2.1 Relatively hyperbolic groups

We first recall definitions and basic properties of relatively hyperbolic groups. More details are given in part I [Reference DussauleDus22]. Consider a finitely generated group $\Gamma$ acting discretely and by isometries on a proper and geodesic hyperbolic space $(X,d)$. We denote the limit set of $\Gamma$ by $\Lambda \Gamma$, that is, the set of accumulation points in the Gromov boundary $\partial X$ of an orbit $\Gamma \cdot o$, $o\in X$. A point $\xi \in \Lambda \Gamma$ is called conical if there is a sequence $(\gamma _{n})$ of $\Gamma$ and distinct points $\xi _1,\xi _2$ in $\Lambda \Gamma$ such that $\gamma _{n}\xi$ converges to $\xi _1$ and $\gamma _{n}\zeta$ converges to $\xi _2$ for all $\zeta \neq \xi$ in $\Lambda \Gamma$. A point $\xi \in \Lambda \Gamma$ is called parabolic if its stabilizer in $\Gamma$ is infinite, fixes exactly $\xi$ in $\Lambda \Gamma$ and contains no loxodromic element. The stabilizer of a parabolic limit point is called a (maximal) parabolic subgroup. A parabolic limit point $\xi$ in $\Lambda \Gamma$ is called bounded parabolic if is stabilizer in $\Gamma$ is infinite and acts cocompactly on $\Lambda \Gamma \setminus \{\xi \}$. Finally, the action is called geometrically finite if the limit set only consists of conical limit points and bounded parabolic limit points.

One might choose different spaces $X$ on which $\Gamma$ can act geometrically finitely. However, different choices of $X$ give rise to equivariantly homeomorphic limit sets $\Lambda \Gamma$. We call this limit set the Bowditch boundary of $\Gamma$ and we denote it by $\partial _B\Gamma$.

Let $\Gamma$ be a relatively hyperbolic group with respect to a collection of subgroups $\Omega$. Let $\Omega _0$ be a set of representatives of conjugacy classes of elements of $\Omega$. Such a set $\Omega _0$ is necessarily finite, according to [Reference BowditchBow12, Proposition 6.15]. Fix a finite generating set $S$ for $\Gamma$. Denote by $\hat {\Gamma }$ the Cayley graph associated with the infinite generating set consisting of the union of $S$ and of all parabolic subgroups $\mathcal {H}\in \Omega _0$. Endowed with the graph distance, that we write $\hat {d}$, the graph $\hat {\Gamma }$ is hyperbolic.

A relative geodesic is a geodesic in the graph $\hat {\Gamma }$. A relative quasi-geodesic is a path of adjacent vertices in $\hat {\Gamma }$, which is a quasi-geodesic for the distance $\hat {d}$. Following Osin [Reference OsinOsi06], a path is called without backtracking if once it has left a coset $\gamma \mathcal {H}$, for $\mathcal {H}\in \Omega _0$, it never goes back to it. Relative geodesic and relative quasi-geodesic satisfy the following property, called the BCP property.

We need to study both geodesics in $\operatorname {Cay}(\Gamma,S)$ and relative geodesics in the following. We use the following terminology. Let $\alpha$ be a geodesic in $\operatorname {Cay}(\Gamma,S)$ and let $\eta _1,\eta _2\geq 0$. A point $\gamma$ on $\alpha$ is called an $(\eta _1,\eta _2)$-transition point if for any coset $\gamma _0\mathcal {H}$ of a parabolic subgroup, the part of $\alpha$ consisting of points at distance at most $\eta _2$ from $\gamma$ is not contained in the $\eta _1$-neighborhood of $\gamma _0\mathcal {H}$.

Transition points are of great importance in relatively hyperbolic groups. They stay close to points on relative geodesics in the following sense.

2.2 Relatively automatic groups

Hyperbolic groups are known to be strongly automatic, meaning that for every generating set $S$, there exists a finite directed graph $\mathcal {G}=(V,E,v_*)$ encoding geodesics. It is quite implicit in Farb's work [Reference FarbFar94, Reference FarbFar98], although not formally stated, that relatively hyperbolic groups are relatively automatic in the following sense.

Let $\Gamma$ be a finitely generated group and let $\Omega$ be a collection of subgroups invariant by conjugacy and such that there is a finite set $\Omega _0$ of conjugacy classes representatives of subgroups in $\Omega$.

Note that the union $S\cup \bigcup _{\mathcal {H}\in \Omega _0}\mathcal {H}$ is not required to be a disjoint union. Actually, the intersection of two distinct subgroups $\mathcal {H},\mathcal {H}'\in \Omega _0$ can be non-empty. Also note that we require the vertex set $V$ to be finite. However, the set of edges is infinite, except if all subgroups $\mathcal {H}$ in $\Omega _0$ are finite.

If there exists a relative automatic structure for $\Gamma$ with respect to $\Omega _0$ and $S$, we say that $\Gamma$ is automatic relative to $\Omega _0$ and $S$. The following was proved in part I.

Along the proof of this theorem, a lot of technical lemmas about relative geodesics were proved in [Reference DussauleDus22]. We use some of them repeatedly in this paper, so we restate them for convenience. We use the same notation as previously and so we fix a relatively hyperbolic group $\Gamma$ and a finite set $\Omega _0$ of conjugacy classes of parabolic subgroups.

2.3 Spectrally degenerate measures

We now consider a group $\Gamma$, hyperbolic relative to a collection of subgroups $\Omega$. We fix a finite collection $\Omega _0=\{\mathcal {H}_1,\ldots,\mathcal {H}_N\}$ of representatives of conjugacy classes of $\Omega$. We assume that $\Gamma$ is non-elementary. Let $\mu$ be a probability measure on $\Gamma$, $R_{\mu }$ the inverse of the spectral radius of the $\mu$-random walk and $G(\gamma,\gamma '|r)$ the associated Green function, evaluated at $r$, for $r\in [0,R_{\mu }]$. If $\gamma =\gamma '$, we simply use the notation $G(r)=G(\gamma,\gamma |r)=G(e,e|r)$.

We denote by $p_k$ the first return transition kernel to $\mathcal {H}_k$. Namely, if $h,h'\in \mathcal {H}_k$, then $p_k(h,h')$ is the probability that the $\mu$-random walk, starting at $h$, eventually comes back to $\mathcal {H}_k$ and that its first return to $\mathcal {H}_k$ is at $h'$. In other words,

\[ p_k(h,h')=\mathbb{P}_h(\exists n\geq 1, X_n=h',X_1,\ldots,X_{n-1}\notin \mathcal{H}_k). \]

More generally, for $r\in [0,R_{\mu }]$, we denote by $p_{k,r}$ the first return transition kernel to $\mathcal {H}_k$ for $r\mu$. Precisely, if $h,h'\in \mathcal {H}_k$, then

\[ p_{k,r}(h,h')=\sum_{n\geq 1}\sum_{\underset{\notin \mathcal{H}_k}{\gamma_1,\ldots,\gamma_{n-1}}}r^{n}\mu(h^{-1}\gamma_1)\mu(\gamma_1^{-1}\gamma_2)\dots\mu(\gamma_{n-2}^{-1}\gamma_{n-1})\mu(\gamma_{n-1}^{-1}h'). \]

We then denote by $p_{k,r}^{(n)}$ the convolution powers of this transition kernel, by $G_{k,r}(h,h'|t)$ the associated Green function, evaluated at $t$ and by $R_k(r)$ the inverse of the associated spectral radius, that is, the radius of convergence of the power series $t\mapsto G_{k,r}(h,h'|t)$. For simplicity, write $R_k=R_k(R_{\mu })$. If $h=h'$, we simply write $G_{k,r}(t)=G_{k,r}(h,h|t)=G_{k,r}(e,e|t)$.

According to [Reference DussauleDus22, Lemma 3.4], for any $r\in [0,R_{\mu }]$, for any $k\in \{1,\ldots,N\}$,

\[ G_{k,r}(h,h'|1)=G(h,h'|r). \]

In addition, because $\Gamma$ is non-elementary, it contains a free group and, hence, is non-amenable. It follows from a result of Guivarc'h (see [Reference Guivarc'hGui80, p. 85, Remark b)]) that $G(R_{\mu })<+\infty$. Thus, $G_{k,R_{\mu }}(1)<+\infty$. In particular, $R_k\geq 1$.

This definition was introduced in [Reference Dussaule and GekhtmanDG21] to study the homeomorphism type of the Martin boundary at the spectral radius. As explained in [Reference DussauleDus22, § 3.3], it should be thought of as a notion of a spectral gap between the spectral radius of the random walk on the whole group and the spectral radii of the induced walks on the parabolic subgroups. Very roughly speaking, the random walk is not spectrally degenerate if the driving probability measure $\mu$ does not give too much weight to the parabolic subgroups. Let us give more details on this intuition now. For simplicity, we only consider the case of free products which was studied by Woess in [Reference WoessWoe86] and by Candellero and Gilch in [Reference Candellero and GilchCG12]. We first need to introduce some notation.

Let $\Gamma =\Gamma _0*\Gamma _1$ be a free product of two groups. Then, $\Gamma$ is hyperbolic relative to the conjugates of the free factors $\Gamma _0$ and $\Gamma _1$. Consider a symmetric probability measure $\mu _0$, respectively $\mu _1$, whose finite support generates $\Gamma _0$, respectively $\Gamma _1$. For any $0< \beta < 1$, set

\[ \mu=\beta\mu_1+(1-\beta)\mu_0. \]

Then, $\mu$ is a symmetric probability measure whose finite support generates $\Gamma$. Such a probability measure is called adapted to the free product structure: the random walk driven by $\mu$ can only move inside one of the free factors at each step.

In this situation, the Green function $G_\mu$ of $\mu$ and the Green function $G_{\mu _0}$ of $\mu _0$ can be related by the following formula. For every $x,y\in \Gamma _0$, for every $r\leq R_\mu$,

(4)\begin{equation} \frac{G_\mu(x,y|r)}{G_\mu(e,e|r)}=\frac{G_{\mu_0}(x,y|\zeta_0(r))}{G_{\mu_0}(e,e|\zeta_0(r))}, \end{equation}

where $\zeta _0$ is a continuous function of $r$, see [Reference WoessWoe00, Proposition 9.18] for an explicit formula. We always have $\zeta _0(R_\mu )\leq R_{\mu _0}$. A similar formula holds for $x,y\in \Gamma _1$.

Actually, because $\mu$ is adapted to the free product structure, the first return kernel $p_{\Gamma _0,r}$ can be written as

\[ p_{\Gamma_0,r}(e,x)=(1-\beta)r\mu_0+w_0\delta_{e,x}, \]

where $w_0$ is the weight of the first return to $e$ associated to $r\mu$, starting with a step in $\Gamma _1$. Thus, [Reference WoessWoe00, Lemma 9.2] shows that for any $x,y\in \Gamma _0$,

(5)\begin{equation} G_{\Gamma_0,r}(x,y|t)=\frac{1}{1-w_0t}G_{\mu_0}\bigg(x,y \biggm| \frac{(1-\beta)rt}{1-w_0t}\bigg). \end{equation}

In particular, for $t=1$,

\[ G_{\Gamma_0,r}(x,y|1)=\frac{1}{1-w_0}G_{\mu_0}\bigg(x,y \biggm| \frac{(1-\beta)r}{1-w_0}\bigg) \]

and so we recover (4) with $\zeta (r)=({(1-\beta )r})/({1-w_0})$.

Following Woess [Reference WoessWoe00], we call the situation where $\zeta _i(R_\mu )< R_i$ (the inverse of the spectral radius of $\mu _i$) the ‘typical case’. We prove that the ‘typical case’ is exactly the case where the measure is not spectrally degenerate.

Proof. Applying (5) to $t=1+\epsilon$ and $r=R_\mu$, we obtain

\[ G_{\Gamma_0,R_\mu}(x,y|1+\epsilon)=\frac{1}{1-w_0(1+\epsilon)}G_{\mu_0}\bigg(x,y \biggm| \frac{(1-\beta)R_\mu(1+\epsilon)}{1-w_0(1+\epsilon)}\bigg). \]

If $\epsilon >0$, then

\[ \frac{(1-\beta)R_\mu(1+\epsilon)}{1-w_0(1+\epsilon)}>\frac{(1-\beta)R_\mu}{1-w_0}=\zeta_0(R_\mu). \]

Thus, there exists $t>1$ such that $G_{\Gamma _0,R_\mu }(x,y|t)$ is finite if and only if there exists $z>\zeta _0(R_\mu )$ such that $G_{\mu _0}(x,y|z)$ is finite, which concludes the proof.

Hence, in the context of adapted random walks on free products, to check whether the random walk is non-spectrally degenerate, one has to check whether $\zeta _i(R_\mu )< R_i$ or not. This problem was studied by Candellero and Gilch. More specifically, in [Reference Candellero and GilchCG12, § 7], they construct both spectrally degenerate and non-spectrally degenerate measures. In particular, in their example A, $\Gamma _i=\mathbb {Z}^{d_i}$, where $d_i\geq 5$. Then, if $\beta$ is close enough to zero, the random walk is spectrally degenerate along $\Gamma _0$. Similarly, if $\beta$ is close enough to one, the random walk is spectrally degenerate along $\Gamma _1$. In the middle case, the random walk is not spectrally degenerate.

Let us also mention their example F, where $\Gamma _0=\mathbb {Z}^{5}$ and $\Gamma _1=\mathbb {Z}^{6}$. They construct measures $\mu _0$ and $\mu _1$ such that the adapted measure $\mu$ is spectrally degenerate for every $\beta$. More precisely, they show there exists a critical parameter $\beta _c$ such that the following classification holds. For $\beta <\beta _c$, $\mu$ is spectrally degenerate along $\Gamma _0$ and is not spectrally positive recurrent. For $\beta >\beta _c$, $\mu$ is spectrally degenerate along $\Gamma _1$ and is not spectrally positive recurrent. On the other hand, for $\beta =\beta _c$, $\mu$ is spectrally positive recurrent. This yields a random walk which is spectrally degenerate, but which is spectrally positive recurrent. In this situation, one has a local limit theorem of the form (1) with $\alpha =3/2$. This leads to the conjecture that spectral positive recurrence is a sufficient condition to obtain $\alpha =3/2$.

Unfortunately, it seems difficult to construct (non-)spectrally degenerate measures on general relatively hyperbolic groups, so we do not know yet any example beyond free products, except low-dimensional Kleinian groups, according to Theorem 1.3. A reasonable approach would be to try to adapt the method of Candellero and Gilch in the following way. Consider a relatively hyperbolic group $\Gamma$ and choose a parabolic subgroup $\mathcal {H}$. Start with an admissible, finitely supported and symmetric measure $\mu _0$ on $\Gamma$ and an admissible, finitely supported and symmetric measure $\mu _\mathcal {H}$ on $\mathcal {H}$. Define then $\mu =\beta \mu _0+(1-\beta )\mu _\mathcal {H}$. One would expect $\mu$ to be spectrally degenerate along $\mathcal {H}$ for small enough $\beta$. However, to actually prove that it is spectrally degenerate, one would need to prove a sort of continuity of the Green function and its derivatives with respect to the driving measure, which would, in turn, require some new material. Similar difficulties occur when trying to construct non-spectrally degenerate measures.

Finally, let us now recall some consequences of spectral degeneracy proved in [Reference DussauleDus22]. Let $\Gamma$ be a relatively hyperbolic group. We introduce the following notation. We write

\[ I^{(k)}(r)=\sum_{\gamma^{(1)},\ldots,\gamma^{(k)}\in \Gamma}G(\gamma,\gamma^{(1)}|r)G(\gamma^{(1)},\gamma^{(2)}|r)\dots G(\gamma^{(k-1)},\gamma^{(k)}|r)G(\gamma^{(k)},\gamma'|r). \]

Then, $I^{(k)}(r)$ is related to the $k$th derivative of the Green function. For a precise statement, we refer to [Reference DussauleDus22, Lemma 3.2]. For instance, we have the following.

Lemma 2.7 [Reference DussauleDus22, Lemma 3.1]

For every $\gamma,\gamma '\in \Gamma$, for every $r\in [0,R_{\mu }]$, we have

\[ \frac{d}{dr}(rG(\gamma_1,\gamma_2|r))=\sum_{\gamma\in \Gamma}G(\gamma_1,\gamma|r)G(\gamma,\gamma_2|r). \]

If $\mathcal {H}$ is a parabolic subgroup, we also write

\[ I^{(k)}_{\mathcal{H}}(r)=\sum_{\gamma^{(1)},\ldots,\gamma^{(k)}\in \mathcal{H}}G(\gamma,\gamma^{(1)}|r)G(\gamma^{(1)},\gamma^{(2)}|r)\dots G(\gamma^{(k-1)},\gamma^{(k)}|r)G(\gamma^{(k)},\gamma'|r). \]

Once $\Omega _0=\{\mathcal {H}_1,\ldots,\mathcal {H}_N\}$ is fixed, we also write $I^{(k)}_{j}(r)=I^{(k)}_{\mathcal {H}_j}(r)$ for simplicity.

One of the main results of part I is that whenever $I^{(2)}_{\mathcal {H}}(r)<+\infty$ for every parabolic subgroup $\mathcal {H}\in \Omega _0$, then

\[ I^{(2)}(r)\asymp ( I^{(1)}(r))^{3}. \]

This is a consequence of the following result.

Proposition 2.8 [Reference DussauleDus22, Proposition 5.6]

For every $r\in [0,R_{\mu })$, we have

\[ \frac{I^{(2)}(r)}{I^{(1)}(r))^{3}}\asymp 1+\sum_{j}I^{(2)}_{j}(r). \]

In particular, if $\mu$ is non-spectrally degenerate, then

\[ I^{(2)}(r)\asymp ( I^{(1)}(r))^{3}. \]

The following results were also proved in part I. Note that we do not need to assume that $\mu$ is non-spectrally degenerate.

Lemma 2.9 [Reference DussauleDus22, Lemma 5.4]

There exists some uniform $C\geq 0$ such that for every $r\in [0,R_{\mu }]$, for every $m$,

\[ \sum_{\gamma\in \hat{S}_m}H(e,\gamma|r)\leq C. \]

Finally, we also use the following.

Proposition 2.11 [Reference DussauleDus22, Proposition 5.8]

If $\mu$ is non-spectrally degenerate, then

\[ \frac{d}{dr}_{|r=R_\mu}G(e,e|R_{\mu})=+\infty. \]

2.4 Relative Ancona inequalities

We consider a finitely generated group $\Gamma$, hyperbolic relative to a collection of subgroups $\Omega$. Let $\Omega _0$ be a finite set of representatives of conjugacy classes. We also consider a probability measure $\mu$ on $\Gamma$, whose finite support generates $\Gamma$ as a semigroup and denote by $R_{\mu }$ the inverse of its spectral radius. As soon as $\Gamma$ is non-elementary, it is non-amenable, so that $R_{\mu }>1$ according to Kesten's results [Reference KestenKes59].

In the case where $\Gamma$ is hyperbolic, Ancona proved that the Green function $G$ is roughly multiplicative along geodesics. Precisely, there exists $C\geq 1$ such that if $x,y,z$ are elements along a geodesic in this order, then

(6)\begin{equation} \frac{1}{C}G(x,y)G(y,z)\leq G(x,z)\leq C G(x,y)G(y,z). \end{equation}

See [Reference AnconaAnc88] for more details. The proof also works for the Green function evaluated at $r$, when $r< R_{\mu }$. Actually, the lower bound is always true, so that the content of Ancona inequalities really is

\[ G(x,z|r)\leq C G(x,y|r)G(y,z|r). \]

In [Reference GouëzelGou14], Gouëzel proved that these inequalities still hold at $r=R_{\mu }$ and that the constant $C$ is uniform in $r$, when the measure is symmetric. He also gave a strengthened version of them. Namely, if $x,x',y,y'$ are four points such that geodesics $[x,y]$ from $x$ to $y$ and $[x',y']$ from $x'$ to $y'$ fellow travel for a time at least $n$, then for $r\in [1,R_{\mu }]$,

(7)\begin{equation} \bigg| \frac{G(x,y|r)G(x',y'|r)}{G(x',y|r)G(x,y'|r)} -1\bigg|\leq C\rho^{n}, \end{equation}

where $C\geq 0$ and $0<\rho <1$ are uniform (see [Reference GouëzelGou14, Theorem 2.9]). This strengthened version of Ancona inequalities was proved at $r< R_{\mu }$ in [Reference Izumi, Neshveyev and OkayasuINO08]. It was also already proved at the spectral radius by Gouëzel and Lalley in the case of co-compact Fuchsian groups (see [Reference Gouëzel and LalleyGL13, Theorem 4.6]). It allowed the authors to obtain Hölder regularity for Martin kernels on the Martin boundary and then to use thermodynamic formalism to deduce local limit theorems in hyperbolic groups in [Reference GouëzelGou14, Reference Gouëzel and LalleyGL13].

Back to relatively hyperbolic groups, inequalities similar to (6) were obtained by Gekhtman, Gerasimov, Potyagailo, and Yang in [Reference Gekhtman, Gerasimov, Potyagailo and YangGGPY21]. Recall that if $\alpha$ is a geodesic in the Cayley graph $\operatorname {Cay}(\Gamma,S)$, a point on $\alpha$ is called a transition point if it is not deep in a parabolic subgroup. It is proved in [Reference Gekhtman, Gerasimov, Potyagailo and YangGGPY21] that if $x,y,z$ are elements on a geodesic in this order and if $y$ is a transition point on this geodesic, then (6) holds. The proof actually works for $G(\cdot,\cdot |r)$ whenever $r< R_{\mu }$.

Because of our automatic structure, it is more convenient for us to work with relative geodesics rather than transition points on actual geodesics. We fix a generating set $S$ and consider the Cayley graph and the graph $\hat {\Gamma }$ associated with $S$.

Definition 2.4 Let $\Gamma$ be a relatively hyperbolic group and let $\mu$ be a probability measure on $\Gamma$ with Green function $G$. Let $R_\mu$ be the inverse of the spectral radius of $\mu$. If $r\in [1,R_{\mu }]$, say that $\mu$ satisfies the weak $r$-relative Ancona inequalities if there exists $C\geq 0$ (which depends on $r$) such that for every $x,y,z\in \Gamma$ such that their images in $\hat {\Gamma }$ lie in this order on a relative geodesic,

\[ \frac{1}{C}G(x,y|r)G(y,z|r)\leq G(x,z|r)\leq C G(x,y|r)G(y,z|r). \]

Say that $\mu$ satisfies the weak relative Ancona inequalities up to the spectral radius if it satisfies the $r$-relative Ancona inequalities for every $r\in [1,R_{\mu }]$ with a constant $C$ not depending on $r$.

We also need the following enhanced version of relative Ancona inequalities.

Definition 2.6 Let $\Gamma$ be a relatively hyperbolic group and let $\mu$ be a probability measure on $\Gamma$ with Green function $G$. Let $R_\mu$ be the inverse of the spectral radius of $\mu$. If $r\in [1,R_{\mu }]$, say that $\mu$ satisfies the strong $r$-relative Ancona inequalities if for every $c\geq 0$, there exist $C\geq 0$ and $0<\rho <1$ such that if $x,x',y,y'$ are four points such that relative geodesics $[x,y]$ from $x$ to $y$ and $[x',y']$ from $x'$ to $y'$ $c$-fellow travel for a time at least $n$, then

\[ \bigg| \frac{G(x,y|r)G(x',y'|r)}{G(x',y|r)G(x,y'|r)} -1\bigg|\leq C\rho^{n}. \]

Say that $\mu$ satisfies the strong relative Ancona inequalities up to the spectral radius if it satisfies the strong $r$-relative Ancona inequalities for every $r\in [1,R_{\mu }]$ with constants $C$ and $\rho$ not depending on $r$.

The following is proved in [Reference Dussaule and GekhtmanDG21].

Actually, these inequalities are stated in [Reference Dussaule and GekhtmanDG21] using the Floyd distance, which is a suitable rescaling of the distance in $\operatorname {Cay}(\Gamma,S)$. However, [Reference Gerasimov and PotyagailoGP16, Corollary 5.10] relates the Floyd distance with transition points and Lemma 2.2 relates transition points with points on a relative geodesic. We deduce the above theorem combining these two results with [Reference Dussaule and GekhtmanDG21, Theorem 1.6].

3.1 Transfer operators with countably many symbols

We follow here the terminology of Sarig and recall some facts proved in [Reference SarigSar99, Reference SarigSar03]. We consider a countable set $\Sigma$ (the set of symbols), and a matrix $A=(a_{s,s'})_{s,s'\in \Sigma }$, with entries zeros and ones (the transition matrix). We then define

\[ \Sigma_A^{*}=\{x=(x_1,\ldots,x_n), x_i\in \Sigma, n\geq 0,\forall i, a_{x_i,x_{i+1}}=1\} \]

and

\[ \partial \Sigma_A^{*}=\{x=(x_1,\ldots,x_n,\ldots), x_i\in \Sigma, \forall i, a_{x_i,x_{i+1}}=1\}. \]

Note that in the definition of $\Sigma _A^{*}$, $n$ can be zero, so that the empty sequence, that we denote by $\emptyset$, is in $\Sigma _A^{*}$. We also define

\[ \overline{\Sigma}_A=\Sigma_A^{*}\cup \partial \Sigma_A^{*}. \]

If $s_1,\ldots,s_k\in \Sigma$, we define the cylinder $[s_1,\ldots,s_k]$ as $\{x\in \overline {\Sigma }_A,x_1=s_1,\ldots,x_k=s_k\}$.

Let $T:\overline {\Sigma }_A\rightarrow \overline {\Sigma }_A$ be given by

\begin{align*} T((x_1,\ldots,x_n))=(x_2,\ldots,x_n) \end{align*}

if $(x_1,\ldots,x_n)\in \Sigma _A^{*}$ and

\begin{align*} T((x_1,\ldots,x_n,\ldots))=(x_2,\ldots,x_n,\ldots) \end{align*}

if $(x_1,\ldots,x_n,\ldots )\in \partial \Sigma _A^{*}$. We call $T$ the shift map and we call the pair $(\overline {\Sigma }_A,T)$ a Markov shift.

We say that the Markov shift is irreducible if for every $s,s'\in \Sigma$, there exists $N_{a,b}$ such that there exists $x\in \overline {\Sigma }_A$ with $x_1=s$ and $x'\in \overline {\Sigma }_A$ with $x'_1=s'$ such that $T^{N_{a,b}}x=x'$. In other words, one can reach any cylinder $[s']$ from any cylinder $[s]$ with a finite number of iterations of the shift. We say it is topologically mixing if for every $s,s'\in \Sigma$, there exists $N_{a,b}$ such that for every $n\geq N_{a,b}$, there exists $x\in \overline {\Sigma }_A$ with $x_1=s$ and $x'\in \overline {\Sigma }_A$ with $x'_1=s'$ such that $T^{n}x=x'$.

In [Reference SarigSar99], everything is stated only using $\partial \Sigma _A^{*}$. However, up to considering a cemetery symbol $x_{{{\dagger}} }$, we can see finite sequences $(x_1,\ldots,x_n)$ in $\Sigma _A^{*}$ as infinite ones, of the form $(x_1,\ldots,x_n,x_{{{\dagger}} },\ldots,x_{{{\dagger}} },\ldots )$. Thus, we can apply the terminology and results of [Reference SarigSar99] to $\overline {\Sigma }_A$. In addition, for technical reasons, it will be convenient to assume that the empty sequence is not a preimage of itself by the shift. This can be done, for example, using a second cemetery symbol.

We also define a metric on $\overline {\Sigma }_A$, setting $d(x,y)=2^{-n}$, where $n$ is the first time that the two sequences $x$ and $y$ differ.

If $\varphi : \overline {\Sigma }_A\rightarrow \mathbb {R}$ is a function, define

\[ V_n(\varphi)=\sup \{|\varphi(x)-\varphi(y)|,x_1=y_1,\ldots,x_n=y_n\}. \]

For $\rho \in (0,1)$, such a function $\varphi$ is called $\rho$-locally Hölder continuous if it satisfies

\[ \exists C, \forall n\geq 1, V_n(\varphi)\leq C\rho^{n}. \]

It is called locally Hölder continuous if it is $\rho$-locally Hölder continuous for some $\rho$. Note that nothing is required for $V_0(\varphi )$ and, in particular, $\varphi$ can be unbounded. We can always change the metric $d$ on $\overline {\Sigma }_A$, defining $d_{\rho }(x,y)=\rho ^{n}$, where $n$ is the first time that the two sequences $x$ and $y$ differ (and $0<\rho <1$). A $\rho$-locally Hölder continuous function is then a locally Lipschitz function for this new metric.

If $\varphi$ is locally Hölder continuous, we denote by $\varphi _n=\sum _{k=0}^{n-1}\varphi \circ T^{k}$ its $n$th Birkhoff sum. We also define its transfer operator $\mathcal {L}_{\varphi }$ as

\[ \mathcal{L}_{\varphi}f(x)=\sum_{Ty=x}{e}^{\varphi(y)}f(y). \]

It acts on several spaces of functions. We are interested in one in particular, described in the following, on which the transfer operator has a spectral gap. By definition, we have

\[ (\mathcal{L}_{\varphi}^{n}f)(x)=\sum_{T^{n}y=x}{e}^{\varphi_n(y)}f(y). \]

We also define, for $s\in \Sigma$,

\begin{align*} Z_n(\varphi,s)=\sum_{\underset{x_1=s}{T^{n}x=x}}{e}^{\varphi_n(x)}. \end{align*}

For every $s$, $({1}/{n})\log Z_n(\varphi,s)$ has a limit $P(\varphi,s)$. If the Markov shift is irreducible, then it is independent of $s$ and we denote it by $P(\varphi )$. Moreover, $P(\varphi,s)>-\infty$ and if $\|\mathcal {L}_{\varphi }1\|_{\infty }<+\infty$, then $P(\varphi,s)<+\infty$. We refer to [Reference SarigSar99, Theorem 1] for a proof. Independence of $s$ is proved under the assumption that the Markov shift is topologically mixing, although the proof only requires that it is irreducible. We call $P(\varphi,s)$ the Gurevic pressure of $\varphi$ at $s$, or simply its pressure.

We say that $\varphi$ is positive recurrent if for every $s\in \Sigma$, there exist $M_{s}\geq 1$ and $\lambda _{s}>0$ such that for every large enough $n$, ${Z_n(\varphi,s)}/{\lambda _{s}^{n}}\in [M_{s}^{-1},M_{s}]$. If it is the case, then one necessarily has $\log \lambda _{s}=P(\varphi,s)$. The main result of [Reference SarigSar99] is that positive recurrence is a necessary and sufficient condition for convergence of the iterates of the transfer operator $\mathcal {L}_{\varphi }^{n}$ (see [Reference SarigSar99, Theorem 4] for a precise statement).

If the set of symbols $\Sigma$ is finite, then every Hölder continuous function is positive recurrent. Actually, we can say a little more in this case. The convergence of $\mathcal {L}_{\varphi }^{n}$ is exponentially fast. Precisely, if the Markov shift is topologically mixing, there exist $\lambda >0$, a positive function $h$ and a measure $\nu$ and constants $C\geq 0$ and $0<\theta <1$ satisfying, for all $\rho$-Hölder continuous function $f$ and all $n\in \mathbb {N}$,

\[ \bigg\|\lambda^{-n}\mathcal{L}_{\varphi}^{n}f-h\int f \,d\nu\bigg\|\leq C\theta^{n}\|f\|. \]

This is the so-called Ruelle–Perron–Frobenius theorem. Equivalently, $\lambda$ is a positive eigenvalue of the operator $\mathcal {L}_{\varphi }$ acting on the space of Hölder continuous functions and the remainder of the spectrum is contained in a disk of radius strictly smaller than $\lambda$. In other words, $\mathcal {L}_{\varphi }$ acts on this space with a spectral gap

When the set of symbols is countable, it can happen that the convergence is not exponentially fast (see [Reference SarigSar99, Example 1]). However, there are sufficient conditions for this to hold, studied by Aaronson, Denker, and Urbański among others (see [Reference Aaronson, Denker and UrbańskiADU93, Reference Aaronson and DenkerAD01]; see also [Reference GouëzelGou04]).

Definition 3.1 Say that the Markov shift $(\overline {\Sigma }_A,T)$ has finitely many images if the set

\[ \{T[s],s \in \Sigma\} \]

is finite. Equivalently, there is only a finite number of different rows (and, thus, a finite number of different columns) in the matrix $A$.

Fix $0<\rho <1$. Let $\beta$ be the partition generated by the image sets, that is, $\beta$ is the $\sigma$-algebra generated by $\{T[s],s \in \Sigma \}$. Then, define for a $\rho$-locally Hölder continuous function $f$,

\[ D_{\rho,\beta} f=\sup_{b\in \beta}\sup_{x,y\in b}\frac{|f(x)-f(y)|}{d_{\rho}(x,y)}, \]

where we recall that $d_{\rho }(x,y)=\rho ^{n}$, where $n$ is the first time that the two sequences $x$ and $y$ differ. Denote then $\|f\|_{\rho,\beta }=\|f\|_{\infty }+D_{\rho,\beta }f$ and define

\[ \mathcal{B}_{\rho,\beta}=\{f, \|f\|_{\rho,\beta}<+\infty\}. \]

Then, $(\mathcal {B}_{\rho,\beta },\|\cdot \|_{\rho,\beta })$ is a Banach space.

Having finitely many images is a sufficient condition to have a spectral gap on $(\mathcal {B}_{\rho,\beta },\|\cdot \|_{\rho,\beta })$. Indeed, Mauldin and Urbański introduced in [Reference Mauldin and UrbańskiMU01] the BIP property, which is automatically satisfied if the shift has finitely many images. Moreover, they proved that the BIP property is a sufficient condition for locally Hölder functions to have a Gibbs measure, whereas Sarig proved in [Reference SarigSar99] that having a Gibbs measure is a sufficient condition to be positive recurrent and to have a spectral gap. In particular, we have the following two results.

Proof. As $\varphi$ is locally Hölder, the sum

\[ \sum_{n\geq 1} V_n(\varphi)=\sum_{n\geq 1}\sup \{|\varphi(x)-\varphi(y)|,x_1=y_1,\ldots,x_n=y_n\}\leq C\sum_{n\geq 1}\rho^{n} \]

is finite. Thus, [Reference SarigSar03, Theorem 1] shows that $\varphi$ has a Gibbs measure. Consequently, [Reference SarigSar99, Theorem 8] shows that $\varphi$ is positive recurrent.

Theorem 3.2 Sarig [Reference SarigSar03, Corollary 3] and [Reference SarigSar99, Theorem 4]

Let $(\overline {\Sigma }_A,T)$ be a topologically mixing countable Markov shift having finitely many images. Let $\varphi$ be a locally Hölder continuous function with finite pressure $P(\varphi )$. Then there exist a $\sigma$-finite measure $\nu$ and a function $h$ bounded away from zero and infinity such that $\mathcal {L}_{\varphi }^{*}\nu ={e}^{P(\varphi )}\nu$ and $\mathcal {L}_{\varphi }h={e}^{P(\varphi )}h$. There also exist $C\geq 0$ and $0<\theta <1$ such that for every $f\in \mathcal {B}_{\rho,\beta }$,

\[ \bigg\|{e}^{-nP(\varphi)}\mathcal{L}_{\varphi}^{n}f-h\int f \,d\nu\bigg\|_{\rho,\beta}\leq C\theta^{n}\|f\|_{\rho,\beta}. \]

Moreover, $\nu$ is supported on $\partial \Sigma _A^{*}$ and both measures $\nu$ and $m$ defined by $dm=h\,d\nu$ are ergodic.

The fact that $\nu$ is ergodic is not stated in Corollary 3 but in Corollary 2 of [Reference SarigSar03]. Ergodicity of $m$ follows (see the remarks after [Reference SarigSar99, Theorem 4]). Finally, the fact that $h$ is bounded away from zero and infinity is deduced from the fact that the shift has finitely many images, see [Reference SarigSar99, Proposition 2].

Actually, we never really use the $\|\cdot \|_{\rho,\beta }$ norm and all our bounds in the following are on the $\rho$-Hölder norm, that is, we both bound $\|\cdot \|_{\infty }$ and $D_{\rho }$, which is defined by

\[ D_{\rho}f=\sup_{x,y\in \overline{\Sigma}_A}\frac{|f(x)-f(y)|}{d_{\rho}(x,y)}. \]

Obviously, a bound on $D_{\rho }$ is stronger than a bound $D_{\rho,\beta }$. Moreover, when bounding $D_{\rho }f$, we have to bound ${|f(x)-f(y)|}/{d_{\rho }(x,y)}$. We always assume that $x$ and $y$ start with the same element, otherwise $d(x,y)=\rho$ and one can, thus, bound ${|f(x)-f(y)|}/{d_{\rho }(x,y)}$ by $2\rho ^{-1}\|f\|_{\infty }$.

One issue we have to deal with, when applying these results to random walks on relatively hyperbolic groups in the next subsection, is that our Markov shift will not be topologically mixing. It will not even be irreducible (but will have only finitely many recurrent classes). This issue was already addressed in [Reference GouëzelGou14] for a Markov shift with finitely many symbols, as we now explain.

In the following, we consider a countable Markov shift $(\overline {\Sigma }_A,T)$ with set of symbols $\Sigma$ and transition matrix $A$ and we assume that it has finitely many images. If the Markov shift is irreducible, but not topologically mixing, then there is a minimal period $p>1$ such that for any symbol $s \in \Sigma$, if $T^{-n}[s]\cap [s]\neq \emptyset$, then $n=pk$ for some $k\geq 0$. Then, one can decompose the set of symbols as a finite union $\Sigma =\Sigma _A^{(1)}\sqcup \Sigma _A^{(2)}\sqcup \cdots \sqcup \Sigma _A^{(p)}$, such that for $i\in \mathbb {Z}/p\mathbb {Z}$, if $a_{s,s'}=1$ and $s\in \Sigma ^{(i)}$, then $s'\in \Sigma ^{(i+1)}$. We call such a decomposition a cyclic decomposition. We denote by $\overline {\Sigma }_A^{(i)}$ the subset of $\overline {\Sigma }_A$ of sequences that begin with an element of $\Sigma _A^{(i)}$, so that the shift map $T$ maps $\overline {\Sigma }_A^{(i)}$ to $\overline {\Sigma }_A^{(i+1)}$. Moreover, in this case, $T^{p}$ acts on $\overline {\Sigma }_A^{(i)}$ and the induced Markov shift is topologically mixing. Using this decomposition together with Theorem 3.2, we get that if $\varphi$ is locally Hölder continuous function with finite pressure $P(\varphi )$ and if the Markov shift has finitely many images, then there are positive functions $h^{(i)}$ on $\overline {\Sigma }_A^{(i)}$ and probability measures $\nu ^{(i)}$ on $\overline {\Sigma }_A^{(i)}$ with $\int h^{(i)}d\nu ^{(i)}=1$ such that for $f\in \mathcal {B}_{\rho,\beta }$,

\[ \bigg\|{e}^{-nP(\varphi)}\mathcal{L}_{\varphi}^{n}f-\sum_{i=1}^{p}h^{(i)}\int f \,d\nu^{((i-n) \text{ mod } p)}\bigg\|_{\rho,\beta}\leq C\theta^{n}\|f\|_{\rho,\beta}. \]

Assume that the Markov shift is not irreducible. Then, because it has finitely many images, one can first decompose $\Sigma$ as $\Sigma =\Sigma _{A,0}\sqcup \Sigma _{A,1}\sqcup \cdots \Sigma _{A,q}$, such that if a path starts at $s\in \Sigma _{A,0}$, then it never reaches $s$ again and for $s,s'\in \Sigma$, one can reach $s'$ starting at $s$ and conversely if and only if $s$ and $s'$ are in the same subset $\Sigma _{A,j}$, $j\geq 1$. More formally, the decomposition of $\Sigma$ satisfies the following properties.

1. (i) If $s\in \Sigma _{A,0}$, then for all $n\geq 1$, $T^{-n}[s]\cap [s]=\emptyset$.

2. (ii) If there exist $n,n'$ such that $T^{-n}[s]\cap [s']\neq \emptyset$ and $T^{-n'}[s']\cap [s]\neq \emptyset$, then there exists $j\geq 1$ such that $s,s'\in \Sigma _{A,j}$.

3. (iii) Conversely, if $s,s'$ lie in the same $\Sigma _{A,j}$, $j\geq 1$, then there exist $n,n'$ such that $T^{-n}[s]\cap [s']\neq \emptyset$ and $T^{-n'}[s']\cap [s]\neq \emptyset$.

We call $\Sigma _{A,0}$ the transient component of $\Sigma$ and the sets $\Sigma _{A,j}$, $j\geq 1$, the biconnected components of $\Sigma$. All the non-trivial dynamical behavior of the Markov shift happens in the biconnected components. We denote by $\overline {\Sigma }_{A,j}$ the subset of $\overline {\Sigma }_A$ of sequences $x$ that stay in $\Sigma _{A,j}$, that is, for every $n$, $x_n\in \Sigma _{A,j}$. We similarly call the sets $\overline {\Sigma }_{A,j},j\geq 1$ the biconnected components of $\overline {\Sigma }_A$.

Then, $\overline {\Sigma }_{A,j}$ is stable under the shift map $T$ and we can apply the above discussion to $\overline {\Sigma }_{A,j}$. If $\varphi$ is a locally Hölder continuous function on $\overline {\Sigma }_A$, denote by $\varphi _j$ its restriction to the component $\overline {\Sigma }_{A,j}$, with associated transfer operator $\mathcal {L}_{\varphi _j}$. Denote the pressure of $\varphi _j$ by $P_j(\varphi )$. Then, $\mathcal {L}_{\varphi _j}$ has a spectral gap and ${e}^{P_j(\varphi )}$ is its dominant eigenvalue. Let $P(\varphi )$ be the maximum of all the $P_j(\varphi )$ and call a component $\overline {\Sigma }_{A,j}$ maximal if $P_j(\varphi )=P(\varphi )$.

Elaborating on ideas of Calegari and Fujiwara from [Reference Calegari and FujiwaraCF10], Gouëzel proved in [Reference GouëzelGou14] a spectral gap theorem for the transfer operator of a semisimple Hölder continuous function, when the set of symbols is finite. His proof works for a countable set of symbols, if the Markov shift has finitely many images and the Hölder continuous function is positive recurrent, because it is based on a spectral decomposition over the sets $\overline {\Sigma }_{A,j}$, on which one applies the Ruelle–Perron–Frobenius theorem (that we replace here with Theorem 3.2). Thus, combining Theorem 3.2 and the proof of [Reference GouëzelGou14, Theorem 3.8], we obtain the following.

Theorem 3.3 Let $(\overline {\Sigma }_A,T)$ be a countable Markov shift with finitely many images. Let $\varphi$ be a locally Hölder continuous function with finite maximal pressure $P(\varphi )$. Assume that $\varphi$ is semisimple. Denote by $\overline {\Sigma }_{A,1},\ldots \overline {\Sigma }_{A,k}$ the maximal components, with corresponding period $p_1,\ldots,p_k$ and consider a cyclic decomposition

\[ \Sigma_{A,j}=\Sigma_{A,j}^{(1)}\sqcup \cdots \Sigma_{A,j}^{(p_j)}. \]

Then, there exist functions $h_j^{(i)}$ and probability measures $\nu _j^{(i)}$ with $\int h_{j}^{(i)}d\nu _j^{(i)}=1$ and such that $\mathcal {L}_{\varphi }^{*}\nu _{j}^{(i)}=\nu _j^{(i-1) \text { mod } p_j}$ and $\mathcal {L}_{\varphi }h_{j}^{(i)}=h_j^{(i-1) \text { mod } p_j}$. Moreover, for $f\in \mathcal {B}_{\rho,\beta }$,

\[ \bigg\|{e}^{-nP(\varphi)}\mathcal{L}_{\varphi}^{n}f-\sum_{j=1}^{k}\sum_{i=1}^{p_j}h_j^{(i)}\int f\, d\nu_j^{((i-n) \text{ mod } p_j)}\bigg\|_{\rho,\beta}\leq C\theta^{n}\|f\|_{\rho,\beta}, \]

for some $C\geq 0$ and $0<\theta <1$. Finally, the functions $h_j^{(i)}$ are bounded away from zero and infinity on the support of $\nu _j^{(i)}$.

We also obtain the following result, again proved in [Reference GouëzelGou14] for finite sets of symbols, which applies in our situation (see [Reference GouëzelGou14, Lemma 3.7]).

Lemma 3.4 Let $(\overline {\Sigma }_A,T)$ be a countable Markov shift with finitely many images. Let $\varphi$ be a locally Hölder continuous function with finite maximal pressure $P(\varphi )$. Let $s\in \Sigma$ and assume that there is a path starting with $s$ that visits $k$ maximal components. Then, for any non-negative function $f$ with $f\geq 1$ on the set of paths starting with $s$, one has

\[ \mathcal{L}^{n}_{\varphi}f(\emptyset)\geq Cn^{k-1}{e}^{nP(\varphi)}, \]

where we recall that $\emptyset$ is the empty sequence in $\Sigma _A^{*}$. In particular, for $k=2$, if $\varphi$ is not semisimple, then

\[ \mathcal{L}^{n}_{\varphi}1(\emptyset)\geq Cn {e}^{nP(\varphi)}. \]

3.2 Perturbation of the pressure

In [Reference GouëzelGou14], Gouëzel proves a perturbation theorem for finite sets of symbols (see precisely [Reference GouëzelGou14, Proposition 3.10]). Its proof remains valid for countable shifts with finitely many images. Denote by $ |||{\cdot }|||_{\rho,\beta }$ the operator norm for operators acting on $(\mathcal {B}_{\rho,\beta },\|\cdot \|_{\rho,\beta })$. To apply Gouëzel's perturbation theorem, one needs to control $ |||{\mathcal {L}|||_{\varphi }-\mathcal {L}_{\psi }}_{\rho,\beta }$. However, estimating this norm will be very difficult in this paper, so we need finer results, that are based on the following theorem, proved by Keller and Liverani [Reference Keller and LiveraniKL99].

Consider a Banach $(V,\|\cdot \|)$ endowed with a norm $|\cdot |_w$, satisfying $|\cdot |_w\leq C \|\cdot \|$ for some uniform $C$. Letting $\mathcal {L}:V\to V$ be a linear operator, let

\[ |||{\mathcal{L}}|||=\sup \big\{\|\mathcal{L}v\|,\|v\|\leq 1\big\} \]

denote the operator norm of $\mathcal {L}$ associated with $\|\cdot \|$ and let

\[ |||{\mathcal{L}}|||_{s\to w}=\sup \big\{|\mathcal{L}v|_w,\|v\|\leq 1\big \} \]

be the operator norm of $\mathcal {L}:(V,\|\cdot \|)\to (V,|\cdot |_w)$.

Theorem 3.5 Consider a family of bounded operators $\mathcal {L}_r:(V,\|\cdot \|)\to (V,\|\cdot \|)$, with $r$ varying in $(0,R]$. Assume there exist $0<\sigma < M$ and $C\geq 0$ and there exists a function $\tau (r)$ converging to zero as $r$ tends to $R$ such that the following hold.

1. (i) For every $n$, for every $v\in V$, $|\mathcal {L}_R^{n}v|_w\leq CM^{n}|v|_w$.

2. (ii) For every $r\leq R$, for every $n$, for every $v\in V$, $\|\mathcal {L}_r^{n} v\|\leq C\sigma ^{n}\|v\|+CM^{n}|v|_w$.

3. (iii) For every $r\leq R$, $ |||{\mathcal {L}|||_r-\mathcal {L}_R}_{s\to w}\leq \tau (r)$.

For fixed $\rho >0$ and $\rho '>0$ let

\[ A_{\rho,\rho'}=\{z\in \mathbb{C},|z|\geq \sigma +\rho,d(z,\operatorname{spec}(\mathcal{L}_R))\geq \rho'\}. \]

Then, for any $\rho,\rho '>0$ there exist $\beta _0<1$ and $K_0\geq 0$ and there exists $r_0$ such that for every $\beta \leq \beta _0$, for every $r\in [r_0,R]$ for every z $z\in A_{\rho,\rho '}$:

1. (a) the operator $zI-\mathcal {L}_r:(V,\|\cdot \|)\to (V,\|\cdot \|)$ is invertible;

2. (b) the operator norm $ |||{(zI-\mathcal {L}_r)^{-1}}|||$ is bounded independently of $r$;

3. (c) the norm $ |||{\cdot }|||_{s\to w}$ satisfies $ |||{(zI-\mathcal {L}_r)^{-1}-(zI-\mathcal {L}_R)^{-1}}|||_{s\to w}\leq K_0 \tau (r)^{\beta }$.

Moreover, $\beta _0$ only depends on $\rho$ and can be explicitly computed whenever $\sigma +\rho \leq M$. Indeed, one can then choose

\[ \beta_0=\frac{\log(({\sigma+\rho})/{\sigma})}{\log({M}/{\sigma})}. \]

In particular, $\beta _0$ converges to zero as $\rho$ tends to zero and converges to one as $\rho$ tends to $M-\sigma$.

For a proof, we refer to [Reference BaladiBal18, A.3]. Note that it is asked there that for every $r$, $|\mathcal {L}_r^{n}v|_w\leq CM^{n}|v|_w$, whereas our condition (i) only requires that this holds for $r=R$. However, the proof in [Reference BaladiBal18, A.3] only uses this inequality for $r=R$.

Let us apply this to transfer operators. Consider a countable shift with finitely many images $(X_A,T)$ and a family of locally Hölder functions $f_r$, for $r\in (0,R]$. Let $\mathcal {L}_r=\mathcal {L}_{f_r}$ be the associated transfer operator. Assume that for every $r$, the maximal pressure $P_r$ of $f_r$ is finite and that $f_r$ is semisimple. Let $h_{j}^{(i)}$ and $\nu _j^{(i)}$ be the functions and measures given by Theorem 3.3, associated with $\mathcal {L}_{R}$. Define the measure $m_j$ as $dm_j= ({1}/{p_j})\sum _{i=1}^{p_j}h_{j}^{(i)}\,d\nu _j^{(i)}$.

Let $m=\sum m_j$. Consider the Banach space $(V=H_{\rho,\beta },\|\cdot \|=\|\cdot \|_{\rho,\beta })$ on $V=H_{\rho,\beta }$, endowed with the norm $|\cdot |_w=\|\cdot \|_{L^{1}(m)}$. As $m$ is finite, we have $|\cdot |_w\leq C \|\cdot \|$. We deduce from Theorem 3.5 the following.

Theorem 3.6 With the same notation as previously, assume there exists $\sigma$ such that $0<\sigma < e^{P_R}$ and that there exist $C\geq 0$ and a function $\tau (r)$ converging to zero as $r$ tends to $R$ such that the following hold.

1. (α) For every $r\leq R$, for every $n$, for every $v\in V$,

   \[ \|\mathcal{L}_r^{n} v\|\leq C\sigma^{n}\|v\|+C{e}^{nP_R}|v|_w. \]

2. (β) For every $r\leq R$, $ |||{\mathcal {L}_r-\mathcal {L}_R}|||_{s\to w}\leq \tau (r)$.

Then, for every $r$ which is close enough to $R$, there exist numbers $\tilde {P}_j(r)$ and eigenfunctions $\tilde {h}_{j,r}^{(i)}$ and eigenmeasures $\tilde {\nu }_{j,r}^{(i)}$ of $\mathcal {L}_r$ associated with the eigenvalue ${e}^{\tilde {P}_j(r)}$ such that

\[ \bigg\|{e}^{-nP_R}\mathcal{L}_{r}^{n}g-\sum_{j=1}^{k}{e}^{n(\tilde{P}_j(r)-P_R)}\sum_{i=1}^{p_j}\tilde{h}_{j,r}^{(i)}\int g \,d\tilde{\nu}_{j,r}^{((i-n) \text{ mod } p_j)}\bigg\|_{\rho,\beta}\leq C\theta^{n}\|g\|_{\rho,\beta}. \]

The functions $\tilde {h}_{j}^{(i)}$ and the measures $\tilde {\nu }_j^{(i)}$ have the same support as $h_j^{(i)}$ and $\nu _j^{(i)}$, respectively. Moreover, $\big \|\tilde {h}_{j,r}^{(i)}\big \|$ is uniformly bounded. Finally, $\big |\tilde {h}_{j,r}^{(i)}-h_j^{(i)}\big |_w$ converges to zero as $r$ tends to $R$ and $\tilde {\nu }_{j,r}^{(i)}$ weakly converges to $\nu _j^{(i)}$ as $r$ tends to $R$.

Note that $\int \tilde {h}_{j,r}^{(i)}\,d\nu _j^{(i)}\neq 0$ for $r$ close enough to $R$, because $\int h_j^{(i)}d\nu _j^{(i)}=1$ and $\big |\tilde {h}_{j,r}^{(i)}-h_j^{(i)}\big |_w$ converges to zero. One can, thus, normalize $\tilde {h}_{j,r}$ declaring $\int \tilde {h}_{j,r}^{(i)}\,d\nu _j^{(i)}=1$. We make this assumption in the following. We still have that $\big \|\tilde {h}_{j,r}^{(i)}\big \|$ is uniformly bounded and that $\big |\tilde {h}_{j,r}^{(i)}-h_j^{(i)}\big |_w$ converges to zero. The following result allows us to obtain a precise asymptotic of $\tilde {P}_{j,r}-P_R$ in the following sections.

Proposition 3.7 Under the assumptions of Theorem 3.6,

\[ {e}^{\tilde{P}_j(r)-P_R}-1= \int ({e}^{f_r-f_R}-1)\,dm_j+ \int ({e}^{f_r-f_R}-1)\frac{1}{p_j}\bigg(\sum_{i=0}^{p_j-1}h_j^{(i)}-\tilde{h}_{j,r}^{(i)}\bigg)\,d\nu_j^{(i)}, \]

where $dm_j= ({1}/{p_j})\sum _{i=1}^{p_j}h_{j}^{(i)}\,d\nu _j^{(i)}$.

Proof. As $\tilde {h}_{j,r}^{(i)}$ are eigenfunctions of $\mathcal {L}_r$ and $\tilde {h}_{j,r}$ is normalized, we have

\[ {e}^{\tilde{P}_j(r)}=\int \mathcal{L}_r \tilde{h}_{j,r}^{(i)}\,d\nu_j^{(i)}. \]

Consequently,

\[ {e}^{\tilde{P}_j(r)}-{e}^{P_R}=\int ( \mathcal{L}_r \tilde{h}_{j,r}^{(i)}-\mathcal{L}_Rh_j^{(i)})\,d\nu_j^{(i)}. \]

Note that for any function $g$, $\mathcal {L}_rg=\mathcal {L}_R ({e}^{f_r-f_R}g)$. In particular,

\[ {e}^{\tilde{P}_j(r)}-{e}^{P_R}=\int \mathcal{L}_R ( {e}^{f_r-f_R}\tilde{h}_{j,r}^{(i)}-h_j^{(i)})\,d\nu_j^{(i)}. \]

Using that $d\nu _j^{(i)}$ is an eigenmeasure of $\mathcal {L}_R$ associated with the eigenvalue ${e}^{P_R}$, we obtain

\begin{align*} {e}^{\tilde{P}_j(r)-P_R}-1&=\int ( {e}^{f_r-f_R}\tilde{h}_{j,r}^{(i)}-h_j^{(i)})\,d\nu_j^{(i)}\\ &=\int ( {e}^{f_r-f_R}-1) (\tilde{h}_{j,r}^{(i)}-h_j^{(i)})\,d\nu_j^{(i)} +\int \tilde{h}_{j,r}^{(i)}\,d\nu_j^{(i)}-\int {e}^{f_r-f_R}h_j^{(i)}\,d\nu_j^{(i)} . \end{align*}

As $\int h_j^{(i)}\,d\nu _j^{(i)}=\int \tilde {h}_{j,r}^{(i)}\,d\nu _j^{(i)}=1$, we thus obtain

\[ {e}^{\tilde{P}_j(r)-P_R}-1= \int ( {e}^{f_r-f_R}-1) (\tilde{h}_{j,r}^{(i)}-h_j^{(i)})\,d\nu_j^{(i)}+\int ({e}^{f_r-f_R}-1)h_j^{(i)}\,d\nu_j^{(i)}. \]

This holds for every $i$, which concludes the proof summing over $i$ and then dividing by $p_j$.

4.1 Transfer operator for the Green function

We choose a generating set $S$ of $\Gamma$ as in Theorem 2.3, so that $\Gamma$ is automatic relative to $\Omega _0$ and $S$, where $\Omega _0$ is a finite set of representatives of conjugacy classes of the parabolic subgroups Let $\mathcal {G}=(V,E,v_*)$ be a graph and $\phi :E\rightarrow S\cup \bigcup _{\mathcal {H}\in \Omega _0}\mathcal {H}$ be a labelling map as in the definition of a relative automatic structure.

The set of vertices $V$ is finite. Moreover, if $\sigma \in \Sigma _0= S\cup \bigcup _{\mathcal {H}\in \Omega _0}\mathcal {H}$ and if $v\in V$, there is at most one edge that leaves $v$ and that is labelled with $\sigma$. Thus, the set of edges $E$ is countable. Set $\Sigma =E$ and consider the transition matrix $A=(a_{s,s'})_{s,s'\in \Sigma }$, defined by $a_{s,s'}=1$ if the edges $s$ and $s'$ are adjacent in $\mathcal {G}$ and $a_{s,s'}=0$ otherwise. We then define $\Sigma _A^{*}$, $\partial \Sigma _A$ and $\overline {\Sigma }_A$ as previously. According to the definition of a relative automatic structure, elements of $\overline {\Sigma }_A$ represent relative geodesics and relative geodesic rays.

We decompose $\Sigma _0= S\cup \bigcup _{\mathcal {H}\in \Omega _0}\mathcal {H}$ as follows. The sets $\mathcal {H}_j\cap \mathcal {H}_k$ are finite if $j\neq k$ (see, e.g., [Reference Druţu and SapirDS05, Lemma 4.7]). We can, thus, consider $\mathcal {H}'_k=\mathcal {H}_k\setminus \cup _{j\neq k}\mathcal {H}_j$ and $\mathcal {H}'_0=\Sigma _0\setminus \cup _k \mathcal {H}'_k$. Then, $\mathcal {H}'_0$ remains finite and the sets $\mathcal {H}'_k$ are disjoint. By analogy with free factors in a free product, we introduce the following terminology.

Paths of length $n$ in $\mathcal {G}$ beginning at $v_*$ are in bijection with the relative sphere $\hat {S}_n$. Moreover, infinite paths in $\mathcal {G}$ starting at $v_*$ give relative geodesic rays starting at $e$. Denote by $E_*\subset E$ the set of edges that starts at $v_*$. The labelling map $\phi$ can be extended to infinite paths. When restricted to infinite words starting in $E_*$, it gives a surjective map from paths beginning at $v_*$ to the Gromov boundary of the graph $\hat {\Gamma }$, which is by definition the set of conical limit points of $\Gamma$, included in the Bowditch boundary. Restricting the distance $d_\rho (x,y)=\rho ^{-n}$ to $E_*$, this induced map is continuous, endowing the Bowditch boundary with the usual topology. A formal way of restricting our attention to elements of the group and to conical limit points is to consider the function $1_{E_*}$ on $\overline {\Sigma }_A$ which takes value one on sequences in $\overline {\Sigma }_A$ beginning with an edge in $E_*$ and that takes value zero elsewhere. This function $1_{E_*}$ is locally Hölder continuous.

We have the following, which proves that every locally Hölder continuous function with finite pressure is positive recurrent, according to Proposition 3.1.

Recall that $R_{\mu }$ is the inverse of the spectral radius of the $\mu$-random walk on $\Gamma$. Also recall that for $r\in [0,R_{\mu }]$, we write $H(e,\gamma |r)=G(e,\gamma |r)G(\gamma,e|r)$. For $r\in [1,R_{\mu }]$, we define the function $\varphi _r$ on $\Sigma _A^{*}$ by $\varphi _r(\emptyset )=1$ and

\[ \varphi_r(x=x_1,\ldots,x_n)=\log \bigg(\frac{H(e,\phi(x)|r)}{H(e,\phi(T x)|r)}\bigg) = \log \bigg(\frac{H(e,\phi(x_1\dots x_n)|r)}{H(e,\phi(x_2\dots x_n)|r)}\bigg). \]

Using equivariance of the Green function, we also have

\[ \varphi_r(x=x_1,\ldots,x_n)=\log \bigg(\frac{H(e,\phi(x_1\dots x_n)|r)}{H(\phi(x_1),\phi(x_1\dots x_n)|r)}\bigg). \]

Proof. Let $n\geq 1$ and let $x,y\in \Sigma _A^{*}$ be such that $x_1=y_1,\ldots,x_n=y_n$. Hence, in $\Gamma$, we have $\phi (x_1)=\phi (y_1),\ldots,\phi (x_n)=\phi (y_n)$. This means that relative geodesics from $e$ to $\phi (x)$ and from $\phi (x_1)=\phi (y_1)$ to $\phi (y)$ fellow-travel for a time at least $n-1$. According to strong relative Ancona inequalities, we thus have, for some $C\geq 0$ and $0<\rho <1$,

\[ \bigg|\frac{G(e,\phi(x)|r)G(\phi(y_1),\phi(y)|r)}{G(\phi(x_1),\phi(x)|r)G(e,\phi(x)|r)} -1\bigg|\leq C\rho^{n}. \]

Weak relative Ancona inequalities also show that

\[ G(e,\phi(x)|r)G(\phi(y_1),\phi(y)|r)\geq \frac{1}{C}G(e,\phi(x_1)|r)G(\phi(x_1),\phi(x)|r)G(\phi(y_1),\phi(y)|r) \]

and because $x_1=y_1$, we obtain

\[ G(e,\phi(x)|r)G(\phi(y_1),\phi(y)|r)\geq \frac{1}{C^{2}}G(e,\phi(y)|r)G(\phi(x_1),\phi(x)|r). \]

Thus, ${G(e,\phi (x)|r)G(\phi (y_1),\phi (y)|r)}/{G(\phi (x_1),\phi (x)|r)G(e,\phi (x)|r)}$ is bounded away from zero, so that

(8)\begin{align} &\bigg| \log \bigg(\frac{H(e,\phi(x)|r)}{H(\phi(x_1),\phi(x)|r)}\bigg) - \log \bigg(\frac{H(e,\phi(y)|r)}{H(\phi(y_1),\phi(y)|r)}\bigg)\bigg|\nonumber\\ &\quad \leq C_1\bigg|\frac{G(e,\phi(x)|r)G(\phi(y_1),\phi(y)|r)}{G(\phi(x_1),\phi(x)|r)G(e,\phi(y)|r)} -1\bigg|\leq C\rho^{n}. \end{align}

This proves that if $x=(x_1,\ldots,x_n,\ldots )\in \partial \Sigma _A^{*}$, then the sequence $\varphi _r(x_1,\ldots,x_k)$ is Cauchy, so that it converges to some well-defined limit $\varphi _r(x)$. This extended function $\varphi _r$ on $\overline {\Sigma }_A$ still satisfies (8), so that it is locally Hölder continuous.

We denote by $\mathcal {L}_r$ the transfer operator associated with the function $\varphi _r$.

Proof. As noted previously, because the Markov shift has finitely many images, Proposition 3.1 shows that any locally Hölder function with finite pressure is positive recurrent. Thus, we only need to prove that $\varphi$ has finite pressure, which is equivalent to proving that $\|\mathcal {L}_r1\|_\infty <+\infty$ by [Reference SarigSar99, Theorem 1]. For $x\in \overline {\Sigma }_A$, let $X_x^{1}$ be the set of symbols that can precede $x$ in the automaton $\mathcal {G}$. Then, by weak relative Ancona inequalities,

\[ \mathcal{L}_r1(x)=\sum_{\sigma\in X_x^{1}}\frac{H(e,\sigma x|r)}{H(\sigma,\sigma x|r)}\lesssim \sum_{k=0}^{N}\sum_{\sigma\in \mathcal{H}_k'}H(e,\sigma|r). \]

This last sum is bounded by Corollary 2.10, which concludes the proof.

Let $P_j(r)$ be the pressure of the restriction of $\varphi _r$ to a component $\overline {\Sigma }_{A,j}$ of the Markov shift and let $P(r)$ be the maximal pressure, that is, the maximum of the $P_j(r)$. Recall that we declared that the empty sequence is not a preimage of the empty sequence. This will simplify the following.

The main reason for introducing this function $\varphi _r$ is that

\[ \mathcal{L}_r^{n}1_{E_*}(\emptyset)=\frac{1}{H(e,e|r)}\sum_{\gamma \in \hat{S}_n}H(e,\gamma|r), \]

where we recall that $1_{E_*}$ is the function on $\overline {\Sigma }_A$ that takes value one on paths that start at $v_*$ in the automaton $\mathcal {G}$ and zero elsewhere. Indeed, to prove Theorem 4.1, we want to understand $I^{(1)}(r)=\sum _{\gamma \in \Gamma }H(e,\gamma |r)$. Thus, we want to understand the behavior of $\sum _{\gamma \in \hat {S}_n}H(e,\gamma |r)$, which is thus the same as understanding the behavior of $\mathcal {L}_r^{n}1_{E_*}(\emptyset )$.

4.2 Continuity properties of the transfer operator

Our goal in this subsection is to prove that the map $r\mapsto \mathcal {L}_r$ is continuous in a weak sense. We begin by the following result.

Lemma 4.5 There exists $C>0$ such that for all $r\in [1,R_{\mu })$,

\[ \frac{1}{C}\frac{1}{\sqrt{R_{\mu}-r}}\leq \sum_{\gamma\in \Gamma}H(e,\gamma|r)\leq C \frac{1}{\sqrt{R_{\mu}-r}}. \]

Proof. Let $I_1(r)=\sum _\gamma H(e,\gamma |r)$ and $F(r)=r^{2}I_1(r)$. According to [Reference DussauleDus22, Lemma 3.2],

\[ F'(r)=2r\sum_{\gamma,\gamma'}G(e,\gamma'|r)G(\gamma',\gamma|r)G(\gamma,e|r). \]

Proposition 2.8 gives

\[ \frac{1}{C'}\leq \frac{F'(r)}{F(r)^{3}}\leq C' \]

and Proposition 2.11 gives $F(R_{\mu })=+\infty$. Thus, integrating the inequality above between $r$ and $R_{\mu }$, we obtain

\begin{align*} \frac{1}{C'}(R_{\mu}-r)\leq \frac{1}{F(r)^{2}}\leq C'(R_{\mu}-r), \end{align*}

which leads to the desired inequality.

We also prove the following.

Proof. If $P(r)$ were positive, then Theorem 3.3 would show that

\[ \mathcal{L}_r1_{E_*}(\emptyset)=\frac{1}{H(e,e|r)}\sum_{\gamma\in \hat{S}_n}H(e,\gamma|r) \]

tends to infinity. However, Lemma 2.9 shows that this quantity is bounded, so we obtain a contradiction.

If $P(R_{\mu })$ were negative, then $\sum _{\gamma \in \hat {S}_n}H(e,\gamma |r)$ would converge to zero exponentially fast, according to Theorem 3.3. In particular, $\sum _{\gamma \in \Gamma }H(e,\gamma |r)$ would be finite, that is, using Lemma 2.7, $ ({d}/{dr})_{|r=R_\mu }G(e,e|r)$ would be finite. This would be a contradiction with Proposition 2.11.

Finally, if $\varphi _{R_{\mu }}$ were not semisimple, then Lemma 3.4 would again show that $\sum _{\gamma \in \hat {S}_n}H(e,\gamma |r)$ would tend to infinity, because we already know that $P(R_{\mu })=0$. Again, Lemma 2.9 shows that this quantity is bounded.

Let $h_{j}^{(i)}$ and $\nu _j^{(i)}$ be the functions and measures given by Theorem 3.3, associated with $\mathcal {L}_{R_\mu }$. Let $m_j$ be the measure defined as $dm_j= ({1}/{p_j})\sum _{i=1}^{p_j}h_{j}^{(i)}\,d\nu _j^{(i)}$. According to [Reference SarigSar99, Proposition 4], $m_j$ is a Gibbs measure. However, we have to apply this proposition to each component $\overline {\Sigma }_{A,j}$ of the shift, so that we do not have that $m_j([x_1,\ldots,x_n])\asymp H(e,x_1\dots x_n|R_\mu )$ for any cylinder $[x_1,\ldots,x_n]$. We still deduce that there exists $C\geq 0$ such that for any $n$, for any cylinder $[x_1,\ldots,x_n]$,

(9)\begin{align} m_j([x_1\dots x_n]) \leq C H(e,x_1\dots x_n|R_\mu). \end{align}

Furthermore, letting $m=\sum _j m_j$, there exists $C\geq 0$ such that for any $n$, for any cylinder $[x_1\dots x_n]$ in the support of one of the measures $m_j$,

(10)\begin{equation} \frac{1}{C} H(e,x_1\dots x_n|R_\mu)\leq m([x_1,\ldots,x_n]) \leq C H(e,x_1\dots x_n|R_\mu). \end{equation}

Proposition 4.7 There exists a non-negative function $\varphi$ on $\overline {X}_A$, possibly taking the value $+\infty$ on $\partial \Sigma _A$, which is integrable with respect to the measure $m$ and such that for every $x\in \Sigma _A$ and for every $1\leq r,r'\leq R_\mu$,

\begin{align*} |\varphi_r(x)-\varphi_{r'}(x)|\leq 2\varphi(x)\sqrt{|r-r'|}. \end{align*}

Integrating over $r$, this proposition is a direct consequence of the following lemma.

Lemma 4.8 There exists a non-negative function $\varphi$ on $\overline {X}_A$, possibly taking the value $+\infty$ on $\partial \Sigma _A$, which is integrable with respect to the measure $m$ and such that for every $x\in \Sigma _A$ and for every $1\leq r< R_\mu$,

\[ \bigg|\frac{d}{dr}\varphi_r(x)\bigg|\leq \varphi(x) \frac{1}{\sqrt{R_\mu-r}}. \]

Proof. Fix $x\in \Sigma _A$. We compute the derivative of $r\mapsto \varphi _r(x)$. To simplify the notation, we identify $x$ with $\phi (x)\in \Gamma$. We obtain

\[ \frac{d}{dr}\varphi_r(x)=\frac{d}{dr}\log \frac{H(e,x|r)}{H(x_1,x|r)}=\frac{d}{dr}\log r^{2}H(e,x|r)-\frac{d}{dr}\log r^{2}H(x_1,x|r). \]

In [Reference DussauleDus22, Lemma 3.2] we showed that

(11)\begin{align} \frac{d}{dr}\varphi_r(x)&=\sum_{y\in \Gamma}\frac{G(e,y|r)G(y,x|r)G(x,e|r)+G(e,x|r)G(x,y|r)G(y,e|r)}{rH(e,x|r)}\nonumber\\ &\quad -\sum_{y\in \Gamma}\frac{G(x_1,y|r)G(y,x|r)G(x,x_1|r)+G(x_1,x|r)G(x,y|r)G(y,x_1|r)}{rH(x_1,x|r)}. \end{align}

We first give an upper bound for

\[ \bigg|\frac{\sum_{y\in \Gamma}G(e,y|r)G(y,x|r)G(x,e|r)}{rH(e,x|r)}-\frac{\sum_{y\in \Gamma}G(x_1,y|r)G(y,x|r)G(x,x_1|r)}{rH(x_1,x|r)}\bigg|. \]

The remaining term in (11) will be bounded in the same way. Putting together these two sums, we obtain

\[ \frac{1}{rH(e,x|r)}\sum_{y\in \Gamma}G(e,y|r)G(y,x|r)G(x,e|r) -G(x_1,y|r)G(y,x|r)G(x,x_1|r)\frac{H(e,x|r)}{H(x_1,x|r)}. \]

We rewrite this as

\[ \frac{1}{rH(e,x|r)}\sum_{y\in \Gamma}G(e,y|r)G(y,x|r)G(x,e|r) \bigg( 1-\frac{G(x_1,y|r)G(e,x|r)}{G(x_1,x|r)G(e,y|r)}\bigg). \]

We decompose the sum over $\Gamma$ in the following way. Let $n=\hat {d}(e,x)$, so that the relative geodesic $[e,x]$ has length $n$. Denote by $e,x_1,\ldots,x_{n}$ successive points on this relative geodesic. In addition, for $0\leq k \leq n$, let $\Gamma _k$ be the set of $y\in \Gamma$ whose projection on $[e,x]$ which is closest to $x$ is exactly at $x_k$.

We use Lemma 2.5 several times. Let us first focus on the sum over $\Gamma _0$. If $y\in \Gamma _0$, then any relative geodesic from $y$ to $x$ passes within a bounded distance of $e$. Weak relative Ancona inequalities show that

\[ G(y,x|r)\lesssim G(y,e|r)G(e,x|r). \]

Similarly, any relative geodesic from $x_1$ to $y$ passes within a bounded distance of $e$, hence

\[ G(x_1,y|r)\lesssim G(x_1,e|r)G(e,y|r). \]

We also have

\[ G(e,x|r)\lesssim G(e,x_1|r)G(x_1,x|r), \]

so that

\begin{align*} &\bigg|\frac{1}{rH(e,x|r)}\sum_{y\in \Gamma_0}G(e,y|r)G(y,x|r)G(x,e|r) \bigg( 1-\frac{G(x_1,y|r)G(e,x|r)}{G(x_1,x|r)G(e,y|r)}\bigg)\bigg|\\ &\quad \lesssim \sum_{y\in \Gamma}H(e,y|r) (1+H(e,x_1|r)). \end{align*}

As $H(e,x_1|r)$ is uniformly bounded, we deduce from Lemma 4.5 that

\begin{align*} &\bigg|\frac{1}{rH(e,x|r)}\sum_{y\in \Gamma_0}G(e,y|r)G(y,x|r)G(x,e|r) \bigg( 1-\frac{G(x_1,y|r)G(e,x|r)}{G(x_1,x|r)G(e,y|r)}\bigg)\bigg |\\ &\quad \lesssim \frac{1}{\sqrt{R_\mu-r}}. \end{align*}

Let us focus on the sum over $\Gamma _1$ now. Let $\mathcal {H}_1$ be the union of parabolic subgroups containing $x_1$. Let $y\in \Gamma _1$ and denote by $\sigma$ its projection on $\mathcal {H}_1$. Then, any relative geodesic from $e$ to $y$ passes within a bounded distance of $\sigma$ and any relative geodesic from $y$ to $x$ passes first to a point within a bounded distance of $\sigma$, then to a point within bounded distance of $x_1$. We thus obtain

\[ G(e,y|r)\lesssim G(e,\sigma|r)G(\sigma,y|r) \]

and

\begin{align*} G(y,x|r)\lesssim G(y,\sigma|r)G(\sigma,x_1|r)G(x_1,x|r). \end{align*}

Similarly,

\[ \frac{G(x_1,y|r)G(e,x|r)}{G(x_1,x|r)G(e,y|r)}\lesssim \frac{G(x_1,\sigma|r)G(e,x_1|r)}{G(e,\sigma|r)}\lesssim 1. \]

Letting $\Gamma _1^{\sigma }$ be the set of $y$ whose projection on $\mathcal {H}_1$ is at $\sigma$, we obtain

\begin{align*} &\bigg|\frac{1}{rH(e,x|r)}\sum_{y\in \Gamma_1}G(e,y|r)G(y,x|r)G(x,e|r) \bigg( 1-\frac{G(x_1,y|r)G(e,x|r)}{G(x_1,x|r)G(e,y|r)}\bigg)\bigg|\\ &\quad\lesssim \sum_{\sigma\in \mathcal{H}_1}\sum_{y\in \Gamma_1^{\sigma}}\frac{G(e,\sigma|r)G(\sigma,x_1|r)}{G(e,x_1|r)}H(\sigma,y|r). \end{align*}

We bound the sum over $y\in \Gamma _1^{\sigma }$ by a sum over $y\in \Gamma$, so that

\begin{align*} &\bigg|\frac{1}{rH(e,x|r)}\sum_{y\in \Gamma_1}G(e,y|r)G(y,x|r)G(x,e|r) \bigg( 1-\frac{G(x_1,y|r)G(e,x|r)}{G(x_1,x|r)G(e,y|r)}\bigg)\bigg|\\ &\quad \lesssim \frac{1}{\sqrt{R_\mu-r}}\sum_{\sigma\in \mathcal{H}_1}\frac{G(e,\sigma|r)G(\sigma,x_1|r)}{G(e,x_1|r)}. \end{align*}

Suppose now that $k\geq 2$ and consider the sum over $\Gamma _k$. For any $y\in \Gamma _k$, relative geodesic from $x_1$ to $y$ and from $e$ to $x$ travel together for a time at least $k-1$. We deduce from strong relative Ancona inequalities that

\[ \bigg|1-\frac{G(x_1,y|r)G(e,x|r)}{G(x_1,x|r)G(e,y|r)}\bigg|\lesssim \rho^{k} \]

for some $0<\rho <1$. Letting $\mathcal {H}_k$ be the union of parabolic subgroups containing $x_{k-1}^{-1}x_{k}$, we also obtain

\begin{align*} &\bigg|\frac{1}{rH(e,x|r)}\sum_{y\in \Gamma_k}G(e,y|r)G(y,x|r)G(x,e|r) \bigg(1-\frac{G(x_1,y|r)G(e,x|r)}{G(x_1,x|r)G(e,y|r)}\bigg)\bigg|\\ &\quad \lesssim \rho^{k}\frac{1}{\sqrt{R_\mu-r}}\sum_{\sigma\in \mathcal{H}_k}\frac{G(x_{k-1},x_{k-1}\sigma|r)G(x_{k-1}\sigma,x_{k}|r)}{G(x_{k-1},x_{k}|r)}. \end{align*}

Putting everything together and letting $x_0=e$, we obtain

(12)\begin{align} &\bigg|\frac{1}{rH(e,x|r)}\sum_{y\in \Gamma}G(e,y|r)G(y,x|r)G(x,e|r) \bigg(1-\frac{G(x_1,y|r)G(e,x|r)}{G(x_1,x|r)G(e,y|r)}\bigg)\bigg|\nonumber\\ &\quad \lesssim \frac{1}{\sqrt{R_\mu-r}} \bigg(1+\sum_{k=0}^{n-1}\rho^{k}\sum_{\sigma\in \mathcal{H}_k}\frac{G(x_k,x_k\sigma|r)G(x_k\sigma,x_{k+1}|r)}{G(x_k,x_{k+1}|r)} \bigg). \end{align}

We now bound

\begin{align*} \sum_{k=0}^{n}\rho^{k}\sum_{\sigma\in \mathcal{H}_k}\frac{G(x_k,x_k\sigma|r)G(x_k\sigma,x_{k+1}|r)}{G(x_k,x_{k+1}|r)} \end{align*}

by an integrable function independently of $r$ and $n$. We first prove the following.

Lemma 4.9 There exists $\Lambda$ such that the following holds. Let $\mathcal {H}$ be a parabolic subgroup. For every $x$ in $\mathcal {H}$ and for any $1\leq r\leq R_\mu$, we have

\[ \sum_{y\in \mathcal{H}}G(e,y|r)G(y,x|r)\leq( \Lambda d(e,x)+\Lambda )G(e,x|r). \]

Proof. We write $G^{(1)}_{r}(e,x)= ({d}/{dt})_{|t=1} (G_{r}(e,x|t))$, where $G_r$ is the Green function associated with the first return kernel $p_{r}$ to $\mathcal {H}$. According to Lemma 2.7, it is enough to prove that

\[ G^{(1)}_{r}(e,x)\leq ( \Lambda d(e,x)+\Lambda )G(e,x|r). \]

As we are assuming that $\mu$ is non-spectrally degenerate along $\mathcal {H}$, there exists $\rho <1$ such that for any $x$, for any $n$,

\[ p_{r}^{(n)}(e,x)\leq p_{R_\mu}^{(n)}(e,x)\leq \rho^{n}, \]

where $p_{r}^{(n)}$ denotes the $n$th power of convolution of $p_{r}$. By definition,

\[ G^{(1)}_{r}(e,x)=\sum_{n\geq 0}np_{r}^{(n)}(e,x). \]

Note then that

\[ \sum_{n\geq \Lambda d(e,x)+\Lambda}np_{r}^{(n)}(e,x)\lesssim (\rho')^{\Lambda d(e,x)+\Lambda}. \]

As $r\geq 1$, for any $x$, $G(e,x|r)\geq p^{d(e,x)}$ for some $p<1$ and so

\[ G^{(1)}_{r}(e,x)\geq G(e,x|r)\geq p^{d(e,x)}. \]

If $\Lambda$ is large enough, we thus have

\[ \sum_{n\geq \Lambda d(e,x)+\Lambda}np_{r}^{(n)}(e,x)\leq \frac{1}{2}G^{(1)}_{r}(e,x), \]

so that

\[ G^{(1)}_{r}(e,x)\leq 2\sum_{n\leq \Lambda d(e,x)+\Lambda}np_{r}^{(n)}(e,x)\leq (2\Lambda d(e,x)+2\Lambda)G(e,x|r). \]

This concludes the proof.

Going back to the proof of Lemma 4.8, we obtain the upper bound

\[ \sum_{k=0}^{n}\rho^{k}\sum_{\sigma\in \mathcal{H}_k}\frac{G(x_k,x_k\sigma|r)G(x_k\sigma,x_{k+1}|r)}{G(x_k,x_{k+1}|r)}\leq \sum_{k=0}^{n}\rho^{k}(\Lambda d(x_{k},x_{k+1})+\Lambda). \]

We fix $N$ and define $\varphi _1^{(N)}$ by

(13)\begin{equation} \varphi_1^{(N)}(x)=1+\sum_{k=0}^{n-1}\rho^{k}(\Lambda d(x_{k},x_{k+1})+\Lambda) \end{equation}

for any word $x$ of length $n\leq N$. We extend $\varphi _1^{(N)}$ to a function on $\overline {\Sigma }_A$ declaring $\varphi _1^{(N)}$ to be constant on cylinders of length $N$. For fixed $x$, the sequence $\varphi _1^{(N)}(x)$ is non-decreasing. Let $\varphi _{1}(x)$ be its limit, possibly infinite if $x\in \partial \Sigma _A$. We now prove that $\varphi _1$ is integrable with respect to $m$. This is based on the following.

Letting $E$ be a set, a transition kernel $p$ is a function $p:E\times E\rightarrow [0,+\infty )$. We fix a base point $x_0\in E$. We say that $p$ is finite if its total mass is finite, that is,

\[ \sum_{x\in E}p(x_0,x)<+\infty. \]

If the total mass is one, then $p$ defines a Markov chain $Z_n$ on $E$. Otherwise, $p$ still defines a chain $Z_n$ with transition given by $p$. We let $z_n$ be the increments of this chain. Whenever $E$ is endowed with a distance $d$, we say that $p$ is $C$-quasi-invariant if there exists $C$ such that for any $k$,

\[ d(Z_k,Z_{k+1})\leq C d(e,z_{k+1}). \]

Lemma 4.10 Let $p$ be a finite $C$-quasi-invariant transition kernel on a countable metric space $(E,d)$. Let $x_0$ be a fixed point in $E$. Assume that $p$ has exponential moments in the sense that

\[ \sum_{x\in E}p(x_0,x){e}^{\alpha d(x_0,x)} \]

for some positive $\alpha$. Then, for any $\beta >0$, there exists $\lambda >0$ and $C_{\lambda }$ such that for any $x\in E$,

\[ \sum_{n\leq d(e,x)/\lambda}p^{(n)}(x_0,x)\leq C_{\lambda}{e}^{-\beta d(e,x)}, \]

where $p^{(n)}$ denotes the $n$th power of convolution of $p$.