j, k)}(u')^{{-}1}) H(u') \]

for all $u \in \tilde {U}$ and $H \in C(\tilde {U}, V_\rho ^{\oplus \dim (\rho )})$.

Let $\xi \in \mathbb {C}$. We define $\tilde {\mathcal {L}}_{\xi \tau } = \tilde {\mathcal {M}}_{\xi \tau , 1}$ and simply call it the transfer operator. For any $\rho \in \hat {M}$, denote by $|_U: C(\tilde {U}, V_\rho ^{\oplus \dim (\rho )}) \to C(U, V_\rho ^{\oplus \dim (\rho )})$ the restriction map. Then for all $\rho \in \hat {M}$, we also define the transfer operator with holonomy $\mathcal {M}_{\xi \tau , \rho } = |_U \circ \tilde {\mathcal {M}}_{\xi \tau , \rho } \circ (|_U)^{-1}$, where $(|_U)^{-1}$ denotes taking any preimage using the Tietze extension theorem, and the transfer operator $\mathcal {L}_{\xi \tau } = \mathcal {M}_{\xi \tau , 1}$.

We recall the Ruelle–Perron–Frobenius theorem along with the theory of Gibbs measures in this setting [Reference BowenBow08, Reference Parry and PollicottPP90].

In light of Theorem 5.2, it is convenient to normalize the transfer operators defined above. Let $a \in \mathbb {R}$. Define $\lambda _a = e^{\operatorname {Pr}_\sigma (-(\delta _\Gamma + a)\tau _{\Sigma })}$ which is the maximal simple eigenvalue of $\mathcal {L}_{-(\delta _\Gamma + a)\tau }$ by Theorem 5.2 and recall that $\lambda _0 = 1$. Define the eigenvectors, the unique positive function $h_a \in C^{\operatorname {Lip}(d)}(U, \mathbb {R})$ and the unique probability measure $\nu _a$ on $U$ with $\nu _a(h_a) = 1$ such that

\[ \mathcal{L}_{-(\delta_\Gamma + a)\tau}(h_a) = \lambda_a h_a,\quad \mathcal{L}_{-(\delta_\Gamma + a)\tau}^{*}(\nu_a) = \lambda_a \nu_a \]

provided by Theorem 5.2. Note that $d\nu _U = h_0 \, d\nu _0$. Now by Theorem A.2 in the Appendix, the eigenvector $h_a \in C^{\operatorname {Lip}(d)}(U, \mathbb {R})$ extends to an eigenvector $h_a \in C^{\infty }(\tilde {U}, \mathbb {R})$ with bounded derivatives for $\tilde {\mathcal {L}}_{-(\delta _\Gamma + a)\tau }$. For all admissible pairs $(j, k)$, we define the smooth map

\[ f_{(j, k)}^{(a)} ={-}(\delta_\Gamma + a)\tau_{(j, k)} + \log(h_0) - \log(h_0 \circ \sigma^{(j, k)}) - \log(\lambda_a). \]

For all $k \in \mathbb {N}$ and admissible sequences $\alpha = (\alpha _0, \alpha _1, \dotsc , \alpha _k)$, we define the smooth map $f_\alpha ^{(a)}: \tilde {\mathtt {C}}[\alpha ] \to \mathbb {R}$ by

\[ f_\alpha^{(a)}(u) = \sum_{j = 0}^{k - 1} f_{(\alpha_j, \alpha_{j + 1})}^{(a)}(\sigma^{(\alpha_0, \alpha_1, \dotsc, \alpha_j)}(u)) \quad \text{for all }u \in \tilde{\mathtt{C}}[\alpha]. \]

For all admissible sequences $\alpha$ with $\operatorname {len}(\alpha ) = 0$, we define $f_\alpha ^{(a)}(u) = 0$. As before, for all $u \in U$, we can also use the notation $f_k^{(a)}(u)$ for any $k \in \mathbb {Z}_{\geq 0}$.

We now normalize the transfer operators for convenience. Let $\xi \in \mathbb {C}$ and $\rho \in \hat {M}$. We define $\tilde {\mathcal {M}}_{\xi , \rho }: C(\tilde {U}, V_\rho ^{\oplus \dim (\rho )}) \to C(\tilde {U}, V_\rho ^{\oplus \dim (\rho )})$ by

\[ \tilde{\mathcal{M}}_{\xi, \rho}(H)(u) = \sum_{\substack{(j, k)\\ u' = \sigma^{-(j, k)}(u)}} e^{(f_{(j, k)}^{(a)} + ib\tau_{(j, k)})(u')} \rho(\vartheta^{(j, k)}(u')^{{-}1}) H(u') \]

for all $u \in \tilde {U}$ and $H \in C(\tilde {U}, V_\rho ^{\oplus \dim (\rho )})$. For all $k \in \mathbb {N}$, its $k$th iteration is

(4)\begin{equation} \tilde{\mathcal{M}}_{\xi, \rho}^{k}(H)(u) = \sum_{\substack{\alpha: \operatorname{len}(\alpha) = k\\ u' = \sigma^{-\alpha}(u)}} e^{f_\alpha^{(a)}(u')} \rho_b(\Phi^{\alpha}(u')^{{-}1}) H(u') \end{equation}

for all $u \in \tilde {U}$ and $H \in C(\tilde {U}, V_\rho ^{\oplus \dim (\rho )})$. Again, we denote $\tilde {\mathcal {L}}_\xi = \tilde {\mathcal {M}}_{\xi , 1}$ and, using the restriction map $|_U$, we get the corresponding normalized operators $\mathcal {M}_{\xi , \rho }: C(U, V_\rho ^{\oplus \dim (\rho )}) \to C(U, V_\rho ^{\oplus \dim (\rho )})$ and $\mathcal {L}_\xi : C(U, \mathbb {C}) \to C(U, \mathbb {C})$. With this normalization, for all $a \in \mathbb {R}$, the maximal simple eigenvalue of $\mathcal {L}_a$ is $1$ with eigenvector ${h_a}/{h_0}$. Moreover, we have $\mathcal {L}_0^{*}(\nu _U) = \nu _U$.

We fix some related constants. By perturbation theory of operators (see [Reference KatoKat95, Chapter 7] and [Reference Parry and PollicottPP90, Proposition 4.6]), we can fix $a_0' > 0$ such that the map $[-a_0', a_0'] \to \mathbb {R}$ defined by $a \mapsto \lambda _a$ and the map $[-a_0', a_0'] \to C(\tilde {U}, \mathbb {R})$ defined by $a \mapsto h_a$ are Lipschitz. We then fix $A_f > 0$ such that $|f_{(j, k)}^{(a)}(u) - f_{(j, k)}^{(0)}(u)| \leq A_f|a|$ for all admissible pairs $(j, k)$, $u \in \tilde {\mathtt {C}}[j, k]$, and $|a| \leq a_0'$. Fix $\bar {\tau } = \max _{(j, k)} \sup _{u \in \tilde {\mathtt {C}}[j, k]} \tau _{(j, k)}(u)$ and $\underline {\tau } = \min _{(j, k)} \inf _{u \in \tilde {\mathtt {C}}[j, k]} \tau _{(j, k)}(u)$. Fix

\[ T_0 >{}\max\biggl(\max_{(j, k)} \|\tau_{(j, k)}\|_{C^{1}}, \max_{(j, k)} \sup_{|a| \leq a_0'} \big\|f_{(j, k)}^{(a)}\big\|_{C^{1}}, \max_{(j, k)} \big\|\vartheta^{(j, k)}\big\|_{C^{1}}\biggr) \]

which is possible by [Reference Petkov and StoyanovPS16, Lemma 4.1].

5.3 Spectral bounds with holonomy

We first introduce some norms and seminorms. Let $\rho \in \hat {M}$ and $H \in C(U, V_\rho ^{\oplus \dim (\rho )})$. We will write $\|H\| \in C(U, \mathbb {R})$ for the function defined by $\|H\|(u) = \|H(u)\|_2$ for all $u \in U$, and if $\rho = 1$, we will write $|H| \in C(U, \mathbb {R})$ for the function defined by $|H|(u) = |H(u)| \in \mathbb {R}$ for all $u \in U$. We define $\|H\|_\infty = \sup \|H\|$. We use similar notation if the domain is $\tilde {U}$. We define the Lipschitz seminorm and the Lipschitz norm by

\[ \operatorname{Lip}_d(H) = \sup_{\substack{u, u' \in U\\ \text{such that } u \neq u'}} \frac{\|H(u) - H(u')\|_2}{d(u, u')}, \quad \|H\|_{\operatorname{Lip}(d)} = \|H\|_\infty + \operatorname{Lip}_d(H), \]

respectively.

As we will mostly use the $C^{1}$ norm, we avoid defining the $C^{k}$ norm for integers $k > 1$. Let $Y$ be a Riemannian manifold and $H \in C^{1}(\tilde {U}, Y)$. We define the $C^{1}$ seminorm and the $C^{1}$ norm by

\[ |H|_{C^{1}} = \sup_{u \in \tilde{U}}\|(dH)_u\|_{\mathrm{op}}, \quad \|H\|_{C^{1}} = \|H\|_\infty + |H|_{C^{1}}, \]

respectively. We can define another useful norm by

\[ \|H\|_{1, b} = \|H\|_\infty + \frac{1}{\max(1, |b|)} |H|_{C^{1}}. \]

Henceforth, by differentiable function spaces on $\tilde {U}$ or its derived suspension spaces, such as $C^{1}(\tilde {U}, Y)$, we will always mean the space of $C^{1}$ functions whose $C^{1}$ norm is bounded.

For all $\rho \in \hat {M}$, we define the Banach spaces

\[ \mathcal{V}_\rho(U) = C^{\operatorname{Lip}(d)}\bigl(U, V_\rho^{{\oplus} \dim(\rho)}\bigr),\quad \mathcal{V}_\rho(\tilde{U}) = C^{1}\bigl(\tilde{U}, V_\rho^{{\oplus} \dim(\rho)}\bigr). \]

Now we can state the following theorem regarding spectral bounds of transfer operators with holonomy.

Theorem 5.3 There exist $\eta > 0$, $C > 0$, $a_0 > 0$, and $b_0 > 0$ such that for all $\xi \in \mathbb {C}$ with $|a| < a_0$, if $(b, \rho ) \in \hat {M}_0(b_0)$, then for all $k \in \mathbb {N}$ and $H \in \mathcal {V}_\rho (\tilde {U})$, we have

\[ \big\|\tilde{\mathcal{M}}_{\xi, \rho}^{k}(H)\big\|_2 \leq Ce^{-\eta k} \|H\|_{1, \|\rho_b\|}. \]

We reduce Theorem 5.3 to Theorem 5.4 which captures the mechanism of Dolgopyat's method in our setting. Similar theorems have appeared in [Reference DolgopyatDol98, Reference NaudNau05, Reference StoyanovSto11, Reference Oh and WinterOW16]. The main difference from previous works is that we deal with holonomy.

We define the cone

\[ K_B(\tilde{U}) = \{h \in C^{1}(\tilde{U}, \mathbb{R}): h > 0, \|(dh)_u\|_{\mathrm{op}} \leq Bh(u) \text{ for all } u \in \tilde{U}\}. \]

Proof Proof that Theorem 5.4 implies Theorem 5.3

Fix $m \in \mathbb {N}, a_0 > 0, b_0 > 0, E > 0$ to be the ones from Theorem 5.4 and $\tilde {\eta } \in (0, 1)$ to be the $\eta$ from Theorem 5.4. Fix

\[ B = \sup_{|a| \leq a_0, \rho \in \hat{M}} \big\|\tilde{\mathcal{M}}_{\xi, \rho}\big\|_{\mathrm{op}} \leq \sup_{|a| \leq a_0} \big\|\tilde{\mathcal{L}}_\xi\big\|_{\mathrm{op}} \leq Ne^{T_0}, \]

viewing the transfer operators as operators on $L^{2}(\tilde {U}, V_\rho ^{\oplus \dim (\rho )})$ and $L^{2}(\tilde {U}, \mathbb {R})$, respectively. Fix $\eta = ({-\log (\tilde {\eta })})/{m}$ and $C = B^{m} \tilde {\eta }^{-1}$. Let $\xi \in \mathbb {C}$ with $|a| < a_0$. Suppose $(b, \rho ) \in \hat {M}_0(b_0)$. Let $k \in \mathbb {N}$ and $H \in \mathcal {V}_\rho (\tilde {U})$. The theorem is trivial if $H = 0$, so suppose that $H \neq 0$. First set $h_0 \in K_{E\|\rho _b\|}(\tilde {U})$ to be the positive constant function defined by $h_0(u) = \|H\|_{1, \|\rho _b\|}$ for all $u \in \tilde {U}$. Denote $H_0 = H$. Then $H_0$ and $h_0$ satisfy properties (3)(1a) and (3)(1b) in Theorem 5.4. Thus, Theorem 5.4 allows us to inductively obtain $h_j = \mathcal {N}_{a, J_{j - 1}}^{H_{j - 1}}(h_{j - 1}) \in K_{E\|\rho _b\|}(\tilde {U})$ for some $J_{j - 1} \in \mathcal {J}(b, \rho )$ and $H_j = \tilde {\mathcal {M}}_{\xi , \rho }^{m}(H_{j - 1})$ which satisfy properties (3)(2a) and (3)(2b) in Theorem 5.4, for all $j \in \mathbb {N}$. Now using property (2) in Theorem 5.4, we have $\bigl\|\tilde {\mathcal {M}}_{\xi , \rho }^{jm}(H)\bigr\|_2 \leq \|h_j\|_2 \leq \tilde {\eta }^{j}\|h_0\|_2 = \tilde {\eta }^{j}\|H\|_{1, \|\rho _b\|}$ for all $j \in \mathbb {Z}_{\geq 0}$. Writing $k = jm + l$ for some $j \in \mathbb {Z}_{\geq 0}$ and $0 \leq l < m$, we have

\[ \big\|\tilde{\mathcal{M}}_{\xi, \rho}^{k}(H)\big\|_2 \leq B^{l}\big\|\tilde{\mathcal{M}}_{\xi, \rho}^{jm}(H)\big\|_2 \leq B^{l}\tilde{\eta}^{j}\|H\|_{1, \|\rho_b\|} \leq Ce^{-\eta k} \|H\|_{1, \|\rho_b\|}. \]

6.1 Local non-integrability condition

First, we will define a map related to Brin–Pesin moves [Reference Brin and PesinBP74, Reference BrinBri82] which will be needed for the LNIC in our setting. We choose unique isometric lifts $\tilde {\mathsf {R}}_j = [\tilde {\mathsf {U}}_j, \mathsf {S}_j] \subset \operatorname {T}^{1}(\mathbb {H}^{n})$ of $\tilde {R}_j$ for all $j \in \mathcal {A}$. Define $\tilde {\mathsf {R}} = \bigsqcup _{j \in \mathcal {A}} \tilde {\mathsf {R}}_j$ and $\tilde {\mathsf {U}} = \bigsqcup _{j \in \mathcal {A}} \tilde {\mathsf {U}}_j$. For all $u \in \tilde {R}$, let $\tilde {u} \in \tilde {\mathsf {R}}$ denote the unique lift in $\tilde {\mathsf {R}}$. We then lift the section $F$ to $\mathsf {F}: \bigsqcup _{\gamma \in \Gamma } \gamma \tilde {\mathsf {R}} \to \operatorname {F}(\mathbb {H}^{n})$ in the natural way.

Definition 6.1 (Associated sequence of frames)

Let $z_1 \in \tilde {R}_1$ be the center. Consider some sequence of tangent vectors $(z_1, z_2, z_3, z_4, z_1) \in (\tilde {R}_1)^{5}$ such that $z_2 \in S_1$, $z_4 \in \tilde {U}_1$ and $z_3 = [z_4, z_2]$. Its lift to the universal cover is $(\tilde {z}_1, \tilde {z}_2, \tilde {z}_3, \tilde {z}_4, \tilde {z}_1) \in (\tilde {\mathsf {R}}_1)^{5} \subset \operatorname {T}^{1}(\mathbb {H}^{n})^{5} \cong (G/M)^{5}$. We define an associated sequence of frames to be the unique sequence $(g_1, g_2, \dotsc , g_5) \in \operatorname {F}(\mathbb {H}^{n})^{5} \cong G^{5}$ where

\begin{align*} g_1 &= \mathsf{F}(\tilde{z}_1), \\ g_2 &= \mathsf{F}(\tilde{z}_2) \in g_1N^{-} \text{ such that } g_2M = \tilde{z}_2 \in \operatorname{T}^{1}(\mathbb{H}^{n}) \cong G/M, \\ g_3 &\in g_2N^{+} \text{ such that } g_3a_tM = \tilde{z}_3 \in \operatorname{T}^{1}(\mathbb{H}^{n}) \cong G/M \text{ for some } t \in (-\underline{\tau}, \underline{\tau}), \\ g_4 &\in g_3N^{-} \text{ such that } g_4a_tM = \tilde{z}_4 \in \operatorname{T}^{1}(\mathbb{H}^{n}) \cong G/M \text{ for some } t \in (-\underline{\tau}, \underline{\tau}), \\ g_5 &\in g_4N^{+} \text{ such that } g_5a_tM = \tilde{z}_1 \in \operatorname{T}^{1}(\mathbb{H}^{n}) \cong G/M \text{ for some } t \in (-\underline{\tau}, \underline{\tau}).\end{align*}

We continue using the notation in the above definition. Define the subsets

\begin{align*} N_1^{+} &= \{n^{+} \in N^{+}: F(z_1)n^{+} \in F(\tilde{U}_1)\} \subset N^{+}, \\ N_1^{-} &= \{n^{-} \in N^{-}: F(z_1)n^{-} \in F(S_1)\} \subset N^{-}, \end{align*}

where the first is open and the second is compact. Now, if the above sequence $(z_1, z_2, z_3, z_4, z_1)$ corresponds to some $n^{+} \in N_1^{+}$ and some $n^{-} \in N_1^{-}$ such that $F(z_4) = F(z_1)n^{+}$ and $F(z_2) = F(z_1)n^{-}$ respectively, then we can define the map $\Xi : N_1^{+} \times N_1^{-} \to AM$ by

\[ \Xi(n^{+}, n^{-}) = g_5^{{-}1}g_1 \in AM. \]

To view it as a function of the first coordinate for a fixed $n^{-} \in N_1^{-}$, we write $\Xi _{n^{-}}: N_1^{+} \to AM$.

Let $z_1 \in \tilde {R}_1$ be the center. Let $j \in \mathbb {N}$ and $\alpha = (\alpha _0, \alpha _2, \dotsc , \alpha _{j - 1}, 1)$ be an admissible sequence. By following definitions, there exists an element which we denote by $n_\alpha \in N_1^{-}$ such that

\[ F(\mathcal{P}^{j}(\sigma^{-\alpha}(z_1))) = F(z_1)n_\alpha. \]

This is well defined since $\sigma ^{-\alpha }(z_1) \in \mathtt {C}[\alpha ] \subset U$.

In order to derive the LNIC in Proposition 6.5, we first provide a few useful lemmas regarding $\Xi$.

Lemma 6.2 Let $j \in \mathbb {N}$, $\alpha = (\alpha _0, \alpha _1, \dotsc , \alpha _{j - 1}, 1)$ be an admissible sequence, and $n^{-} = n_\alpha \in N_1^{-}$. Let $u \in \tilde {U}_1$ and $n^{+} \in N_1^{+}$ such that $F(u) = F(z_1)n^{+}$ where $z_1 \in \tilde {R}_1$ is the center. Then we have

\[ \Xi(n^{+}, n^{-}) = \Phi^{\alpha}(\sigma^{-\alpha}(z_1))^{{-}1}\Phi^{\alpha}(\sigma^{-\alpha}(u)). \]

Proof. Let $z_1 \in \tilde {R}_1$ be the center. Let $j \in \mathbb {N}$, $\alpha = (\alpha _0, \alpha _1, \dotsc , \alpha _{j - 1}, 1)$ be an admissible sequence and $n^{-} = n_\alpha \in N_1^{-}$. Let $u \in \tilde {U}_1$ and $n^{+} \in N_1^{+}$ such that $F(u) = F(z_1)n^{+}$. Let $s \in S_1$ such that $F(s) = F(z_1)n^{-}$. To calculate $\Xi (n^{+}, n^{-})$, we first consider $(z_1, z_2, z_3, z_4, z_1) = (z_1, s, [u, s], u, z_1) \in (\tilde {R}_1)^{5}$ and then compute the associated sequence of frames $(g_1, g_2, g_3, g_4, g_5) \in G^{5}$. Firstly, $g_1 = \mathsf {F}(\tilde {z}_1)$. Then using definitions, we have

\[ g_2 = g_1n^{-} = \mathsf{F}(\tilde{z}_1)n^{-} = \mathsf{F}(\mathcal{P}^{j}(\sigma^{-\alpha}(\tilde{z}_1))) = \mathsf{F}(\sigma^{-\alpha}(\tilde{z}_1))\Phi^{\alpha}(\sigma^{-\alpha}(z_1)). \]

Now $g_3 = g_2n_2$ for some $n_2 \in N^{+}$. So

\[ g_3 = \mathsf{F}(\sigma^{-\alpha}(\tilde{z}_1))\Phi^{\alpha}(\sigma^{-\alpha}(z_1))n_2 = \mathsf{F}(\sigma^{-\alpha}(\tilde{z}_1)) n_2' \Phi^{\alpha}(\sigma^{-\alpha}(z_1)), \]

where $n_2' = \Phi ^{\alpha }(\sigma ^{-\alpha }(z_1)) n_2 \Phi ^{\alpha }(\sigma ^{-\alpha }(z_1))^{-1} \in N^{+}$. But the frame $\mathsf {F}(\sigma ^{-\alpha }(\tilde {z}_1)) n_2'$ must be based at $\tilde {v} \in \gamma \tilde {\mathsf {U}}_{\alpha _0}$ for some $\gamma \in \Gamma$ such that $\tilde {v}a_{\tau _\alpha (\sigma ^{-\alpha }(\tilde {z}_1)) + t} = [\tilde {u}, \tilde {s}]$ for some $t \in (-\underline {\tau }, \underline {\tau })$. So $\tilde {v} = \sigma ^{-\alpha }(\tilde {u})$ and $t = \tau _\alpha (\sigma ^{-\alpha }(\tilde {u})) - \tau _\alpha (\sigma ^{-\alpha }(\tilde {z}_1))$. Moreover, $\mathsf {F}(\sigma ^{-\alpha }(\tilde {z}_1)) n_2' = \mathsf {F}(\sigma ^{-\alpha }(\tilde {u}))$ and hence $g_3 = \mathsf {F}(\sigma ^{-\alpha }(\tilde {u}))\Phi ^{\alpha }(\sigma ^{-\alpha }(z_1))$. From definitions, $\tilde {v}a_{\tau _\alpha (\sigma ^{-\alpha }(\tilde {u}))} = [\tilde {u}, \tilde {s}]$ implies $\mathsf {F}(\sigma ^{-\alpha }(\tilde {u})) = \mathsf {F}([\tilde {u}, \tilde {s}])\Phi ^{\alpha }(\sigma ^{-\alpha }(u))^{-1}$. Thus

\[ g_3 = \mathsf{F}([\tilde{u}, \tilde{s}])\Phi^{\alpha}(\sigma^{-\alpha}(u))^{{-}1}\Phi^{\alpha}(\sigma^{-\alpha}(z_1)). \]

Now $g_4 = g_3n_3$ for some $n_3 \in N^{+}$. So

\begin{align*} g_4 &= \mathsf{F}([\tilde{u}, \tilde{s}])\Phi^{\alpha}(\sigma^{-\alpha}(u))^{{-}1}\Phi^{\alpha}(\sigma^{-\alpha}(z_1))n_3\\ &= \mathsf{F}([\tilde{u}, \tilde{s}])n_3'\Phi^{\alpha}(\sigma^{-\alpha}(u))^{{-}1}\Phi^{\alpha}(\sigma^{-\alpha}(z_1)), \end{align*}

where $n_3' = \Phi ^{\alpha }(\sigma ^{-\alpha }(u))^{-1}\Phi ^{\alpha }(\sigma ^{-\alpha }(z_1)) n_3 (\Phi ^{\alpha }(\sigma ^{-\alpha }(u))^{-1}\Phi ^{\alpha }(\sigma ^{-\alpha }(z_1)))^{-1} \in N^{-}$. By similar arguments, the frame $\mathsf {F}([\tilde {u}, \tilde {s}])n_3'$ must be based at $\tilde {u} \in \tilde {\mathsf {U}}_1$. Thus

\[ g_4 = \mathsf{F}(\tilde{u})\Phi^{\alpha}(\sigma^{-\alpha}(u))^{{-}1}\Phi^{\alpha}(\sigma^{-\alpha}(z_1)). \]

Finally, $g_5 = g_4n_4$ for some $n_4 \in N^{+}$. So

\[ g_5 = \mathsf{F}(\tilde{u})\Phi^{\alpha}(\sigma^{-\alpha}(u))^{{-}1}\Phi^{\alpha}(\sigma^{-\alpha}(z_1))n_4 = \mathsf{F}(\tilde{u})n_4'\Phi^{\alpha}(\sigma^{-\alpha}(u))^{{-}1}\Phi^{\alpha}(\sigma^{-\alpha}(z_1)), \]

where $n_4' = \Phi ^{\alpha }(\sigma ^{-\alpha }(u))^{-1}\Phi ^{\alpha }(\sigma ^{-\alpha }(z_1)) n_4 (\Phi ^{\alpha }(\sigma ^{-\alpha }(u))^{-1}\Phi ^{\alpha }(\sigma ^{-\alpha }(z_1)))^{-1} \in N^{-}$. Again by similar arguments, the frame $\mathsf {F}(\tilde {u})n_4'$ must be based at $\tilde {z}_1 \in \tilde {\mathsf {U}}_1$. Thus

\[ g_5 = \mathsf{F}(\tilde{z}_1)\Phi^{\alpha}(\sigma^{-\alpha}(u))^{{-}1}\Phi^{\alpha}(\sigma^{-\alpha}(z_1)). \]

Then by definition, we have the calculation

\[ \Xi(n^{+}, n^{-}) = g_5^{{-}1}g_1 =\Phi^{\alpha}(\sigma^{-\alpha}(z_1))^{{-}1}\Phi^{\alpha}(\sigma^{-\alpha}(u)). \]

Recall from definitions that $e \in N_1^{+}$ where $N_1^{+} \subset N^{+}$ is an open subset and hence $\operatorname {T}_e(N_1^{+}) = \operatorname {T}_e(N^{+}) = \mathfrak {n}^{+}$. Note that $AMN^{+}N^{-} \subset G$ is an open dense subset and hence we have the vector space decomposition $\mathfrak {g} = \mathfrak {a} \oplus \mathfrak {m} \oplus \mathfrak {n}^{+} \oplus \mathfrak {n}^{-}$. Denote the projection onto $\mathfrak {a} \oplus \mathfrak {m}$ with respect to this decomposition by $\pi : \mathfrak {g} \to \mathfrak {a} \oplus \mathfrak {m}$. We then have the following lemma where $\epsilon _0$ is as in § 3.1 and $N_{1, \epsilon _0}^{-} = \{n^{-} \in N^{-}: F(z_1)n^{-} \in F(W_{\epsilon _0}^{\mathrm {ss}}(z_1))\}$ where $z_1 \in \tilde {R}_1$ is the center.

Lemma 6.3 For all $n^{-} \in N_1^{-}$, we have

\[ (d\Xi_{n^{-}})_e = \pi \circ \operatorname{Ad}_{n^{-}}|_{\mathfrak{n}^{+}} \circ (dh_{n^{-}})_e, \]

where $h_{n^{-}}: N_1^{+} \to N^{+}$ is a diffeomorphism onto its image which is also smooth in $n^{-} \in N_{1, \epsilon _0}^{-}$ and satisfies $h_e = \operatorname {Id}_{N_1^{+}}$. Consequently, its image is $(d\Xi _{n^{-}})_e(\mathfrak {n}^{+}) = \pi (\operatorname {Ad}_{n^{-}}(\mathfrak {n}^{+})) \subset \mathfrak {a} \oplus \mathfrak {m}$.

Proof. Let $z_1 \in \tilde {R}_1$ be the center. Let $n^{-} \in N_1^{-}$. For all $n^{+} \in N_1^{+}$, consider the sequence $(z_1, z_2, z_3(n^{+}), z_4(n^{+}), z_1) \in (\tilde {R}_1)^{5}$ such that $F(z_2) = F(z_1)n^{-}$ and $F(z_4(n^{+})) = F(z_1)n^{+}$ and let $(g_1, g_2, g_3(n^{+}), g_4(n^{+}), g_5(n^{+}))$ be the associated sequence of frames. Then for all $n^{+} \in N_1^{+}$, the frame $g_5(n^{+})a_t$ is based at $\tilde {z}_1$ for some $t \in (-\underline {\tau }, \underline {\tau })$. By transversality of the smooth foliations, the implicit function theorem on a coordinate chart gives smooth functions $f_{n^{-}, 2}: N_1^{+} \to N^{+}, f_{n^{-}, 3}: N_1^{+} \to N^{-}$, and $f_{n^{-}, 4}: N_1^{+} \to N^{+}$ which are also smooth in $n^{-} \in N_{1, \epsilon _0}^{-}$ such that

\[ g_5(n^{+}) = g_1n^{-}f_{n^{-}, 2}(n^{+})f_{n^{-}, 3}(n^{+})f_{n^{-}, 4}(n^{+}). \]

Note that $f_{n^{-}, 2}$ is a diffeomorphism onto its image. Let $h_{n^{-}} = h_{n^{-}, 2}: N_1^{+} \to N^{+}, h_{n^{-}, 3}: N_1^{+} \to N^{-}$, and $h_{n^{-}, 4}: N_1^{+} \to N^{+}$ be smooth functions which are also smooth in $n^{-} \in N_{1, \epsilon _0}^{-}$ defined by the group inverses $h_{n^{-}, j}(n^{+}) = f_{n^{-}, j}(n^{+})^{-1}$ for all $n^{+} \in N_1^{+}$ and $2 \leq j \leq 4$. Then $h_{n^{-}, 2}$ is a diffeomorphism onto its image and

\[ \Xi_{n^{-}}(n^{+}) = g_5(n^{+})^{{-}1}g_1 = h_{n^{-}, 4}(n^{+})h_{n^{-}, 3}(n^{+})h_{n^{-}, 2}(n^{+})(n^{-})^{{-}1}. \]

Note that $h_{n^{-}, 2}(e) = h_{n^{-}, 4}(e) = e$ and $h_{n^{-}, 3}(e) = n^{-}$. Let $m_g^{\mathrm {R}}: G \to G$ be the right multiplication map and $C_g: G \to G$ be the conjugation map by $g \in G$. Let $\omega \in \mathfrak {n}^{+}$. The product rule gives

\begin{align*} (d\Xi_{n^{-}})_e(\omega) &={} \bigl(\big(dm_{(n^{-})^{{-}1}}^{\mathrm{R}}\big)_{n^{-}} \circ \big(dm_{n^{-}}^{\mathrm{R}}\big)_e\bigr)((dh_{n^{-}, 4})_e(\omega)) + \bigl(dm_{(n^{-})^{{-}1}}^{\mathrm{R}}\bigr)_{n^{-}}((dh_{n^{-}, 3})_e(\omega))\\ &\quad {}+ (dC_{n^{-}})_e((dh_{n^{-}, 2})_e(\omega)) \\ &={} (dh_{n^{-}, 4})_e(\omega) + \bigl(dm_{(n^{-})^{{-}1}}^{\mathrm{R}}\bigr)_{n^{-}}((dh_{n^{-}, 3})_e(\omega)) + \operatorname{Ad}_{n^{-}}((dh_{n^{-}, 2})_e(\omega)). \end{align*}

Since $(d\Xi _{n^{-}})_e(\omega ) \in \mathfrak {a} \oplus \mathfrak {m}$, $(dh_{n^{-}, 4})_e(\omega ) \in \mathfrak {n}^{+}$, and $(dm_{(n^{-})^{-1}}^{\mathrm {R}})_{n^{-}}((dh_{n^{-}, 3})_e(\omega )) \in \mathfrak {n}^{-}$, we have $(d\Xi _{n^{-}})_e(\omega ) = \pi (\operatorname {Ad}_{n^{-}}((dh_{n^{-}, 2})_e(\omega )))$. Noting that $(dh_{n^{-}, 2})_e$ surjects onto $\mathfrak {n}^{+}$, we also have $(d\Xi _{n^{-}})_e(\mathfrak {n}^{+}) = \pi (\operatorname {Ad}_{n^{-}}(\mathfrak {n}^{+}))$.

Throughout the paper we often use the upper half space model $\mathbb {H}^{n} \cong \{(x_1, x_2, \dotsc , x_n) \in \mathbb {R}^{n}: x_n > 0\}$ with boundary at infinity $\partial _\infty (\mathbb {H}^{n}) \cong \{(x_1, x_2, \dotsc , x_n) \in \mathbb {R}^{n}: x_n = 0\} \cup \{\infty \} \cong \mathbb {R}^{n - 1} \cup \{\infty \}$. We also use the isomorphism $\operatorname {T}(\mathbb {H}^{n}) \cong \mathbb {H}^{n} \times \mathbb {R}^{n}$. Let $(e_1, e_2, \dotsc , e_n)$ be the standard basis on $\mathbb {R}^{n}$. We assume without loss of generality that the identifications are made such that the reference vector is $v_o = (e_n, e_n)$ and the reference frame is $F_o = ((e_n, e_1), (e_n, e_2), \dotsc , (e_n, e_n))$, where the first entries of the tangent vectors are their basepoints. Let $d_{\mathrm {E}}$ denote the Euclidean distance. Let $B^{\mathrm {E}}_\epsilon (x) \subset \mathbb {R}^{n - 1}$ denote the open Euclidean ball of radius $\epsilon > 0$ centered at $x \in \mathbb {R}^{n - 1}$.

Lemma 6.4 There exist $n_1^{-}, n_2^{-}, \dotsc , n_{j_{\mathrm {m}}}^{-} \in N_1^{-}$ for some $j_{\mathrm {m}} \in \mathbb {N}$ and $\delta > 0$ such that if $\eta _1^{-}, \eta _2^{-}, \dotsc , \eta _{j_{\mathrm {m}}}^{-} \in N_1^{-}$ with $d_{N^{-}}(\eta _j^{-}, n_j^{-}) \leq \delta$ for all $1 \leq j \leq j_{\mathrm {m}}$, then

\[ \sum_{j = 1}^{j_{\mathrm{m}}} \pi\big(\operatorname{Ad}_{\eta_j^{-}}(\mathfrak{n}^{+})\big) = \mathfrak{a} \oplus \mathfrak{m}. \]

Proof. First we show that $\sum _{n^{-} \in N^{-}} \pi (\operatorname {Ad}_{n^{-}}(\mathfrak {n}^{+})) = \mathfrak {a} \oplus \mathfrak {m}$, that is, there exist $n_1^{-}, n_2^{-}, \dotsc , n_{j_{\mathrm {m}}}^{-} \in N^{-}$ for some $j_{\mathrm {m}} \in \mathbb {N}$ such that $\sum _{j = 1}^{j_{\mathrm {m}}} \pi (\operatorname {Ad}_{n_j^{-}}(\mathfrak {n}^{+})) = \mathfrak {a} \oplus \mathfrak {m}$. We use the formula

\[ \operatorname{Ad}_{e^{n^{-}}}(n^{+}) = e^{\mathrm{ad}_{n^{-}}}(n^{+}) = \sum_{j = 0}^{\infty} \frac{1}{j!}(\mathrm{ad}_{n^{-}})^{j}(n^{+}) = n^{+} + [n^{-}, n^{+}] + \frac{1}{2!}[n^{-}, [n^{-}, n^{+}]] + \dotsb \]

for all $n^{+} \in \mathfrak {n}^{+}$ and $n^{-} \in \mathfrak {n}^{-}$. Note that $\exp : \mathfrak {n}^{-} \to N^{-}$ is surjective since $N^{-} \cong \mathbb {R}^{n - 1}$. Examining the formula above term by term, our first objective follows if we show that there exist $n_1^{+}, n_2^{+}, \dotsc , n_{j_{\mathrm {m}}}^{+} \in \mathfrak {n}^{+}$ and $n_1^{-}, n_2^{-}, \dotsc , n_{j_{\mathrm {m}}}^{-} \in \mathfrak {n}^{-}$ for some $j_{\mathrm {m}} \in \mathbb {N}$ such that $\operatorname {span}\{[n_j^{-}, n_j^{+}] \in \mathfrak {g}: j \in \{1, 2, \dotsc , j_{\mathrm {m}}\}\} = \mathfrak {a} \oplus \mathfrak {m}$ and $[n_j^{-}, [n_j^{-}, n_j^{+}]] \in \mathfrak {n}^{-}$ for all $1 \leq j \leq j_{\mathrm {m}}$.

We use the upper half space model. Recall $M = \operatorname {Stab}_G(v_o) \cong \operatorname {SO}(n - 1)$ whose elements act on $\mathbb {H}^{n}$ by rotations in $\mathbb {R}^{n}$ which keep the $n$th coordinate fixed. It is a fact that for any chosen basis of $\mathbb {R}^{n}$, any rotation can be expressed as a composition of planar rotations where the planes are generated by any two distinct basis vectors. Using our standard basis $(e_1, e_2, \dotsc , e_n)$, this means that $M$ is generated by the subgroups $M_{j, k} = \{m \in M: m(e_l) = e_l \text { for all } l \in \{1, 2, \dotsc , n\} \setminus \{j, k\}\} \cong \operatorname {SO}(2)$ for all $1 \leq j, k \leq n - 1$ with $j \neq k$. Then we have the corresponding sum of vector spaces $\mathfrak {m} = \sum _{\substack {1 \leq j, k \leq n - 1,\ j \neq k}} \mathfrak {m}_{j, k}$ where $\mathfrak {m}_{j, k} \cong \mathfrak {so}(2) \cong \mathbb {R}$ is the Lie algebra of $M_{j, k}$ for all $1 \leq j, k \leq n - 1$ with $j \neq k$. Let $1 \leq j, k \leq n - 1$ with $j \neq k$ and consider the totally geodesic submanifold $P_{j, k} = \operatorname {span}(e_j, e_k, e_n) \cap \mathbb {H}^{n} \subset \mathbb {H}^{n}$. Let $H_{j, k} = \{g \in G: g(P_{j, k}) = P_{j, k}\} < G$ be the subgroup of isometries of $P_{j, k}$. Then, $P_{j, k} \cong \mathbb {H}^{3}$ is the upper half space of $\operatorname {span}(e_j, e_k, e_n) \cong \mathbb {R}^{3}$ which induces the canonical identifications of Lie groups and their Lie algebras, $H_{j, k} \cong \operatorname {PSL}_2(\mathbb {C})$ and $\mathfrak {h}_{j, k} \cong \mathfrak {sl}_2(\mathbb {C})$. Let $N_{j, k}^{+} = \{n^{+} \in N^{+}: n^{+} e_n \in P_{j, k}\} = N^{+} \cap H_{j, k}$ and $N_{j, k}^{-} = \{n^{-} \in N^{-}: n^{-} e_n \in P_{j, k}\} = N^{-} \cap H_{j, k}$ be the expanding and contracting horospherical subgroups of $H_{j, k}$, with corresponding Lie algebras $\mathfrak {n}_{j, k}^{+} \subset \mathfrak {n}^{+}$ and $\mathfrak {n}_{j, k}^{-} \subset \mathfrak {n}^{-}$, respectively. Note that $M_{j, k}, N_{j, k}^{+}, N_{j, k}^{-} < H_{j, k}$ and hence also $\mathfrak {m}_{j, k}, \mathfrak {n}_{j, k}^{+}, \mathfrak {n}_{j, k}^{-} \subset \mathfrak {h}_{j, k}$. Now it suffices to show that $[\mathfrak {n}_{j, k}^{-}, \mathfrak {n}_{j, k}^{+}] = \mathfrak {a} \oplus \mathfrak {m}_{j, k}$ and $[\mathfrak {n}_{j, k}^{-}, \mathfrak {a} \oplus \mathfrak {m}_{j, k}] = \mathfrak {n}_{j, k}^{-}$. This is simply a matter of calculations. Using the canonical identifications, we can explicitly write the Lie algebras as

\begin{align*} \mathfrak{a} &\cong \biggl\{ \begin{pmatrix} t & 0 \\ 0 & -t \end{pmatrix} : t \in \mathbb{R}\biggr\}, \quad \mathfrak{m}_{j, k} \cong \biggl\{ \begin{pmatrix} i\theta & 0 \\ 0 & -i\theta \end{pmatrix} : \theta \in \mathbb{R}\biggr\}, \\ \mathfrak{n}_{j, k}^{+} &\cong \biggl\{ \begin{pmatrix} 0 & 0 \\ n^{+} & 0 \end{pmatrix} : n^{+} \in \mathbb{C}\biggr\}, \quad \mathfrak{n}_{j, k}^{-} \cong \biggl\{ \begin{pmatrix} 0 & n^{-} \\ 0 & 0 \end{pmatrix} : n^{-} \in \mathbb{C}\biggr\}. \end{align*}

Now for all $n^{+} \in \mathfrak {n}_{j, k}^{+}$ and $n^{-} \in \mathfrak {n}_{j, k}^{-}$, using the corresponding matrices $\big (\begin {smallmatrix}0 & n^{-} \\ 0 & 0 \end {smallmatrix} \big )$ and $\big (\begin {smallmatrix} 0 & 0 \\ n^{+} & 0\end {smallmatrix} \big )$, we have the calculation

\[ \begin{pmatrix} 0 & n^{-} \\ 0 & 0 \end{pmatrix} \begin{pmatrix} 0 & 0 \\ n^{+} & 0 \end{pmatrix} - \begin{pmatrix} 0 & 0 \\ n^{+} & 0 \end{pmatrix} \begin{pmatrix} 0 & n^{-} \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} n^{-} n^{+} & 0 \\ 0 & -n^{-} n^{+} \end{pmatrix} \]

for the matrix corresponding to $[n^{-}, n^{+}]$. Thus, $[\mathfrak {n}_{j, k}^{-}, \mathfrak {n}_{j, k}^{+}] = \mathfrak {a} \oplus \mathfrak {m}_{j, k}$. Similarly, for all $n^{-} \in \mathfrak {n}_{j, k}^{-}$ and $a + m \in \mathfrak {a} \oplus \mathfrak {m}_{j, k}$, using the corresponding matrices $\big ( \begin {smallmatrix}0 & n^{-} \\ 0 & 0 \end {smallmatrix}\big )$ and $\big (\begin {smallmatrix} z & 0 \\ 0 & -z \end {smallmatrix} \big )$, we have the calculation

\[ \begin{pmatrix} 0 & n^{-} \\ 0 & 0 \end{pmatrix} \begin{pmatrix} z & 0 \\ 0 & -z \end{pmatrix} - \begin{pmatrix} z & 0 \\ 0 & -z \end{pmatrix} \begin{pmatrix} 0 & n^{-} \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & -2zn^{-} \\ 0 & 0 \end{pmatrix} \]

for the matrix corresponding to $[n^{-}, a + m]$. Thus, $[\mathfrak {n}_{j, k}^{-}, \mathfrak {a} \oplus \mathfrak {m}_{j, k}] = \mathfrak {n}_{j, k}^{-}$.

Now we show that we can choose $n_1^{-}, n_2^{-}, \dotsc , n_{j_{\mathrm {m}}}^{-}$ to be in $N_1^{-}$. By way of contradiction, suppose this is false. Then $V = \sum _{n^{-} \in N_1^{-}} \pi (\operatorname {Ad}_{n^{-}}(\mathfrak {n}^{+})) \subset \mathfrak {a} \oplus \mathfrak {m}$ is a proper subspace. Hence, there is a functional $L: \mathfrak {a} \oplus \mathfrak {m} \to \mathbb {R}$ with $\ker (L) = V$. But we have already proved that $\sum _{n^{-} \in N^{-}} \pi (\operatorname {Ad}_{n^{-}}(\mathfrak {n}^{+})) = \mathfrak {a} \oplus \mathfrak {m}$ and so we can choose $\hat {n}^{+} \in \mathfrak {n}^{+}$ and $\hat {n}^{-} \in N^{-}$ such that $\pi (\operatorname {Ad}_{\hat {n}^{-}}(\hat {n}^{+})) \in \mathfrak {a} \oplus \mathfrak {m} \setminus V$. Let $z_1 \in \tilde {R}_1$ be the center. Without loss of generality, we can assume $\mathsf {F}(\tilde {z}_1) = e$. Consider the map $N^{-} \to \mathbb {R}^{n - 1}$ defined by $n^{-} \mapsto (n^{-})^{-}$ which is just mapping the frame $n^{-}$ to its backward limit point $(n^{-})^{-} \in \mathbb {R}^{n - 1} \subset \partial _\infty (\mathbb {H}^{n})$. Its inverse is a Lie group isomorphism $\mathbb {R}^{n - 1} \to N^{-}$. Since $\exp : \mathbb {R}^{n - 1} \to \mathbb {R}^{n - 1}$ is simply the identity map, the previous Lie group isomorphism induces the Lie algebra isomorphism $\mathbb {R}^{n - 1} \to \mathfrak {n}^{-}$ where we can still view the domain as the boundary at infinity $\mathbb {R}^{n - 1} \subset \partial _\infty (\mathbb {H}^{n})$. Denote the image of $x \in \mathbb {R}^{n - 1}$ under this map by $n_x^{-} \in \mathfrak {n}^{-}$. Now consider the function $P: \mathbb {R}^{n - 1} \to \mathbb {R}$ defined by $P(x) = L (\pi (\operatorname {Ad}_{e^{n_x^{-}}}(\hat {n}^{+})))$ for all $x \in \mathbb {R}^{n - 1}$. Since using the basis $(e_1, e_2, \dotsc , e_n)$ above was arbitrary, we can see from the calculations that in fact $\pi (\operatorname {Ad}_{e^{n^{-}}}(n^{+})) = [n^{-}, n^{+}]$ for all $n^{+} \in \mathfrak {n}^{+}$ and $n^{-} \in \mathfrak {n}^{-}$, and so in particular $\pi (\operatorname {Ad}_{e^{n_x^{-}}}(\hat {n}^{+})) = [n_x^{-}, \hat {n}^{+}] = -\mathrm {ad}_{\hat {n}^{+}}(n_x^{-})$ for all $x \in \mathbb {R}^{n - 1}$. Then $P(x) = -L(\mathrm {ad}_{\hat {n}^{+}}(n_x^{-}))$ for all $x \in \mathbb {R}^{n - 1}$, which is a composition of linear maps. Now $\Lambda (\Gamma ) \cap B_\epsilon ^{\mathrm {E}}((\tilde {z}_1)^{-}) \subset \ker (P)$ for some $\epsilon > 0$, but $P$ is non-trivial because $P(x_0) \neq 0$ where $x_0 \in \mathbb {R}^{n - 1}$ such that $\hat {n}^{-} = e^{n_{x_0}^{-}}$. But this is a contradiction by [Reference WinterWin15, Proposition 3.12] since $\Gamma < G$ is Zariski dense. Finally, it is clear that $n_1, n_2, \dotsc , n_{j_{\mathrm {m}}}$ satisfying the result is an open set and so the lemma follows.

We fix $j_{\mathrm {m}} \in \mathbb {N}$ as in Lemma 6.4 for the rest of the paper.

The following proposition is the required LNIC in our setting.

Proposition 6.5 (LNIC)

There exist $\epsilon \in (0, 1)$, $m_0 \in \mathbb {N}$, $j_{\mathrm {m}} \in \mathbb {N}$, and an open neighborhood $\mathcal {U} \subset \tilde {U}_1$ of the center $z_1 \in \tilde {R}_1$ with $\mathcal {U} \cap \Omega \subset U_1$ such that for all $m \geq m_0$, there exist sections $v_j = \sigma ^{-\alpha _j}: \tilde {U}_1 \to \tilde {U}_{\alpha _{j, 0}}$ for some admissible sequences $\alpha _j = (\alpha _{j, 0}, \alpha _{j, 1}, \dotsc , \alpha _{j, m - 1}, 1)$ for all integers $0 \leq j \leq j_{\mathrm {m}}$ such that for all $u \in \mathcal {U}$ and $\omega \in \mathfrak {a} \oplus \mathfrak {m}$ with $\|\omega \| = 1$, there exist $1 \leq j \leq j_{\mathrm {m}}$ and $Z \in \operatorname {T}_u(\tilde {U}_1)$ with $\|Z\| = 1$ such that

\[ |\langle (d\operatorname{BP}_{j, u})_u(Z), \omega \rangle| \geq \epsilon, \]

where we define $\operatorname {BP}_j: \tilde {U}_1 \times \tilde {U}_1 \to AM$ by

\[ \operatorname{BP}_j(u, u') = \Phi^{\alpha_0}(v_0(u))^{{-}1} \Phi^{\alpha_0}(v_0(u')) \Phi^{\alpha_j}(v_j(u'))^{{-}1} \Phi^{\alpha_j}(v_j(u)) \]

and we denote $\operatorname {BP}_{j, u} = \operatorname {BP}_j(u, \cdot )$ for all $u, u' \in \tilde {U}_1$ and $1 \leq j \leq j_{\mathrm {m}}$. Moreover, the images $v_0(\mathcal {U}), v_1(\mathcal {U}), \dotsc , v_{j_{\mathrm {m}}}(\mathcal {U})$ are mutually disjoint.

Proof. By Lemma 6.4, there exist distinct non-trivial $n_1^{-}, n_2^{-}, \dotsc , n_{j_{\mathrm {m}}}^{-} \in N_1^{-}$ for some $j_{\mathrm {m}} \in \mathbb {N}$ and a sufficiently small $\delta _1 > 0$ such that for all $\eta _1^{-}, \eta _2^{-}, \dotsc , \eta _{j_{\mathrm {m}}}^{-} \in N_1^{-}$ with $d_{N^{-}}(\eta _j^{-}, n_j^{-}) \leq \delta _1$ for all $1 \leq j \leq j_{\mathrm {m}}$, we have

\[ \sum_{j = 1}^{j_{\mathrm{m}}} \pi\big(\operatorname{Ad}_{\eta_j^{-}}(\mathfrak{n}^{+})\big) = \mathfrak{a} \oplus \mathfrak{m}. \]

This property allows us to define a positive constant

(5)\begin{equation} \epsilon_1 ={}\inf_{\substack{\eta_1^{-}, \eta_2^{-}, \dotsc, \eta_{j_{\mathrm{m}}}^{-} \in N_1^{-} \text{ such that}\\ d_{N^{-}}(\eta_j^{-}, n_j^{-}) \leq \delta_1 \, \forall j \in \{1, 2, \dotsc, j_{\mathrm{m}}\}}} \inf_{\substack{\omega \in \mathfrak{a} \oplus \mathfrak{m}\\ \text{such that}\\ \|\omega\| = 1}} \sup_{\substack{n^{+} \in \mathfrak{n}^{+}\\ \text{such that } \|n^{+}\| = 1,\\ j \in \{1, 2, \dotsc, j_{\mathrm{m}}\}}} \bigl|\bigl\langle \pi\bigl(\operatorname{Ad}_{\eta_j^{-}}(n^{+})\bigr), \omega \bigr\rangle\bigr|. \end{equation}

Let $z_1 \in \tilde {R}_1$ be the center. Define the diffeomorphism $\psi : \tilde {U}_1 \to N_1^{+}$ by $\psi (u) = n^{+}$ such that $F(u) = F(z_1)n^{+}$ for all $u \in \tilde {U}_1$. Fix $C_1 = \|(d\psi )_{z_1}\|_{\mathrm {op}} > 0$. Recall $h_{n^{-}}: N_1^{+} \to N^{+}$ from Lemma 6.3 which is a diffeomorphism onto its image which is also smooth in $n^{-} \in N_{1, \epsilon _0}^{-}$ and satisfies $h_e = \operatorname {Id}_{N_1^{+}}$. Since $N_1^{-}$ is compact, there exists $C_2 > 1$ such that ${1}/{C_2} \leq \|(dh_{n^{-}})_e\|_{\mathrm {op}} \leq C_2$ for all $n^{-} \in N_1^{-}$. Fix $\epsilon \in (0, \min ({C_1\epsilon _1}/{4C_2}, 1))$ and $\epsilon _2 \in (0, {\epsilon }/{C_1C_2})$. Observe that $(\pi \circ \operatorname {Ad}_{n^{-}})|_{\mathfrak {n}^{+}}: \mathfrak {n}^{+} \to \mathfrak {a} \oplus \mathfrak {m}$ is linear and also smooth in $n^{-} \in N_{1, \epsilon _0}^{-}$. Since $(\pi \circ \operatorname {Ad}_e)|_{\mathfrak {n}^{+}} = 0$, it follows that there exists $\delta _2 > 0$ such that $\|(\pi \circ \operatorname {Ad}_{n^{-}})|_{\mathfrak {n}^{+}}\|_{\mathrm {op}} \leq \epsilon _2$ for all $n^{-} \in N^{-}$ with $d_{N^{-}}(n^{-}, e) \leq \delta _2$. Without loss of generality, we can assume that $\# \mathcal{A}=N\geq j_{\rm m}$. Now, using the Markov property and the topological mixing property of $T$, we can fix $m_0 \in \mathbb {N}$ such that, given any $m \geq m_0$, there exist distinct $\eta _0^{-}, \eta _1^{-}, \dotsc , \eta _{j_{\mathrm {m}}}^{-} \in N_1^{-}$ with $d_{N^{-}}(\eta _0^{-}, e) < \delta _2$ and $d_{N^{-}}(\eta _j^{-}, n_j^{-}) < \delta _1$ for all $1 \leq j \leq j_{\mathrm {m}}$, such that $\mathcal {P}^{m}(u_j) = s_j \in S_1$ with $F(s_j) = F(z_1)\eta _j^{-}$, for some $u_j \in \hat {U}_{\alpha _{j, 0}}$ and $\alpha _{j, 0} \in \mathcal {A}$ for all $0\leq j\leq j_{\rm m}$, where $\alpha_{0,0},\alpha_{1,0}, \dotsc, \alpha_{j_{\rm m},0}$ are distinct. Then for all $0 \leq j \leq j_{\mathrm {m}}$, the associated trajectory of the geodesic flow of $u_j$ through the Markov section gives a corresponding admissible sequence $\alpha _j = (\alpha _{j, 0}, \alpha _{j, 1}, \dotsc , \alpha _{j, m - 1}, 1)$. We define the sections $v_j = \sigma ^{-\alpha _j}: \tilde {U}_1 \to \tilde {U}_{\alpha _{j, 0}}$ for all $0 \leq j \leq j_{\mathrm {m}}$. Note that the last criterion of the proposition will be automatically satisfied. We define $\operatorname {BP}_j: \tilde {U}_1 \times \tilde {U}_1 \to AM$ for all $1 \leq j \leq j_{\mathrm {m}}$ as in the proposition. The equation

\[ F(\mathcal{P}^{m}(\sigma^{-\alpha_j}(z_1))) = F(\mathcal{P}^{m}(v_j(z_1))) = F(\mathcal{P}^{m}(u_j)) = F(s_j) = F(z_1)\eta_j^{-} \]

implies that $\eta _j^{-} = n_{\alpha _j}$ for all $0 \leq j \leq j_{\mathrm {m}}$. Hence by Lemma 6.2, we have

(6)\begin{align} \operatorname{BP}_j(u, u') = \Xi_{\eta_0^{-}}(\psi(u))^{{-}1}\Xi_{\eta_0^{-}}(\psi(u'))\Xi_{\eta_j^{-}}(\psi(u'))^{{-}1}\Xi_{\eta_j^{-}}(\psi(u)) \end{align}

for all $u, u' \in \tilde {U}_1$ and $1 \leq j \leq j_{\mathrm {m}}$. Starting with the case $u = z_1 \in \tilde {U}_1$, we have

\[ \operatorname{BP}_{j, z_1}(u') = \Xi_{\eta_0^{-}}(\psi(u'))\Xi_{\eta_j^{-}}(\psi(u'))^{{-}1} \quad \text{for all }u' \in \tilde{U}_1\ \text{and}\ 1 \leq j \leq j_{\mathrm{m}}. \]

By Lemma 6.3, the differential $(d\operatorname {BP}_{j, z_1})_{z_1}: \operatorname {T}_{z_1}(\tilde {U}_1) \to \mathfrak {a} \oplus \mathfrak {m}$ is given by

\begin{align*} (d\operatorname{BP}_{j, z_1})_{z_1}(Z) &= \big(d\Xi_{\eta_0^{-}}\big)_e((d\psi)_{z_1}(Z)) - \big(d\Xi_{\eta_j^{-}}\big)_e((d\psi)_{z_1}(Z))\\ &= \pi\big(\operatorname{Ad}_{\eta_0^{-}}\big(\big(dh_{\eta_0^{-}}\big)_e((d\psi)_{z_1}(Z))\big)\big) - \pi\big(\operatorname{Ad}_{\eta_j^{-}}\big(\big(dh_{\eta_j^{-}}\big)_e((d\psi)_{z_1}(Z))\big)\big) \end{align*}

for all $Z \in \operatorname {T}_{z_1}(\tilde {U}_1)$ and $1 \leq j \leq j_{\mathrm {m}}$. Define $S(\mathfrak{a} \oplus \mathfrak{m})= \{\omega \in \mathfrak {a} \oplus \mathfrak {m}: \|\omega \| = 1\}$ and let ${\omega \in S(\mathfrak{a} \oplus \mathfrak{m})}$. Using the above formula, we have

(7)\begin{align} |\langle (d\operatorname{BP}_{j, z_1})_{z_1}(Z), \omega \rangle| & \geq{}\bigl|\bigl\langle \pi\big(\operatorname{Ad}_{\eta_j^{-}}\big(\big(dh_{\eta_j^{-}}\big)_e((d\psi)_{z_1}(Z))\big)\big), \omega \bigr\rangle\bigr|\nonumber\\ &\quad {}- \bigl|\bigl\langle \pi\big(\operatorname{Ad}_{\eta_0^{-}}\big(\big(dh_{\eta_0^{-}}\big)_e((d\psi)_{z_1}(Z))\big)\big), \omega \bigr\rangle\bigr| \end{align}

for all $Z \in \operatorname {T}_{z_1}(\tilde {U}_1)$ and $1 \leq j \leq j_{\mathrm {m}}$. We first deal with the first term in (7). Since $d_{N^{-}}(\eta _j^{-}, n_j^{-}) < \delta _1$ for all $1 \leq j \leq j_{\mathrm {m}}$, using (5), there exist $1 \leq j_\omega \leq j_{\mathrm {m}}$ and $n_\omega ^{+} \in \mathfrak {n}^{+}$ with $\|n_\omega ^{+}\| = 1$ such that $|\langle \pi (\operatorname {Ad}_{\eta _{j_\omega }^{-}}(n_\omega ^{+})), \omega \rangle | \geq \epsilon _1$. Since $(dh_{\eta _{j_\omega }^{-}})_e$ and $(d\psi )_{z_1}$ are invertible linear maps, there exists $Z_\omega \in \operatorname {T}_{z_1}(\tilde {U}_1)$ with $\|Z_\omega \| = 1$ such that $(dh_{\eta _{j_\omega }^{-}})_e((d\psi )_{z_1}(Z_\omega ))$ is a scalar multiple of $n_\omega ^{+}$. The operator norm bounds on the linear maps give

\[ \bigl|\bigl\langle \pi\bigl(\operatorname{Ad}_{\eta_{j_\omega}^{-}}\bigl(\bigl(\bigl(dh_{\eta_{j_\omega}^{-}}\bigr)_e((d\psi)_{z_1} (Z_\omega))\bigr)\bigr), \omega \bigr\rangle\bigr| \geq C_1C_2^{{-}1}\epsilon_1 \geq 4\epsilon. \]

Now we turn to the second term in (7). Since $d_{N^{-}}(\eta _0^{-}, e) < \delta _2$, we use the Cauchy–Schwarz inequality and then again the operator norm bounds to get

\[ \bigl|\bigl\langle \pi\bigl(\operatorname{Ad}_{\eta_0^{-}}\bigl(\bigl(dh_{\eta_0^{-}}\bigr)_e((d\psi)_{z_1}(Z_\omega))\bigr)\bigr), \omega \bigr\rangle\bigr| \leq C_1C_2\epsilon_2 \leq \epsilon. \]

With these bounds, we conclude that for all $\omega \in S(\mathfrak{a} \oplus \mathfrak{m})$, there exist $1 \leq j_\omega \leq j_{\mathrm {m}}$ and $Z_\omega \in \operatorname {T}_{z_1}(\tilde {U}_1)$ with $\|Z_\omega \| = 1$ such that $|\langle (d\operatorname {BP}_{j_\omega , z_1})_{z_1}(Z_\omega ), \omega \rangle | \geq 3\epsilon$. Since the map $S(\mathfrak{a} \oplus \mathfrak{m}) \to \mathbb {R}$ defined by $\omega \mapsto |\langle \omega ', \omega \rangle |$ is continuous for all $\omega ' \in \mathfrak {a} \oplus \mathfrak {m}$ and $S(\mathfrak{a} \oplus \mathfrak{m})$ is compact, there exist some absolute $k_0\in \mathbb{N}$ and $\omega _1, \omega _2, \dotsc , \omega _{k_0} \in S(\mathfrak{a} \oplus \mathfrak{m})$ contained in corresponding open sets $V_1, V_2, \dotsc , V_{k_0} \subset S(\mathfrak{a} \oplus \mathfrak{m})$ which cover $S(\mathfrak{a} \oplus \mathfrak{m})$ such that $|\langle (d\operatorname {BP}_{j_{\omega _k}, z_1})_{z_1}(Z_{\omega _k}), \omega \rangle | \geq 2\epsilon$ for all $\omega \in \overline {V_k}$ and $1 \leq k \leq k_0$. Now, for all $1 \leq k \leq k_0$, extend $Z_{\omega _k} \in \operatorname {T}_{z_1}(\tilde {U}_1)$ to any smooth unit vector field $Z_k: \mathcal{U}'\to \operatorname {T}(\tilde {U}_1)$ for some open set $\mathcal{U}' \subset \tilde {U}_1$ containing $z_1$, that is, $Z_k$ satisfies $Z_k(z_1) = Z_{\omega _k}$ and $\|Z_k(u)\| = 1$ for all $u \in \mathcal{U}'$. We can ensure that their $C^1$ norms are always bounded above by an absolute constant. Since $\operatorname {BP}_{j_{\omega _k}}$ is smooth, the map $\mathcal{U}' \times \overline {V_k} \to \mathbb {R}$ defined by $(u, \omega ) \mapsto |\langle (d\operatorname {BP}_{j_{\omega _k}, u})_u(Z_k(u)), \omega \rangle |$ is also smooth for all $1 \leq k \leq k_0$. Hence, by compactness of $\overline {V_k}$, there exists an open subset $\mathcal {U} \subset \mathcal{U}'$ containing $z_1$ such that we can extend the inequality to

\[ |\langle (d\operatorname{BP}_{j_{\omega_k}, u})_u(Z_k(u)), \omega \rangle| \geq \epsilon \quad \text{for all }u \in \mathcal{U},\ \omega \in \overline{V_k},\ \text{and}\ 1 \leq k \leq k_0. \]

Note that $\mathcal{U}$ is absolute and, in particular, is not dependent on the sections $v_0, v_1, \dotsc, v_{j_{\mathrm{m}}}$ nor their length m due to (6) and compactness of $N_1^-$. Shrinking it so that $\mathcal{U} \cap \Omega \subset U_1$ finishes the proof.

Fix $\varepsilon _2 \in (0, 1)$, $m_0 \in \mathbb {N}$, $j_{\mathrm {m}} \in \mathbb {N}$, and the open subset $\mathcal {U} \subset \tilde {U}_1$ containing the center $z_1 \in \tilde {R}_1$ to be the $\epsilon$, $m_0$, $j_{\mathrm {m}}$, and $\mathcal {U}$ provided by Proposition 6.5.

6.2 Non-concentration property

In the upper half space model, applying an appropriate isometry, we assume that the vectors in $\tilde {\mathsf {U}}_1$ have direction $\pi _2(\tilde {\mathsf {U}}_1) = -e_n$ and their basepoints lie on the hyperplane $\langle \pi _1(\tilde {\mathsf {U}}_1), e_n\rangle = 1$. We will often view the limit set as $\Lambda (\Gamma ) \subset \mathbb {R}^{n - 1} \cup \{\infty \}$ in the rest of the paper. The following proposition is the required NCP.

Proof. By way of contradiction, suppose the proposition is false. Then for all $j \in \mathbb {N}$, taking $\delta _j = {1}/{j}$, there exist $\epsilon _j \in (0, 1)$, $w_j \in \mathbb {R}^{n - 1}$ with $\|w_j\| = 1$, and $x_j \in \Lambda (\Gamma ) \cap \mathbb {R}^{n - 1}$ such that $|\langle y - x_j, w_j \rangle | \leq \epsilon _j \delta _j = \epsilon _j/j$ for all $y \in \Lambda (\Gamma ) \cap B_{\epsilon _j}^{\mathrm {E}}(x_j)$. Hence, we can rewrite this as

(8)\begin{equation} \Lambda(\Gamma) \cap B_{\epsilon_j}^{\mathrm{E}}(x_j) \subset \biggl\{y \in \mathbb{R}^{n - 1}: |\langle y - x_j, w_j \rangle| \leq \frac{\epsilon_j}{j}\biggr\} \quad \text{for all }j \in \mathbb{N}. \end{equation}

We want to use the self-similarity property of the fractal set $\Lambda (\Gamma )$. Note that $A < G$ is such that $a_t$ acts on $\overline {\mathbb {H}^{n}}$ by dilation by $e^{t}$ for all $t \in \mathbb {R}$, and elements of $N^{-}$ act on $\overline {\mathbb {H}^{n}}$ by translation. For all $x \in \mathbb {R}^{n - 1}$, denote by $n_x^{-} \in N^{-}$ the element which acts on $\overline {\mathbb {H}^{n}}$ by translation by $x$. For all $j \in \mathbb {N}$, we have $(n_{x_j}^{-})^{+} = \infty \in \Lambda (\Gamma )$ and $(n_{x_j}^{-})^{-} = x_j \in \Lambda (\Gamma )$, and hence $\Gamma n_{x_j}^{-} \in \Omega _{\operatorname {F}} \subset \Gamma \backslash G$ which we recall is compact. Also recalling that $\Omega _{\operatorname {F}}$ is invariant under the frame flow, we have $n_{x_j}^{-} a_t \in \Gamma \Omega _0$ for all $t \in \mathbb {R}$ and $j \in \mathbb {N}$, where $\Omega _0 \subset G$ is some compact subset. Then for all $j \in \mathbb {N}$, setting $t_j = \log (\epsilon _j)$, there exist $\gamma _j \in \Gamma$ and $g_j \in \Omega _0$ such that $n_{x_j}^{-} a_{t_j} = \gamma _j g_j$. Now, for all $j \in \mathbb {N}$, we have $g_j a_{-t_j} n_{-x_j}^{-} = \gamma _j^{-1}$ whose action on $\partial _\infty (\mathbb {H}^{n})$ preserves $\Lambda (\Gamma )$. This captures the self-similarity property of the fractal set $\Lambda (\Gamma )$.

Now applying $g_j a_{-t_j} n_{-x_j}^{-}$ in (8) gives

\[ \Lambda(\Gamma) \cap g_j \cdot B_1^{\mathrm{E}}(0) \subset g_j \cdot \biggl\{y \in \mathbb R^{n - 1}: |\langle y, w_j \rangle| \leq \frac{1}{j}\biggr\} \quad \text{for all}\ j \in \mathbb N. \]

By compactness, we can pass to subsequences so that $\lim _{j \to \infty } w_j = w \in \mathbb {R}^{n - 1}$ with $\|w\| = 1$ and $\lim _{j \to \infty } g_j = g \in \Omega _0$. In the limit $j \to \infty$, we have $\Lambda(\Gamma) \cap g \cdot B_1^{\mathrm{E}}(0) \subset g \cdot \bigl\{y \in \mathbb R^{n - 1}: \langle y, w \rangle = 0\bigr\}$, which contradicts [Reference WinterWin15, Proposition 3.12] since $\Gamma < G$ is Zariski dense.

Fix $\varepsilon _3 \in (0, 1)$ henceforth to be the $\delta$ provided by Proposition 6.6.