2.1 Topological dynamics

A continuous action $\Gamma \times Z\to Z$ of a topological group on a topological space is minimal if Z contains no proper closed $\Gamma $ -invariant subsets or, equivalently, if every $\Gamma $ -orbit is dense in Z. A point $z\in Z$ is a wandering point for an action $\Gamma \times Z\to Z$ if there exists a neighborhood U of z such that $g U\cap U=\emptyset $ for all but finitely many $g\in \Gamma $ . If the space Z is metrizable, then a point $z\in Z$ is not a wandering point if and only if there exist a sequence $(g_k)$ of distinct elements in $\Gamma $ and a sequence $(z_k)$ in Z converging to z such that $g_kz_k\to z$ . For further discussion of dynamical relations between points under group actions, see [Reference Kapovich and Leeb22, §4.3].

A continuous action $\Gamma \times Z\to Z$ of a discrete group $\Gamma $ on a compact metrizable topological space Z is a convergence action if the product action of $\Gamma $ on $Z^3$ restricts to a properly discontinuous action on

$$\begin{align*}T(Z)=\{(z_1, z_2, z_3)\in Z^3\mid \mathrm{card}\{z_1, z_2, z_3\}=3\}. \end{align*}$$

Equivalently, a continuous action of a discrete group is a convergence action if every sequence $(g_k)$ contains a subsequence $(g_{k_j})$ which is constant or converges to a point $z_+\in Z$ uniformly on compacts in $Z-\{z_-\}$ for some $z_-\in Z$ ; see [Reference Bowditch6, Proposition 7.1]. In this situation, the inverse sequence $(g_{k_j}^{-1})$ converges to $z_-$ uniformly on compacts in $Z-\{z_+\}$ . The set of such limit points $z_+$ is the limit set $\Lambda $ of the action of $\Gamma $ ; this is a closed $\Gamma $ -invariant subset of Z. Observe that a convergence action need not be faithful but it necessarily has finite kernel, provided that $T(Z)\ne \emptyset $ . A convergence action on Z is called uniform if it is cocompact on $T(Z)$ .

Item (1) of the following theorem can be found in [Reference Tukia36, Theorem 2S]; for item (2) see [Reference Tukia37, Theorem 1A] for instance.

2.2 Coarse geometry

A metric space $(X,d)$ is proper if the closed ball $\overline {B}_r(x)$ is compact for every $x\in X$ and every $r>0$ . Note that proper metric spaces are complete. A metric space $(X,d)$ is called a geodesic space if every pair of points $x,y \in X$ can be joined by a geodesic segment $xy$ , that is, an isometric embedding of an interval into X joining x to y.

Definition 2.5 (Quasi-geodesic)

Let I be an interval of $\mathbb {R}$ (or its intersection with $\mathbb {Z}$ ) and $(X,d)$ a metric space. A map $c:I\to X$ is called an $(A,C)$ -quasi-geodesic with constants $A\geq 1$ and $C\geq 0$ if for all $t,t'\in I$ ,

$$\begin{align*}\frac{1}{A}|t-t'|-C\le d(c(t),c(t'))\le A|t-t'|+C. \end{align*}$$

Let X be a proper $\delta $ -hyperbolic space. Two geodesic rays $\mathbb {R}_{\ge 0}\to X$ are said to be asymptotic if the Hausdorff distance between their images is finite. Being asymptotic is an equivalence relation on the set of geodesic rays. The set of equivalence classes of geodesic rays in X is called the visual boundary of X and denoted by $\partial _{\infty } X$ . In view of the Morse lemma for hyperbolic spaces (see [Reference Bridson and Haeiger7, Theorem III.H.1.7] or [Reference Druţu and Kapovich15, Lemma 11.105] for example), one can also define $\partial _{\infty } X$ as the set of equivalence classes of quasi-geodesic rays $\mathbb {R}_{\ge 0}\to X$ . We will use the notation $x\xi $ for a geodesic ray in X emanating from x and representing the point $\xi \in \partial _{\infty } X$ .

Fix $r>2\delta $ and let $c_0:\mathbb {R}_{\ge 0}\to X$ be a geodesic ray representing $\xi \in \partial _{\infty } X$ with $c_0(0)=x$ . A topology on $\partial _{\infty } X$ is given by setting the basis of neighborhoods of $\xi $ to be the collection $\{V_k(\xi )\}_{k\in \mathbb {N}}$ , where $V_k(\xi )$ is the set of equivalence classes of geodesic rays c such that $c(0)=x$ and $d(c(k),c_0(k))<r$ . This topology extends to the visual compactification of X

$$\begin{align*}\overline{X}:= X\cup \partial_{\infty} X, \end{align*}$$

which is a compact metrizable space. We refer to [Reference Bridson and Haeiger7, III.H.3.6] for details.

Let $x,y,z \in X$ . The Gromov product of y and z with respect to x is defined by

$$\begin{align*}(y \cdot z)_x := \frac{1}{2}(d(x,y)+d(x,z)-d(y,z)). \end{align*}$$

The Gromov product is extended to $X\cup \partial _{\infty } X$ by

$$\begin{align*}(y\cdot z)_x:=\sup \liminf_{k,j\to \infty} (y_k\cdot z_j)_x, \end{align*}$$

where the supremum is taken over all sequences $(y_k)$ and $(z_j)$ in X such that $\lim y_k=y$ and $\lim z_j=z$ .

Definition 2.7 (Visual metric)

Let X be a hyperbolic space with base point $x\in X$ . A metric $d_a$ on $\partial _{\infty } X$ is called a visual metric with parameter $a>1$ if there exist constants $k_1, k_2>0$ such that

$$\begin{align*}k_1a^{-(\xi\cdot \xi')_x} \leq d_a(\xi,\xi') \leq k_2a^{-(\xi\cdot \xi')_x} \end{align*}$$

for all $\xi ,\xi '\in \partial _{\infty } X$ .

For every $a>1$ sufficiently close to $1$ , a proper hyperbolic space admits a visual metric $d_a$ which induces the same topology as the topology on $\partial _{\infty } X$ described above. We refer to [Reference Bridson and Haeiger7, Chapter III.H.3] for more details on constructing visual metrics.

In the rest of the section, we discuss hyperbolic groups and their relation to convergence actions.

The Gromov boundary $\partial _{\infty }\Gamma $ of a hyperbolic group $\Gamma $ is defined as the visual boundary of a Cayley graph X of $\Gamma $ . The closure of $\Gamma \subset X$ in the visual compactification $\overline {X}$ equals $\Gamma \cup \partial _{\infty } \Gamma $ and is denoted $\overline \Gamma $ ; it is the visual compactification of $\Gamma $ .

Every hyperbolic group $\Gamma $ acts on its visual compactification $\overline {\Gamma }$ by homeomorphisms. This action is a convergence action; see [Reference Tukia36, Theorem 3.A] and [Reference Freden17]. If a sequence $(c_k)$ in $\Gamma $ represents a quasi-geodesic ray within bounded distance from a geodesic ray $g\xi $ ( $g\in \Gamma $ , $\xi \in \partial _{\infty }\Gamma $ ), then this sequence, regarded as a sequence of maps $\overline \Gamma \to \overline \Gamma $ , converges to $\xi $ uniformly on compacts in $\overline \Gamma -\{\xi '\}$ for some $\xi '\in \partial _{\infty }\Gamma $ . We will use the following refinement of this property later in the proof of Theorem 6.2:

Furthermore, the action of $\Gamma $ on $\partial _{\infty }\Gamma $ is a uniform convergence action. In particular, if $\Gamma $ is nonelementary, then this action has finite kernel (the unique maximal finite normal subgroup of $\Gamma $ ), is minimal and $\partial _{\infty }\Gamma $ is a perfect topological space; compare Theorem 2.4. We refer to [Reference Druţu and Kapovich15, Lemma 11.130] for more details.

Conversely, Bowditch [Reference Bowditch6] gave a topological characterization of hyperbolic groups and their Gromov boundaries as uniform convergence actions $\Gamma \times Z\to Z$ of discrete groups on perfect metrizable topological spaces:

2.3 Symmetric spaces

We collect several definitions regarding the geometry of symmetric spaces that are necessary for our presentation in Sections 5, 6 and 7. For more details, readers may refer to [Reference Ballmann, Gromov and Schroeder1], [Reference Eberlein16] or [Reference Kapovich, Leeb and Porti26, Section 2.]

Throughout Sections 2.3 and 2.4, let G be a semisimple Lie group and X the associated symmetric space.

The visual boundary $\partial _{\infty } X$ of X has a topological spherical building structure, the Tits building associated to X. Let $a_{\mathrm {mod}}$ denote the model apartment of this building and W the Weyl group acting isometrically on $a_{\mathrm {mod}}$ . We will fix a chamber ${\sigma _{\mathrm {mod}}}\subset a_{\mathrm {mod}}$ and call it the spherical model Weyl chamber; it is a fundamental domain for the W-action on $a_{\mathrm {mod}}$ . Let $w_0\in W$ denote the unique element sending ${\sigma _{\mathrm {mod}}}$ to the opposite chamber $-{\sigma _{\mathrm {mod}}}$ . Then the opposition involution $\iota $ of the model chamber ${\sigma _{\mathrm {mod}}}$ is defined as $\iota =-w_0$ .

We denote by $\Delta :=\Delta _{\mathrm {mod}}$ the model Weyl chamber corresponding to ${\sigma _{\mathrm {mod}}}$ , a fundamental domain for the W-action on the model (maximal) flat $F_{\mathrm {mod}}$ of X. We will use the notation $d_{\Delta }$ for the G-invariant $\Delta $ -valued distance function on X.

Consider the induced action of G on $\partial _{\infty } X$ . Every orbit intersects every chamber exactly once, so there is a natural identification $\partial _{\infty } X/G\cong {\sigma _{\mathrm {mod}}}$ . The projection $\theta :\partial _{\infty } X\to \partial _{\infty } X/G$ is called the type map. Let ${\tau _{\mathrm {mod}}}$ be a face of ${\sigma _{\mathrm {mod}}}$ . We will assume, in what follows, that the simplex ${\tau _{\mathrm {mod}}}$ is $\iota $ -invariant. The ${\tau _{\mathrm {mod}}}$ -flag manifold $\mathrm {Flag}({\tau _{\mathrm {mod}}})$ is the space of simplices of type ${\tau _{\mathrm {mod}}}$ in $\partial _{\infty } X$ . It has a structure of a compact smooth manifold and can be identified with the quotient space $G/P_{\tau }$ , where $P_{\tau }<G$ is the stabilizer subgroup of a simplex $\tau \subset \partial _{\infty } X$ of type $\theta (\tau )={\tau _{\mathrm {mod}}}$ . Two simplices $\tau _1$ and $\tau _2$ in $\partial _{\infty } X$ are said to be antipodal if they are opposite in an apartment containing both of them; their types are related by $\theta (\tau _2)=\iota \theta (\tau _1)$ . A subset E of $\mathrm {Flag}({\tau _{\mathrm {mod}}})$ is said to be antipodal if any two distinct elements of E are antipodal. A map into $\mathrm {Flag}({\tau _{\mathrm {mod}}})$ is antipodal if it is injective and has an antipodal image. We shall always assume that $\mathrm {Flag}({\tau _{\mathrm {mod}}})$ is equipped with an auxiliary Riemannian metric.

Let $\tau \subset \partial _{\infty } X$ be a simplex of type ${\tau _{\mathrm {mod}}}$ . The star $\mathrm {st}(\tau )\subset \partial _{\infty } X$ is the union of all chambers containing $\tau $ . We denote by $\partial \,\mathrm {st}(\tau )$ the union of all simplices in $\mathrm {st}(\tau )$ which do not contain $\tau $ , and define $\mathrm {ost}(\tau ):=\mathrm {st}(\tau )-\partial \,\mathrm {st}(\tau )$ . We denote by $W_{{\tau _{\mathrm {mod}}}}$ the W-stabilizer of ${\tau _{\mathrm {mod}}}$ . A subset $\Theta \subset {\sigma _{\mathrm {mod}}}$ is $W_{\tau _{\mathrm {mod}}}$ -convex if its symmetrization $W_{\tau _{\mathrm {mod}}}\Theta \subset a_{\mathrm {mod}}$ is convex; for such a subset $\Theta $ , we define $\mathrm {st}_{\Theta }(\tau ):=\mathrm {st}(\tau )\cap \theta ^{-1}(\Theta )$ .

For $x\in X$ and $S\subset \partial _{\infty } X$ , the Weyl cone $V(x,S)\subset X$ is defined to be the union of rays $x\xi $ ( $\xi \in S$ ). For a flat $f\subset X$ the parallel set, $P(f)$ is the union of all flats $f'$ such that $\partial _{\infty } f'=\partial _{\infty } f$ . For a pair of antipodal simplices $\tau _1,\tau _2\subset \partial _{\infty } X$ , we define $P(\tau _1,\tau _2)$ to be the parallel set of the unique minimal flat whose ideal boundary contains $\{\tau _1,\tau _2\}$ ; equivalently, it is the union of all geodesics in X which are forward/backward asymptotic to points in $\tau _1$ and $\tau _2$ .

Let $xy$ be a ${\tau _{\mathrm {mod}}}$ -regular oriented geodesic segment in X and $x\xi $ the geodesic ray extending $xy$ . We denote $\tau (xy)$ the unique simplex $\tau \in \mathrm {Flag}({\tau _{\mathrm {mod}}})$ such that $\xi \in \mathrm {ost}(\tau )$ . Let $\Theta \subset \mathrm {int}_{\tau _{\mathrm {mod}}}({\sigma _{\mathrm {mod}}})$ . Then $xy$ is $\Theta $ -regular if $\theta (\xi )\in \Theta $ . If $x_1x_2$ is $\Theta $ -regular, we define its $\Theta $ -diamond by

$$\begin{align*}\Diamond_{\Theta}(x_1,x_2)=V(x_1,\mathrm{st}_{\Theta}(\tau(x_1x_2)))\cap V(x_2,\mathrm{st}_{\Theta}(\tau(x_2x_1))). \end{align*}$$

It is shown in [Reference Kapovich, Leeb and Porti26, Theorem 5.53] that $(\Theta ,R)$ -Morse quasi-geodesic rays (resp. lines) are uniformly Hausdorff close to a uniformly ${\tau _{\mathrm {mod}}}$ -regular Finsler geodesic rays (resp. lines).

2.4 Anosov subgroups

We recall the definitions of Anosov and nonuniformly Anosov subgroups given by Kapovich, Leeb and Porti in [Reference Kapovich, Leeb and Porti26, Definitions 5.43 and 5.62]. Many other equivalent characterizations of the Anosov subgroups are established in [Reference Kapovich, Leeb and Porti26, Theorem 5.47], and their equivalence with the original definitions given by Labourie [Reference Labourie28] and Guichard and Wienhard [Reference Guichard and Wienhard19] is proven in [Reference Kapovich, Leeb and Porti26, Section 5.11].

3.1 The expansion condition

In order to define the expansion condition we need a little preparation.

Let f be a homeomorphism of M. Given $\lambda>1$ and $U\subset M$ , we say f is $(\lambda ,U)$ -expanding (or $\lambda $ -expanding on U) if

$$\begin{align*}d(f(x),f(y))\ge\lambda\cdot d(x,y) \end{align*}$$

for all $x,y\in U$ . In this case, we also say U is a $(\lambda ,f)$ -expanding subset. Note that f is $(\lambda ,U)$ -expanding if and only if $f^{-1}$ is $(\lambda ^{-1},f(U))$ -contracting, that is,

$$\begin{align*}d(f^{-1}(x),f^{-1}(y))\le\frac{1}{\lambda}\cdot d(x,y) \end{align*}$$

for all $x,y\in f(U)$ .

Given $\delta>0$ , a $(\lambda ,U)$ -expanding homeomorphism f is said to be $(\lambda ,U;\delta )$ -expanding if

$$\begin{align*}B_{\lambda\eta}(f(x))\subset f(B_{\eta}(x))\; \ \mathrm{whenever} \ B_{\eta}(x)\subset U \ \mathrm{ and } \ \eta\le\delta. \end{align*}$$

Clearly, if f is $(\lambda ,U;\delta )$ -expanding, then it is also $(\lambda ,U;\delta ')$ -expanding for every $\delta '\le \delta $ .

Proof. Suppose to the contrary that there exists a point

$$\begin{align*}y\in B_{\lambda\eta}(f(x))\setminus f(B_{\eta}(x)). \end{align*}$$

Let c be a geodesic path in M connecting $f(x)$ and y; the length of this geodesic is less than $\lambda \eta $ . Since $f(B_{\eta }(x))$ is open in M, the path c crosses the boundary of $f(B_{\eta }(x))$ at a point z. Because f is $\lambda $ -expanding on the set U containing $B_{\eta }(x)$ and $f^{-1}(z)\in \partial B_{\eta }(x)$ , we have

$$\begin{align*}d(f(x), y)\ge d(f(x), z)\ge \lambda d(x, f^{-1}(z))=\lambda \eta. \end{align*}$$

This contradicts the assumption that $y\in B_{\lambda \eta }(f(x))$ , that is, $d(f(x), y)< \lambda \eta $ .

The implication in the lemma does not hold for general metric spaces. As a simple example, consider a compact metric space M of diameter D, which is not a singleton, but contains an isolated point x. Consider $f=\mathrm {Id}_M$ and take a neighborhood U of x and $\eta>0$ such that $\{x\}=B_{\eta }(x)\subset U=\{x\}$ . Then f is $(\lambda ,U)$ -expanding for any $\lambda>1$ . Thus, taking $\lambda $ such that $\lambda \eta \ge D$ , we see that

$$\begin{align*}M=B_{\lambda\eta}(f(x))\subsetneq f(B_{\eta}(x))=B_{\eta}(x)=\{x\}. \end{align*}$$

However, we note the following fact:

We are now ready to define the expansion condition. Let $\mathrm {LHomeo}(M)$ denote the group of locally bi-Lipschitz homeomorphisms of M, that is, homeomorphisms of M whose restrictions to compact subsets are bi-Lipschitz.

If $\rho :\Gamma \to \mathrm {LHomeo}(M)$ is an expanding action, let $|\cdot |_{\varSigma }$ (resp. $d_{\varSigma }$ ) denote the word length (resp. the word metric) on the group $\Gamma $ with respect to the generating set $\varSigma $ from Definition 3.4. Then the L-Lipschitz property (i) implies that

(3.5) $$ \begin{align} \mathrm{the \ map} \ \rho(g)\ \mathrm{is} \ L^k\mathrm{-Lipschitz \ on} \ N_{\delta/L^{k-1}}(\Lambda) \end{align} $$

for every $g\in \Gamma $ with $|g|_{\varSigma }=k\in \mathbb {N}$ .

It is possible that an action $\rho :\Gamma \to \mathrm {LHomeo}(M)$ is expanding with different expansion data. In such a case, we define a relation on the expansion data as follows.

Definition 3.7. Let $\rho :\Gamma \to \mathrm {LHomeo}(M)$ be expanding at $\Lambda $ . Let $\mathcal {D}_0=(\mathcal {I}_0,\mathcal {U}_0,\varSigma _0, \delta _0,L_0,\lambda _0)$ and $\mathcal {D}_1=(\mathcal {I}_1,\mathcal {U}_1,\varSigma _1, \delta _1,L_1,\lambda _1)$ be its expansion data, where $\mathcal {U}_k=\{U_{k,i} \mid i\in \mathcal {I}_k\}$ and $\varSigma _k=\{s_{k,i} \mid i\in \mathcal {I}_k\}$ for $k=0,1$ . We write $\mathcal {U}_0 \prec \mathcal {U}_1$ (resp. $\varSigma _0 \prec \varSigma _1$ ) if $\mathcal {I}_0\subset \mathcal {I}_1$ and $U_{0,i} \subset U_{1,i}$ (resp. $s_{0,i}=s_{1,i}$ ) for every $i\in \mathcal {I}_0$ . Furthermore, we write $\mathcal {D}_0\prec \mathcal {D}_1$ and say $\mathcal {D}_1$ is a refinement of $\mathcal {D}_0$ if

$$\begin{align*}\mathcal{I}_0\subset\mathcal{I}_1,\; \mathcal{U}_0\prec\mathcal{U}_1,\; \varSigma_0\prec\varSigma_1,\; \delta_0>\delta_1,\; L_0\le L_1,\; \lambda_0\ge\lambda_1. \end{align*}$$

Note the strict inequality $\delta _0>\delta _1$ , which will be pertinent to Section 4.6.

3.2 Toy examples of expanding actions

The most basic example of an expanding action is a cyclic hyperbolic group of Möbius transformations acting on the unit circle $S^1$ . Namely, we consider the Poincaré (conformal) disk model of $\mathbb {H}^2$ in $\mathbb {C}=\mathbb {R}^2$ and endow $M=S^1=\partial _{\infty }\mathbb {H}^2$ with the induced Euclidean metric. Let $\gamma \in \mathrm {Isom}(\mathbb {H}^2)$ be a hyperbolic element and $\Gamma = \langle \gamma \rangle \cong \mathbb {Z}$ . The limit set of the group $\Gamma $ is $\Lambda =\{\gamma _-, \gamma _+\}$ , with $\gamma _+$ the attracting and $\gamma _-$ the repelling fixed points of $\gamma $ in M. Expanding subsets $U_{\alpha }, U_{\beta }$ for $\gamma , \gamma ^{-1}$ are sufficiently small arcs containing $\gamma _-, \gamma _+$ , respectively.

Explicit expanding subsets can be found by considering the isometric circles $I_{\gamma }$ and $I_{\gamma ^{-1}}$ of $\gamma $ and $\gamma ^{-1}$ , respectively. See Figure 1(Left). (For the definition of isometric circles (or spheres) and their relation to Ford and Dirichlet fundamental domains, we refer to [Reference Maskit31, IV.G].) The arc of $I_{\gamma }$ in $\mathbb {H}^2$ is a complete geodesic which is the perpendicular bisector of the points o and $\gamma ^{-1}(o)$ , where o denotes the Euclidean center of the Poincaré disk. Then we obtain a $(\lambda ,\gamma )$ -expanding subset $U_{\alpha }$ (with $\lambda>1$ ) by cutting down slightly the open arc of $S^1=\partial _{\infty }\mathbb {H}^2$ inside $I_{\gamma }$ . See also the discussion in the beginning of Section 7.2.

Figure 1 Expanding arcs for $\gamma $ and $\gamma ^{-1}$ are colored gray in both examples. (Left) A hyperbolic transformation $\gamma $ of $\mathbb {H}^2$ . (Right) A covering of degree $3$ .

Example 3.9 (k-fold nontrivial covering)

A more interesting example is obtained by taking a degree $k>1$ covering $p: S^1\to S^1$ of the preceding example. See Figure 1(Right) for the case of $k=3$ . The preimage of $\Lambda =\{\gamma _-, \gamma _+\}$ consists of $2k$ points, and we can lift $\gamma $ to a diffeomorphism $\widetilde \gamma : S^1\to S^1$ fixing all these points. Let $\widetilde \rho : \Gamma =\langle \gamma \rangle \cong \mathbb {Z}\to \mathrm {Diff}(S^1)$ be the homomorphism sending the generator $\gamma $ of $\Gamma $ to $\widetilde \gamma $ . The preimages $p^{-1}(U_{\alpha })$ , $p^{-1}(U_{\beta })$ break into connected components

$$\begin{align*}\widetilde{U}_{\alpha_i},\;\widetilde{U}_{\beta_i}\quad(i=1,\ldots,k), \end{align*}$$

and the mappings $\widetilde \gamma $ and $\widetilde \gamma ^{-1}$ act as expanding maps on each of these components. Therefore, we set

$$\begin{align*}\mathcal{I}= \{\alpha_1,\ldots,\alpha_k, \beta_1,\ldots,\beta_k\} \end{align*}$$

and define the map $s:\mathcal {I}\to \varSigma $ ,

$$\begin{align*}\alpha_i\mapsto s_{\alpha_i}= \gamma,\quad \beta_i\mapsto s_{\beta_i}= \gamma^{-1} \end{align*}$$

from this index set to the generating set $\varSigma =\{\gamma , \gamma ^{-1}\}$ of $\Gamma $ . Then $\widetilde {U}_{\alpha _i}$ , $\widetilde {U}_{\beta _i}$ will be expanding subsets for the actions $\widetilde \rho (s_{\alpha _i})$ , $\widetilde \rho (s_{\beta _i})$ on $S^1$ . (Note that the entire preimage $p^{-1}(U_{\alpha })$ (resp. $p^{-1}(U_{\beta })$ ) is not an expanding subset for the action $\widetilde \rho (s_{\alpha _i})$ (resp. $\widetilde \rho (s_{\beta _i})$ ).)

The same construction works for surface group actions; see Example 7.3.

A trivial example where $U_i=\emptyset $ for an index $i\in \mathcal {I}$ is the action of a cyclic group $\Gamma = \langle \gamma \rangle \cong \mathbb {Z}$ generated by a loxodromic transformation $\gamma (z)=mz$ , $|m|>1$ , on $M=\mathbb {C}$ (with the standard Euclidean metric) and $\Lambda =\{0\}$ . Any open subset of M containing $0$ is an expanding subset for $\gamma $ , while the expanding subset for $\gamma ^{-1}$ is empty.

3.3 Expansion enables encoding

Given an expanding action $\rho :\Gamma \to \mathrm {LHomeo}(M)$ with an invariant subset $\Lambda $ the expansion condition (Definition 3.4) enables us to encode points of $\Lambda $ by sequences in the finite index set $\mathcal {I}$ . Roughly speaking, for each $x\in \Lambda $ , we keep zooming in toward x using the ‘microscope’, namely, the expanding maps $\rho (s_{\alpha (k)}^{-1})$ . This is the heuristic Sullivan himself used in his talk according to F. Bonahon. (We thank Bonahon for telling us his personal anecdote.)

Definition 3.10 (Codes)

Suppose an action $\rho :\Gamma \to \mathrm {LHomeo}(M)$ is expanding at $\Lambda $ with a datum $\mathcal {D}=(\mathcal {I},\mathcal {U},\varSigma , \delta ,L,\lambda )$ . Let $x\in \Lambda $ and fix a real number $\eta $ such that $0<\eta \le \delta $ . Given a sequence $\alpha :\mathbb {N}_0\to \mathcal {I},\;k\mapsto \alpha (k)$ , define a sequence $p:\mathbb {N}_0\to \Lambda ,\;k\mapsto p_k$ inductively by

$$ \begin{align*} p_0&=x,\\ p_{k+1}&=\rho(s_{\alpha(k)}^{-1})(p_k). \end{align*} $$

The sequence $\alpha $ (or the pair $(\alpha ,p)$ of sequences) is called a $(\mathcal {D},\eta )$ -code for x if

$$\begin{align*}B_{\eta}(p_k)\subset U_{\alpha(k)}\end{align*}$$

for every $k\in \mathbb {N}$ . (When $\mathcal {D}$ is understood, we simply say an $\eta $ -code in order to emphasize the use of $\eta $ -balls). A $(\mathcal {D},\eta )$ -code $\alpha $ for x is said to be special if $\alpha (0)$ satisfies $B_{\eta }(x)\subset U_{\alpha (0)}$ . We denote by $\mathrm {Code}_x(\mathcal {D},\eta )$ the set of all $(\mathcal {D},\eta )$ -codes for $x\in \Lambda $ .

Every $\eta $ -ball centered at a point on $\Lambda $ is contained in some member of $\mathcal {U}$ since $\eta \le \delta \le \delta _{\mathcal {U}}$ . This enables us to construct codes. A code gives rise to a family of $\eta $ -balls centered at points in $\Lambda $ each of which is a $\lambda $ -expanding domain for the action on M of some element of $\Gamma $ .

Definition 3.12 (Rays)

Let $\rho :\Gamma \to \mathrm {LHomeo}(M)$ be expanding at $\Lambda $ with the datum $\mathcal {D}$ . Given a $(\mathcal {D},\eta )$ -code $\alpha $ for $x\in \Lambda $ , the $(\mathcal {D},\eta )$ -ray (or simply $\eta $ -ray) associated to $\alpha $ is a sequence $c^{\alpha }:\mathbb {N}_0\to \Gamma $ defined by

$$ \begin{align*} c^{\alpha}_k=s_{\alpha(0)}s_{\alpha(1)}\cdots s_{\alpha(k)}. \end{align*} $$

We set $\mathrm {Ray}_x(\mathcal {D},\eta ):=\{c^{\alpha }\mid \alpha \in \mathrm {Code}_x(\mathcal {D},\eta )\}$ and interpret c as a map

$$\begin{align*}c:\mathrm{Code}_x(\mathcal{D},\eta)\to\mathrm{Ray}_x(\mathcal{D},\eta),\quad\alpha\mapsto c^{\alpha}. \end{align*}$$

Now, we will see that, for every $x\in \Lambda $ , each $\eta $ -code for x gives rise to a nested sequence of neighborhoods of x whose diameters tend to $0$ exponentially fast.

Lemma 3.14. Let $\rho :\Gamma \to \mathrm {LHomeo}(M)$ be expanding at $\Lambda $ with a datum $\mathcal {D}=(\mathcal {I},\mathcal {U},\varSigma , \delta ,L,\lambda )$ , and let $0<\eta \le \delta $ . If $(\alpha ,p)$ is a $(\mathcal {D},\eta )$ -code for $x\in \Lambda $ , then the sequence of neighborhoods of x

$$ \begin{align*} x\in\rho(c^{\alpha}_k)[B_{\eta}(p_{k+1})]\quad(k\in\mathbb{N}_0) \end{align*} $$

is nested and exponentially shrinking. More precisely, we have

1. (i) $\rho (c^{\alpha }_k)[B_{\eta }(p_{k+1})]\subset \rho (c^{\alpha }_{k-1})[B_{\eta }(p_k)]$ for $k\in \mathbb {N}$ , and

2. (ii) $\rho (c^{\alpha }_k)[B_{\eta }(p_{k+1})] \subset B_{L\eta /\lambda ^{k}}(x)$ for all $k\in \mathbb {N}_0$ .

Consequently, we have the equality

$$\begin{align*}\{x\}=\bigcap_{k=0}^{\infty}\rho(c^{\alpha}_k)[B_{\eta}(p_{k+1})]. \end{align*}$$

The map $\rho (c^{\alpha }_k)^{-1}$ is $\lambda ^k/L$ -expanding on the neighborhood $\rho (c^{\alpha }_k)[B_{\eta }(p_{k+1})]$ of x for all $k\in \mathbb {N}_0$ , and the map

$$\begin{align*}\rho(c^{\alpha}_k)^{-1}\rho(c^{\alpha}_j)=\rho(s_{\alpha(k)}^{-1}s_{\alpha(k-1)}^{-1}\cdots s_{\alpha(j+1)}^{-1}) \end{align*}$$

is $\lambda ^{k-j}$ -expanding on $\rho (c^{\alpha }_j)^{-1}\big [\rho (c^{\alpha }_k)[B_{\eta }(p_{k+1})]\big ]$ for all $0\le j< k$ .

Proof. Suppose $(\alpha ,p)$ is an $\eta $ -code for $x\in \Lambda $ , and let $k\in \mathbb {N}$ . Since $B_{\eta }(p_k)\subset U_{\alpha (k)}$ and the homeomorphism $\rho (s_{\alpha (k)}^{-1})$ is $(\lambda ,U_{\alpha (k)};\delta )$ -expanding with $\rho (s_{\alpha (k)}^{-1})(p_k)=p_{k+1}$ , we have by definition

(3.15) $$ \begin{align} B_{\lambda\eta}(p_{k+1}) \subset \rho(s_{\alpha(k)}^{-1})[B_{\eta}(p_k)] \subset \rho(s_{\alpha(k)}^{-1})[U_{\alpha(k)}]. \end{align} $$

On the other hand, the inverse map $\rho (s_{\alpha (k)})$ is $(\lambda ^{-1},\rho (s_{\alpha (k)}^{-1})[U_{\alpha (k)}])$ -contracting as we saw in Remark 3.6(d). Since $B_{\eta }(p_{k+1})\subset \rho (s_{\alpha (k)}^{-1})[U_{\alpha (k)}]$ , we obtain

(3.16) $$ \begin{align} \rho(s_{\alpha(k)})[B_{\eta}(p_{k+1})] \subset B_{\eta/\lambda}(p_k) \subset B_{\eta}(p_k). \end{align} $$

See Figure 2. By a similar reasoning, we inductively obtain

$$ \begin{align*} \rho(s_{\alpha(k-1)}s_{\alpha(k)})[B_{\eta}(p_{k+1})] &\subset B_{\eta/\lambda^2}(p_{k-1}),\\ &\;\;\vdots \\ \rho(s_{\alpha(1)}\cdots s_{\alpha(k-1)}s_{\alpha(k)})[B_{\eta}(p_{k+1})] &\subset B_{\eta/\lambda^{k}}(p_1). \end{align*} $$

Lastly, the map $\rho (s_{\alpha (0)})$ is L-Lipschitz on $B_{\eta /\lambda ^{k}}(p_1)\subset N_{\delta }(\Lambda )$ by Definition 3.4(i). Thus, we see that

(3.17) $$ \begin{align} \rho(s_{\alpha(0)}\cdots s_{\alpha(k-1)}s_{\alpha(k)})[B_{\eta}(p_{k+1})] \subset B_{L\eta/\lambda^{k}}(p_0). \end{align} $$

Figure 2 Actions of $\rho (s_{\alpha (k)}^{-1})$ and $\rho (s_{\alpha (k)})$ for $k\in \mathbb {N}$ .

Since $\rho (c^{\alpha }_k)(p_{k+1})=p_0=x$ , the inclusion (3.17) can be written as

$$ \begin{align*} x\in\rho(c^{\alpha}_k)[B_{\eta}(p_{k+1})] \subset B_{L\eta/\lambda^{k}}(x) \end{align*} $$

for all $k\in \mathbb {N}_0$ . Moreover, from equation (3.16) we have

(3.18) $$ \begin{align} \begin{aligned} \rho(c^{\alpha}_k)[B_{\eta}(p_{k+1})] &=\rho(s_{\alpha(0)}\cdots s_{\alpha(k-1)} s_{\alpha(k)})[B_{\eta}(p_{k+1})] \\ &\subset \rho(s_{\alpha(0)}\cdots s_{\alpha(k-1)})[B_{\eta}(p_k)]\\ &= \rho(c^{\alpha}_{k-1})[B_{\eta}(p_k)] \end{aligned} \end{align} $$

for all $k\in \mathbb {N}$ . From these follows the rest of the lemma.

A number of corollaries will follow.

We say an action $\rho :\Gamma \to \mathrm {Homeo}(M)$ with an invariant subset $\Lambda $ has an expansivity constant $c>0$ if for every distinct pair of points $x,y\in \Lambda $ there exists an element $g\in \Gamma $ such that $d(\rho (g)(x),\rho (g)(y))\ge c$ . Compare [Reference Coornaert and Papadopoulos11, Proposition 2.2.4].

Proof. Let $x,y\in \Lambda $ be distinct, and consider a $\delta $ -code $(\alpha ,p)$ for x. By Lemma 3.14, the sequence of neighborhoods of x

$$ \begin{align*} \rho(c^{\alpha}_k)[B_{\delta}(p_{k+1})]\quad (k\in\mathbb{N}_0) \end{align*} $$

is nested and exponentially shrinking. Thus, there exists an $n\in \mathbb {N}_0$ such that

$$ \begin{align*} y&\notin\rho(c^{\alpha}_n)[B_{\delta}(p_{n+1})], \ \mathrm{that \ is,} \ \rho(c^{\alpha}_n)^{-1}(y) \notin B_{\delta}(p_{n+1})=B_{\delta}(\rho(c^{\alpha}_n)^{-1}(x)). \end{align*} $$

Therefore, $d(\rho (c^{\alpha }_n)^{-1}(x),\rho (c^{\alpha }_n)^{-1}(y))\ge \delta $ as desired.

The following corollary will be crucial for Definition 6.1, which in turn plays an essential role when we discuss actions of hyperbolic groups in Section 6. The corollary is also the reason why we need the assumption that no point of $\Lambda $ is isolated in M.

Proof. Let $(\alpha ,p)$ be an $\eta $ -code for $x\in \Lambda $ . Since no point of $\Lambda $ is isolated in M, for each $k\in \mathbb {N}_0$ we can choose a point $y_k$ such that

$$\begin{align*}x\neq y_k\in \rho(c^{\alpha}_k)[B_{\eta}(p_{k+1})]\cap N_{\delta/L^{k}}(\Lambda). \end{align*}$$

Let $k,j\in \mathbb {N}_0$ be such that $0\le j<k$ , and set $r_{kj}:=(c^{\alpha }_k)^{-1}c^{\alpha }_j$ . By (the last statement of) Lemma 3.14, the map $\rho (r_{kj})$ is $\lambda ^{k-j}$ -expanding on $\rho (c^{\alpha }_j)^{-1}\big [\rho (c^{\alpha }_k)[B_{\eta }(p_{k+1})]\big ]\supset \{\rho (c^{\alpha }_j)^{-1}(y_k),\rho (c^{\alpha }_j)^{-1}(x)\}$ , hence

$$\begin{align*}d(\rho(r_{kj})(y_{kj}),\rho(r_{kj})(x_{j})) \ge\lambda^{k-j}d(y_{kj},x_{j}), \end{align*}$$

where we set $y_{kj}:=\rho (c^{\alpha }_j)^{-1}(y_k)$ and $x_{j}:=\rho (c^{\alpha }_j)^{-1}(x)$ . On the other hand, we have

$$\begin{align*}L^{|r_{kj}|_{\varSigma}}d(y_{kj},x_{j})\ge d(\rho(r_{kj})(y_{kj}),\rho(r_{kj})(x_{j})) \end{align*}$$

from equation (3.5), since $x_{j}\in \Lambda $ while $y_{kj}\in N_{\delta /L^{k-j-1}}(\Lambda )$ and $|r_{kj}|_{\varSigma }\le k-j$ . From these two inequalities, we obtain

$$\begin{align*}d_{\varSigma}(c^{\alpha}_k,c^{\alpha}_j)=|r_{kj}|_{\varSigma}\ge\frac{\log\lambda}{\log L}\cdot(k-j). \end{align*}$$

Therefore, the $\eta $ -ray $c^{\alpha }$ is a $(\frac {\log L}{\log \lambda },0)$ -quasi-geodesic ray.

Another consequence of the encoding concerns the dynamics of the action of $\Gamma $ on $\Lambda $ . The action of $\Gamma $ on $\Lambda $ need not be minimal in general even if $\Gamma $ is a nonelementary hyperbolic group (see Example 7.4). Nevertheless, the action of $\Gamma $ on $\Lambda $ has no wandering points:

Proof. (1) By the compactness of $\Lambda $ , the sequence $p=(p_k)$ contains a subsequence $(p_{k_j+1})$ converging to some point $y\in \Lambda $ . The point y is then covered by infinitely many balls $B_{\eta /2}(p_{k_j+1})$ , $j\in \mathbb {N}$ . By Lemma 3.14, the corresponding elements $g_j:= c^{\alpha }_{k_j}\in \Gamma $ will send $B_{\eta /2}(y)\subset B_{\eta }(p_{k_j+1})$ to a subset $\rho (g_j)[B_{\eta /2}(y)]$ of diameter at most $L\eta /\lambda ^{k_j}$ containing x. From this, we conclude that the sequence $(\rho (g_j))$ converges to x uniformly on $B_{\eta /2}(y)$ .

(2) On the other hand, since, in particular, $p_{k_1+1}\in B_{\eta /2}(y)$ , we obtain

$$\begin{align*}\lim_{j\to\infty} \rho( c^{\alpha}_{k_j} ) (p_{k_1+1})= x. \end{align*}$$

Since $\rho (c^{\alpha }_{k_1})^{-1}(x)=p_{k_1+1}$ , it follows that for $h_j:= c^{\alpha }_{k_j} (c^{\alpha }_{k_1})^{-1}$ we have

$$\begin{align*}\lim_{j\to\infty} \rho(h_j)(x)=x. \end{align*}$$

3.4 The meandering hyperbolicity condition

We continue the discussion from the previous section. In order to define the meandering hyperbolicity condition, we need to introduce an equivalence relation $\sim ^N_{(\mathcal {D},\eta )}$ on the set $\mathrm {Ray}_x(\mathcal {D},\eta )$ of rays in $\Gamma $ associated to $(\mathcal {D},\eta )$ -codes for $x\in \Lambda $ ; recall Definitions 3.10 and 3.12.

We are now ready to define the meandering hyperbolicity condition. First, we recall the notion of S-hyperbolicity which Sullivan defined in [Reference Sullivan35].

By weakening the $(\mathcal {D},\eta ;N)$ -fellow-traveling condition in the definition of S-hyperbolicity to the $(\mathcal {D},\eta ;N)$ -equivalence, as well as considering a refined expansion datum (Definition 3.7), we define the notion of meandering hyperbolicity as follows.

4.1 Statement of the theorem: Meandering hyperbolicity implies stability

Suppose $\rho :\Gamma \to \mathrm {LHomeo}(M)$ is expanding at $\Lambda $ . Following Sullivan’s remark (at the very end of his paper [Reference Sullivan35]), we would like to talk about small perturbations of $\rho $ which are still expanding. For this, we only need information on a neighborhood of the compact subset $\Lambda $ .

We equip $\mathrm {LHomeo}(M)$ with what we call the compact-open Lipschitz topology. Given a compact subset $K\subset M$ , a positive number $\epsilon>0$ and $f\in \mathrm {LHomeo}(M)$ , let $U(f;K,\epsilon )$ denote the set of all $g\in \mathrm {LHomeo}(M)$ that are $\epsilon $ -close to f on K with respect to the metric $d_{\mathrm {Lip},K}$ defined by

(4.1) $$ \begin{align} d_{\mathrm{Lip},K}(f,g):= \sup_{x\in K} d(f(x),g(x))+ \sup_{\substack{x,y\in K \\ x\neq y}} \left|\frac{d(f(x),f(y))}{d(x,y)}-\frac{d(g(x),g(y))}{d(x,y)}\right| <\epsilon. \end{align} $$

We define the compact-open Lipschitz topology on $\mathrm {LHomeo}(M)$ as the topology generated by the collection of all such $U(f;K,\epsilon )$ . This topology enables us to control the image as well as the Lipschitz constant on a neighborhood of the compact subset $\Lambda $ .

Suppose $\rho :\Gamma \to \mathrm {LHomeo}(M)$ is expanding at $\Lambda $ with a datum $\mathcal {D}=(\mathcal {I},\varSigma )$ (Definition 3.4). Given a compact subset $K\subset M$ and $\epsilon>0$ , we say that a homomorphism $\rho ':\Gamma \to \mathrm {LHomeo}(M)$ is a $(K,\epsilon )$ -perturbation of $\rho $ if $\rho '(s_i)$ is $\epsilon $ -close to $\rho (s_i)$ on K for all $i\in \mathcal {I}$ . The set of all $(K,\epsilon )$ -perturbations of $\rho $ will be denoted by

(4.3) $$ \begin{align} U(\rho;K,\epsilon). \end{align} $$

Accordingly, we topologize $\mathrm {Hom}(\Gamma ,\mathrm {LHomeo}(M))$ via the topology of ‘algebraic convergence’ by identifying it, via the map $\rho \mapsto (\rho (s_i))_{i\in \mathcal {I}}$ , with a subset of $[\mathrm {LHomeo}(M)]^{\mathcal {I}}$ equipped with the subspace topology. Then the subset $U(\rho ;K,\epsilon )$ of $\mathrm {Hom}(\Gamma ,\mathrm {LHomeo}(M))$ is an open neighborhood of $\rho $ . Note that, when the ambient space M itself is compact, we can set $K=M$ and simply talk about $\epsilon $ -perturbations.

Now, we are able to state our structural stability theorem for meandering-hyperbolic actions.

As an immediate consequence of Theorem 4.4(1), we have:

We note, however, that faithfulness of the $\rho $ -action on M does not imply faithfulness of nearby actions: See Example 7.7.

In view of Remarks 3.6(c), 3.27(b) and 4.2, we obtain Sullivan’s structural stability theorem [Reference Sullivan35, Theorem II] in the $C^1$ -setting as a corollary of Theorem 4.4(1):

4.2 Specifying small perturbations $\rho '$

We now proceed to prove Theorem 4.4. The assertion (1) will be proved in Sections 4.2–4.7, and the assertions (2)-(4) in Sections 4.8–4.10, respectively.

Let $\rho :\Gamma \to \mathrm {LHomeo}(M)$ be a meandering-hyperbolic action (Definition 3.26) at a compact invariant subset $\Lambda \subset M$ with a datum $(\mathcal {D}_0\prec \mathcal {D};N)$ , where

$$ \begin{align*} (\mathcal{I}_0,\mathcal{U}_0,\varSigma_0,\delta_0,L_0,\lambda_0)=\mathcal{D}_0\prec\mathcal{D}=(\mathcal{I},\mathcal{U},\varSigma,\delta,L,\lambda),\\ \mathrm{that \ is,} \ \mathcal{I}_0\subset\mathcal{I},\; \mathcal{U}_0\prec\mathcal{U},\; \varSigma_0\prec\varSigma,\; \delta_0>\delta,\; L_0\le L,\; \lambda_0\ge\lambda. \end{align*} $$

Recall, in particular, that $\rho $ is expanding with respect to the refined datum $\mathcal {D}$ .

For convenience, we summarize all the important constants for the proof as follows:

The compact set $K\supset \Lambda $ and the constant $\epsilon =\epsilon (\mathcal {D},N)>0$ specify the open set $U(\rho ;K,\epsilon )$ of all $(K,\epsilon )$ -perturbations $\rho '$ of $\rho $ . Note that K is compact since $(M,d)$ is assumed to be proper. Since $\delta \le \delta _{\mathcal {U}}$ is a Lebesgue number of $\mathcal {U}$ , it follows that $\mathcal {U}$ is a cover of $N_{\delta }(\Lambda )$ .

Now, we suppose

$$\begin{align*}\rho'\in U(\rho;K,\epsilon). \end{align*}$$

Then, by the definitions (4.1) and (4.3), we have that for all $s\in \varSigma $ ,

(4.8) $$ \begin{align} d_{\mathrm{Lip},K}(\rho(s),\rho'(s))<\epsilon. \end{align} $$

We first observe that for each $i\in \mathcal {I}$ ,

(4.9) $$ \begin{align} \rho'(s_i^{-1}) \ \mathrm{is} \ \lambda'\mathrm{-expanding\ on} \ U_i\cap K. \end{align} $$

Indeed, since $\rho (s_i^{-1})$ is $\lambda $ -expanding on $U_i$ , we see from equation (4.8) that for all distinct $x,y\in U_i\cap K$ ,

$$ \begin{align*} \frac{d(\rho'(s_i^{-1})(x),\rho'(s_i^{-1})(y))}{d(x,y)}>\frac{d(\rho(s_i^{-1})(x),\rho(s_i^{-1})(y))}{d(x,y)}-\epsilon \ge \lambda-\epsilon =\lambda'. \end{align*} $$

Moreover, we also note that

(4.10) $$ \begin{align} \rho'(s_i) \ \mathrm{is} \ L'\mathrm{-Lipschitz \ on} \ N_{\delta}(\Lambda) \end{align} $$

for every $i\in \mathcal {I}$ . To see this, recall that the maps $\rho (s_i)$ are L-Lipschitz on $N_{\delta }(\Lambda )$ by Definition 3.4(i), thus by equation (4.8) again

$$ \begin{align*} \frac{d(\rho'(s_i)(x),\rho'(s_i)(y))}{d(x,y)} <\frac{d(\rho(s_i)(x),\rho(s_i)(y))}{d(x,y)}+\epsilon \le L+\epsilon=L' \end{align*} $$

for all distinct $x,y\in N_{\delta }(\Lambda )\subset K$ . Note that $L'=L+\epsilon \ge \lambda +\epsilon>\lambda -\epsilon =\lambda '$ .

For later use, we prove the following lemma.

Lemma 4.11. For every $w\in \Gamma $ such that $|w|_{\varSigma }=n\le N$ , we have

$$\begin{align*}d(\rho(w)(y),\rho'(w)(y))<nL^{n-1}\epsilon \end{align*}$$

for all $y\in N_{\kappa } (\Lambda )$ .

Proof. We prove this by induction on $n\leq N$ . If $|w|_{\varSigma }=1$ , the claim is true by equation (4.8) since $N_{\kappa }(\Lambda )\subset N_{\delta }(\Lambda )$ . Suppose $w=st$ with $s\in \varSigma $ , $|t|_{\varSigma }=n-1$ and $|w|_{\varSigma }=n$ . Since $\rho (t)$ is $L^{n-1}$ -Lipschitz on $N_{\delta /L^{n-2}}(\Lambda )\supset N_{\kappa }(\Lambda )$ by equation (3.5), we have $\rho (t)(y)\in N_{L^{n-1}\kappa }(\Lambda )\subset N_{\delta }(\Lambda )$ . By the induction hypothesis $d(\rho (t)(y),\rho '(t)(y))<(n-1)L^{n-2}\epsilon $ , we then have $\rho '(t)(y)\in N_{\delta }(\Lambda )$ as well, since $\epsilon =\frac {\lambda -1}{2}\kappa <L\kappa $ and thus

$$\begin{align*}L^{n-1}\kappa+(n-1)L^{n-2}\epsilon <L^{n-1}\kappa+(n-1)L^{n-2}L\kappa =nL^{n-1}\kappa<\delta. \end{align*}$$

As $\rho (s)$ is L-Lipschitz on $N_{\delta }(\Lambda )$ , we obtain

$$\begin{align*}d\big(\rho(s)[\rho(t)(y)],\rho(s)[\rho'(t)(y)]\big) < L\cdot(n-1)L^{n-2}\epsilon. \end{align*}$$

Furthermore, we have from equation (4.8)

$$\begin{align*}d\big(\rho(s)[\rho'(t)(y)],\rho'(s)[\rho'(t)(y)]\big)<\epsilon \end{align*}$$

since $\rho '(t)(y)\in N_{\delta }(\Lambda )$ . Thus,

$$ \begin{align*} d(\rho(w)(y),\rho'(w)(y)) &\le d\big(\rho(s)[\rho(t)(y)],\rho(s)[\rho'(t)(y)]\big) +d\big(\rho(s)[\rho'(t)(y)],\rho'(s)[\rho'(t)(y)]\big)\\ &< (n-1)L^{n-1}\epsilon + \epsilon\\ &< (n-1)L^{n-1}\epsilon + L^{n-1}\epsilon = nL^{n-1}\epsilon, \end{align*} $$

and the claim is proved.

4.3 Definition of $\phi $

We first construct a map $\phi :\Lambda \to M$ .

Let $x\in \Lambda $ . In order to define $\phi (x)$ , choose a $(\mathcal {D}_0,\delta _0)$ -code $(\alpha ,p)$ for x as in Section 3.3. Since $\delta <\delta _0$ , we may regard $\alpha $ as a $(\mathcal {D},\delta )$ -code as in Remark 3.11(c)(d). Then, as we know from Lemma 3.14, the point x has an exponentially shrinking nested sequence of neighborhoods $\rho (c^{\alpha }_k)[B_{\delta }(p_{k+1})]$ ( $k\in \mathbb {N}_0$ ) such that

$$\begin{align*}\{x\}=\bigcap_{k=0}^{\infty}\rho(c^{\alpha}_k)[B_{\delta}(p_{k+1})]. \end{align*}$$

Now, consider a perturbation $\rho '\in U(\rho ;K,\epsilon )$ of $\rho $ as specified in Section 4.2. We claim that the sequence of perturbed subsets $\rho '(c^{\alpha }_k)[B_{\delta }(p_{k})]$ ( $k\in \mathbb {N}_0$ ) is also nested and exponentially shrinking. Since M is complete, the intersection of this collection of subsets is a singleton in M and we can define $\phi _{\alpha }(x)$ by the formula

$$ \begin{align*} \{\phi_{\alpha}(x)\} =\bigcap_{k=0}^{\infty}\rho'(c^{\alpha}_k)[B_{\delta}(p_{k+1})]. \end{align*} $$

The claim will be proved in the following lemma, where, in fact, we shall find the values of $\eta \le \delta $ for which the sequence $\rho '(c^{\alpha }_k)[B_{\eta }(p_{k})]$ is nested and exponentially shrinking. This lemma will later be used often, for example, when we show that $\phi $ is well-defined (Section 4.4) and is continuous (Section 4.6).

Lemma 4.12. Let $\rho :\Gamma \to \mathrm {LHomeo}(M)$ be a meandering-hyperbolic action with the datum $(\mathcal {D}_0\prec \mathcal {D};N)$ . Consider a number $0<t\le \kappa =\min \{\frac {\delta }{(N+1)L^N},1\}$ (so that $\frac {\lambda -1}{2}t\le \epsilon $ ), and let

$$\begin{align*}\rho'\in U(\rho;K,\textstyle{\frac{\lambda-1}{2}t})\subset U(\rho;K,\epsilon). \end{align*}$$

If $(\alpha ,p)$ is a $(\mathcal {D},\delta )$ -code for $x\in \Lambda $ , then for every $\eta \in [t,\delta ]$ the sequence of subsets

$$\begin{align*}\rho'(c^{\alpha}_k)[B_{\eta}(p_{k+1})]\quad(k\in\mathbb{N}_0) \end{align*}$$

is nested and exponentially shrinking. Consequently, we have

$$ \begin{align*} \{\phi_{\alpha}(x)\} =\bigcap_{k=0}^{\infty}\rho'(c^{\alpha}_k)[B_{\eta}(p_{k+1})] =\bigcap_{k=0}^{\infty}\rho'(c^{\alpha}_k)[B_{\delta}(p_{k+1})] \end{align*} $$

and thus, for every $k\in \mathbb {N}_0$ ,

$$\begin{align*}\rho'(c^{\alpha}_k)^{-1}(\phi_{\alpha}(x)) \in B_{\eta}(p_{k+1}). \end{align*}$$

If the $\delta $ -code $(\alpha ,p)$ is special, then

$$\begin{align*}d(x,\phi_{\alpha}(x))<t. \end{align*}$$

Proof. Since $t\le \eta $ , we see from the assumption $\rho '\in U(\rho ;K,\frac {\lambda -1}{2}t)$ that

(4.13) $$ \begin{align} \sup_{x\in K} d(\rho(s_i^{-1})(x),\rho'(s_i^{-1})(x)) \le d_{\mathrm{Lip},K}(\rho(s_i^{-1}),\rho'(s_i^{-1})) < \frac{\lambda-1}{2}t \le \frac{1}{2}(\lambda\eta-\eta) \end{align} $$

for all $i\in \mathcal {I}$ . Since $\eta \le \delta $ , we have by Definition 3.4(ii) the inclusion (3.15) for $\rho $

(4.14) $$ \begin{align} B_{\eta}(p_{k+1}) \subset B_{\lambda\eta}(p_{k+1}) \subset \rho(s_{\alpha(k)}^{-1})[B_{\eta}(p_k)] \subset \rho(s_{\alpha(k)}^{-1})[U_{\alpha(k)}] \end{align} $$

for all $k\in \mathbb {N}$ , while $B_{\eta }(p_k)\subset B_{\delta }(p_k)\subset U_{\alpha (k)}\cap K$ as $\alpha $ is a $\delta $ -code.

In particular, we see from equation (4.14) that the boundary of $\rho (s_{\alpha (k)}^{-1})[B_{\eta }(p_k)]$ lies outside $B_{\lambda \eta }(p_{k+1})$ , and from equation (4.13) that the boundary of $\rho '(s_{\alpha (k)}^{-1})[B_{\eta }(p_k)]$ lies outside the ball centered at $p_{k+1}$ of radius $\lambda \eta -\frac {1}{2}(\lambda \eta -\eta )=\frac {1}{2}\eta (\lambda +1)>\eta $ . Moreover, $\rho '(s_{\alpha (k)}^{-1})(p_k)$ lies in the ball centered at $p_{k+1}$ of radius $\frac {1}{2}\eta (\lambda -1)$ , which implies $\rho '(s_{\alpha (k)}^{-1})[B_{\eta }(p_k)]\cap B_{\frac {1}{2}\eta (\lambda +1)}(p_{k+1})\neq \emptyset $ . Note that every open ball is path-connected in a geodesic metric space M. Thus, we deduce that $\rho '(s_{\alpha (k)}^{-1})[B_{\eta }(p_k)]$ is a connected component of $M\setminus \partial \rho '(s_{\alpha (k)}^{-1})[B_{\eta }(p_k)]$ and hence $B_{\frac {1}{2}\eta (\lambda +1)}(p_{k+1})\subset \rho '(s_{\alpha (k)}^{-1})[B_{\eta }(p_k)]$ . Therefore, we conclude that

(4.15) $$ \begin{align} B_{\eta}(p_{k+1}) \subset B_{\frac{1}{2}\eta(\lambda+1)}(p_{k+1}) \subset \rho'(s_{\alpha(k)}^{-1})[B_{\eta}(p_k)] \subset \rho'(s_{\alpha(k)}^{-1})[U_{\alpha(k)}\cap K]. \end{align} $$

Thus,

(4.16) $$ \begin{align} \rho'(s_{\alpha(k)})[B_{\eta}(p_{k+1})]\subset B_{\eta}(p_k) \end{align} $$

for all $k\in \mathbb {N}$ , and we check as in equation (3.18) the nesting property

$$\begin{align*}\rho'(c^{\alpha}_k)[B_{\eta}(p_{k+1})]\subset\rho'(c^{\alpha}_{k-1})[B_{\eta}(p_k)] \end{align*}$$

for all $k\in \mathbb {N}$ .

Furthermore, the diameter of $ \rho '(c^{\alpha }_k)[B_{\eta }(p_{k+1})] =\rho '(s_{\alpha (0)}s_{\alpha (1)}\cdots s_{\alpha (k)})[B_{\eta }(p_{k+1})]$ is at most $2\eta (L+\epsilon )/(\lambda ')^k$ , because we have equation (4.15) and each $\rho '(s_{\alpha (j)}) (1\le j\le k)$ is $(1/\lambda ')$ -contracting on $\rho '(s_{\alpha (j)}^{-1})[U_{\alpha (j)}\cap K]$ by equation (4.9), and the last map $\rho '(s_{\alpha (0)})$ is $(L+\epsilon )$ -Lipschitz on $B_{\eta }(p_1)\subset N_{\delta }(\Lambda )$ by equation (4.10). Hence, the exponentially shrinking property also holds.

If the $\delta $ -code $(\alpha ,p)$ is special, then the inclusion equation (4.15) as well as equation (4.16) hold for $k=0$ . Thus,

$$\begin{align*}\phi_{\alpha}(x) \in\rho'(c^{\alpha}_0)[B_{\eta}(p_{1})] =\rho'(s_{\alpha(0)})[B_{\eta}(p_{1})] \subset B_{\eta}(x) \end{align*}$$

for all $\eta \in [t,\delta ]$ . It follows that $d(x,\phi _{\alpha }(x))<t$ .

4.4 $\phi $ is well-defined

To show that $\phi $ is well defined, we need to show that $\phi _{\alpha }(x)=\phi _{\beta }(x)$ for any two $(\mathcal {D}_0,\delta _0)$ -codes $(\alpha ,p)$ and $(\beta ,q)$ . Since $\rho $ is meandering-hyperbolic with the datum $(\mathcal {D}_0\prec \mathcal {D};N)$ , the corresponding $(\mathcal {D}_0,\delta _0)$ -rays $c^{\alpha }$ and $c^{\beta }$ are $(\mathcal {D},\delta ;N)$ -equivalent, that is, $c^{\alpha }\sim ^N_{(\mathcal {D},\delta )} c^{\beta }$ in $\mathrm {Ray}_x(\mathcal {D},\delta )$ . By definition, the equivalence relation $\sim ^N_{(\mathcal {D},\delta )}$ is generated by the relation $\approx ^N_{(\mathcal {D},\delta )}$ (see Definition 3.23(a)). Thus, it suffices to show the equality $\phi _{\alpha }(x)=\phi _{\beta }(x)$ when $c^{\alpha }$ and $c^{\beta }$ are $(\mathcal {D},\delta )$ -rays and satisfy $c^{\alpha }\approx ^N_{(\mathcal {D},\delta )} c^{\beta }$ , that is, there exist infinite subsets $P,\,Q\subset \mathbb {N}_0$ such that the subsets $c^{\alpha }(P)$ and $c^{\beta }(Q)$ are within Hausdorff distance N from each other in $(\Gamma ,d_{\varSigma })$ .

Suppose to the contrary that

$$\begin{align*}\bigcap_{k\in P} \rho'(c^{\alpha}_k)[B_{\delta}(p_{k+1})] = \{\phi_{\alpha}(x)\}\neq\{\phi_{\beta}(x)\}=\bigcap_{j\in Q} \rho'(c^{\beta}_j)[B_{\delta}(q_{j+1})]. \end{align*}$$

Since the open sets $\rho '(c^{\beta }_j)[B_{\delta }(q_{j+1})]$ shrink to $\phi _{\beta }(x)$ , there exists an integer $n\in Q$ such that $\phi _{\alpha }(x)\notin \rho '(c^{\beta }_n)[B_{\delta }(q_{n+1})]$ , that is,

(4.17) $$ \begin{align} \rho'(c^{\beta}_{n})^{-1}(\phi_{\alpha}(x))\notin B_{\delta}(q_{n+1}). \end{align} $$

Since the Hausdorff distance between $\{c^{\alpha }_k\}_{k\in P}$ and $\{c^{\beta }_j\}_{j\in Q}$ is at most N, there is an integer $m\in P$ such that $d_{\varSigma }(c^{\beta }_n,c^{\alpha }_m)\le N$ . Set

$$\begin{align*}r=(c^{\beta}_{n})^{-1}c^{\alpha}_{m} \end{align*}$$

so that $|r|_{\varSigma }\le N$ . Note from Remark 3.13(b) that $\rho (r)$ maps $p_{m+1}$ to $q_{n+1}$ :

$$\begin{align*}\rho(r)(p_{m+1})=\rho(r)[\rho(c^{\alpha}_{m})^{-1}(x)]=\rho(c^{\beta}_{n})^{-1}(x)=q_{n+1}. \end{align*}$$

See Figure 3.

Figure 3 The points $\phi _{\alpha }(x)$ and $\phi _{\beta }(x)$ .

In view of Lemma 4.12 (with $t=\eta =\kappa =\min \{\frac {\delta }{(N+1)L^N},1\}$ ), we may assume that we used $\kappa $ -balls in the definition of $\phi _{\alpha }(x)$ so that

$$\begin{align*}\rho'(c^{\alpha}_{m})^{-1}(\phi_{\alpha}(x))\in B_{\kappa}(p_{m+1}). \end{align*}$$

Now, if we show that $\rho '(r)$ maps $B_{\kappa }(p_{m+1})$ into $B_{\delta }(q_{n+1})$ and hence

$$ \begin{align*} \rho'(c^{\beta}_{n})^{-1}(\phi_{\alpha}(x)) &=\rho'(r)[\rho'(c^{\alpha}_{m})^{-1}(\phi_{\alpha}(x))]\\ &\in\rho'(r)[B_{\kappa}(p_{m+1})]\subset B_{\delta}(q_{n+1}), \end{align*} $$

then we are done since we are in contradiction with equation (4.17).

To show $\rho '(r)[B_{\kappa }(p_{m+1})]\subset B_{\delta }(q_{n+1})$ , let $l=|r|_{\varSigma }\le N$ . If $y\in B_{\kappa }(p_{m+1})\subset N_{\kappa }(\Lambda )$ , then

$$ \begin{align*} d(\rho'(r)(y),q_{n+1}) &= d(\rho'(r)(y),\rho(r)(p_{m+1}))\\ &\le d(\rho'(r)(y),\rho(r)(y)) +d(\rho(r)(y),\rho(r)(p_{m+1}))\\ &< k L^{l-1}\epsilon +L^l\kappa\\ &< N L^{N-1}L\kappa +L^N\kappa\le \delta, \end{align*} $$

where the second inequality holds by Lemma 4.11 and since $\rho (r)$ is $L^l$ -Lipschitz on $N_{\delta /L^{l-1}}(\Lambda )\supset B_{\kappa }(p_{m+1})$ by equation (3.5), and the third inequality holds since $\epsilon =\frac {\lambda -1}{2}\kappa <L\kappa $ .

This completes the proof of the equality $\phi _{\alpha }(x)=\phi _{\beta }(x)$ .

From now on, we may write $\phi (x)$ for $x\in \Lambda $ without ambiguity. An immediate consequence of this is that we are henceforth free to choose a special $\delta $ -code for x. Without loss of generality, we may assume that $N\geq 4$ . Then from Lemma 4.12 (with $t=\kappa =\min \{\frac {\delta }{(N+1)L^N},1\}$ ), we conclude that

(4.18) $$ \begin{align} d(x,\phi(x))<\kappa\le\frac{\delta}{(N+1)L^N}<\frac{\delta}{5} \end{align} $$

for all $x\in \Lambda $ .

4.5 $\phi $ is equivariant

To show the equivariance of $\phi $ , it suffices to check it on the generating set $\varSigma $ of $\Gamma $ .

Given $x\in \Lambda $ and $s\in \varSigma $ , set $y=\rho (s^{-1})(x)$ . Let $(\beta ,q)$ be a special $\delta $ -code for y so that $B_{\delta }(y)\subset U_{\beta (0)}$ (see Definition 3.10). Then we consider a $\delta $ -code $(\alpha ,p)$ for x defined by $s_{\alpha (0)}=s$ and

$$ \begin{align*} {\alpha(k)}&={\beta(k-1)},\\ p_k&=q_{k-1} \end{align*} $$

for $k\in \mathbb {N}$ : Indeed, we verify the requirement that

$$\begin{align*}B_{\delta}(p_{k})=B_{\delta}(q_{k-1}) \subset U_{\beta(k-1)}=U_{\alpha(k)} \end{align*}$$

for $k\in \mathbb {N}$ . The associated rays $\{c^{\alpha }_k\}_{k\in \mathbb {N}_0}$ and $\{c^{\beta }_k\}_{k\in \mathbb {N}_0}$ (see Definition 3.12) are related by

$$\begin{align*}c^{\alpha}_k =s_{\alpha(0)}(s_{\alpha(1)}\cdots s_{\alpha(k)}) =s(s_{\beta(0)}\cdots s_{\beta(k-1)}) =s\,c^{\beta}_{k-1} \end{align*}$$

for $k\in \mathbb {N}$ . Therefore, we have

$$ \begin{align*} \{\phi[\rho(s)(y)]\}=\{\phi(x)\} &=\bigcap_{k=0}^{\infty} \rho'(c^{\alpha}_k)[B_{\delta}(p_{k+1})] \\ &=\bigcap_{k=1}^{\infty} \rho'(c^{\alpha}_k)[B_{\delta}(p_{k+1})] \\ &=\bigcap_{k=1}^{\infty} \rho'(s)\rho'(c^{\beta}_{k-1})[B_{\delta}(q_k)] \\ &=\rho'(s)\left[\bigcap_{k=0}^{\infty} \rho'(c^{\beta}_{k})[B_{\delta}(q_{k+1})] \right] =\rho'(s)\{\phi(y)\}, \end{align*} $$

which implies the equivariance of $\phi $ .