1. Introduction
Zimmer's conjecture for actions of higher-rank lattice on compact manifolds says that if the group is large with respect to the dimension of the manifold, then any such action should factor through a finite group. This conjecture is motivated by a long history of research, including the local rigidity results of Selberg [Reference SelbergSel60] and Weil [Reference WeilWei62] on linear representation theory, the global rigidity results of Mostow [Reference MostowMos73], the superrigidity theorem of Margulis [Reference MargulisMar91], and the cocycle superrigidity theorem of Zimmer [Reference ZimmerZim80]. Since its introduction, Zimmer's conjecture has attracted considerable interests.
For 
$C^{0}$ actions on the circle, the above conjecture is confirmed by Lifschitz, and Witte Morris [Reference Lifschitz and Witte MorrisLW08, Reference Witte MorrisWit94] for many non-uniform lattices. For 
$C^{1}$ actions on the circle, Burger and Monod [Reference Burger and MonodBM02] and Ghys [Reference GhysGhy99] showed similar results for many other cases, including all lattices in higher rank simple Lie groups. For 
$C^{1}$ area-preserving actions on closed orientable surface with genus at least 
$2$, Zimmer's conjecture is proved by Polterovich [Reference PolterovichPol02] for non-uniform lattices. His result is then generalized by Franks and Handel in [Reference Franks and HandelFH06] to any 
$C^{1}$ action which preserves a Borel measure. For analytic actions, Ghys [Reference GhysGhy99] studied the case where the manifold is a circle; Farb and Shalen [Reference Farb and ShalenFS99] studied this conjecture under additional assumptions on the group and the manifold. For a very detailed survey on other earlier results on Zimmer's program, we refer the reader to [Reference FisherFis11].
In recent breakthroughs [Reference Brown, Fisher and HurtadoBFH16, Reference Brown, Fisher and HurtadoBFH20], Brown, Fisher and Hurtado proved the 
$C^{2}$ versionFootnote 1 of Zimmer's conjecture for all co-compact latticesFootnote 2 in real split simple Lie group and 
${\rm SL}(n,{{\mathbb {Z}}})$ using some previous progress made by Brown, Rodriguez Hertz and Wang in [Reference Brown, Rodriguez Hertz and WangBRW16b, Reference Brown, Rodriguez Hertz and WangBRW16a] and Lafforgue, de Laat and de la Salle in [Reference LafforgueLaf08, Reference de Laat and de la SalledLdlS15, Reference de la SalledlS19]. We refer the reader to Fisher's paper [Reference FisherFis17] for an excellent survey of the history and recent progress on Zimmer's conjecture. The purpose of the present paper is to extend the results in [Reference Brown, Fisher and HurtadoBFH16, Reference Brown, Fisher and HurtadoBFH20, Reference Brown, Fisher and HurtadoBFH21] to 
$C^{1}$ actions, when the rank of the acting group is sufficiently large.
Compared to the previous results, there are two new ideas here. First is that while many results in Non-uniform Hyperbolic Theory fail or remain unknown in the 
$C^{1}$ setting, some of them continue to hold under the presence of suitable continuous splitting. In our case, we can apply a variant of Avila and Viana's invariance principle to an element in the kernel of all Lyapunov functionals to obtain the extra invariance needed to conclude the proof. For 
$C^{2}$ action, the idea to use action by an element in the kernel of all fiberwise exponents was originally due to Sebastian Hurtado and appears in the Bourbaki notes of Cantat [Reference CantatCan19]. The second one is that we use the information extracted by using strong property (T) to control the 
$L^{p}$ norms of the derivatives for sufficiently large 
$p$. This allows us to show that 
$C^{1}$ action is uniformly bounded under certain Hölder norm. Then we use the resolution of the Hilbert–Smith conjecture for sufficiently Hölder actions to conclude the proof.
2. Statement of the main results
We first recall the statement of Zimmer's conjecture.
For a real semisimple Lie group 
$G$ with Lie algebra 
$\mathfrak {g}$, let:
•
$v(G)$ denote the minimal codimension of proper parabolic subalgebras of $\mathfrak {g}$;•
$d(G)$ denote the minimal codimension of proper subalgebras of the compact real form of $\mathfrak {g}_\mathbb {C}$;•
$n(G)$ denote the minimal dimension of non-trivial real representations of $\mathfrak {g}$.
It is proved in [Reference StuckStu91] that 
$v(G)< n(G)$.Footnote 3
Conjecture 1 Let 
$G$ be a connected real semisimple Lie group with finite center and without almost-simple factors of real rank less than 
$2$. Let 
$\Gamma < G$ be a lattice, 
$M$ be a compact manifold, 
$\alpha :\Gamma \to \text {Diff}(M)$ be an action.
(1) If
$\dim (M)< v(G)$, then $\alpha$ preserves a Riemannian metric.(2) If
$\dim (M)<\min \{v(G),d(G)\}$, then $\alpha (\Gamma )$ is finite.(3) If
$\dim (M)< n(G)$ and $\alpha$ preserves a volume density, then $\alpha$ preserves a Riemannian metric.(4) If
$\dim (M)<\min \{n(G),d(G)\}$ and $\alpha$ preserves a volume density, then $\alpha (\Gamma )$ is finite.
The main result of this paper is the following generalization of the results in [Reference Brown, Fisher and HurtadoBFH16, Reference Brown, Fisher and HurtadoBFH21] to 
$C^{1}$ regularity.
Theorem 1 Let 
$M$ be a compact manifold. Let 
$G$ be an almost simple real Lie group with finite center and with real rank at least 
$2$, and let 
$\Gamma < G$ be a lattice. Let 
$\alpha : \Gamma \to \operatorname {Diff}^{1}(M)$ be a group homomorphism. Assume that 
$\Gamma$ is an uniform lattice or 
$\Gamma = {\rm SL}(n, {{\mathbb {Z}}})$, and assume either that 
$\dim M < \min ({{\rm rank}}_{{\mathbb {R}}}(G), d(G))$, or that 
$\dim M \leq \min ({{\rm rank}}_{{\mathbb {R}}}(G), d(G)-1)$ and 
$\alpha (\Gamma ) \subset \operatorname {Diff}^{1}(M, {\rm vol})$. Then 
$\alpha$ has finite image.
Compared to the main result in [Reference Brown, Fisher and HurtadoBFH16, Reference Brown, Fisher and HurtadoBFH21] for almost-simple real Lie groups, in Theorem 1 we have posed a different requirement on the dimension of the manifold. Indeed, we can deduce from [Reference Brown, Fisher and HurtadoBFH16, Theorem 2.7] that for a group homomorphism 
$\alpha : \Gamma \to \operatorname {Diff}^{2}(M)$, the conclusion of Theorem 1 is true if 
${{\rm rank}}_{{\mathbb {R}}}G$ is replaced by the minimal resonant codimension 
$r(G)$ (see [Reference Brown, Fisher and HurtadoBFH16, Definition 2.1]). We remark that under the conditions of Theorem 1, we always have that
\[ r(G) \geq {{\rm rank}}_{{\mathbb{R}}}G. \]Corollary 1 Let 
$M$ be a compact manifold. Let 
$\Gamma < G$ be a lattice. Let 
$\alpha : \Gamma \to \operatorname {Diff}^{1}(M)$ (respectively 
$\operatorname {Diff}^{1}(M, {\rm vol})$) be a group homomorphism. Assume that 
$\Gamma$ is an uniform lattice or 
$\Gamma = {\rm SL}(n, {{\mathbb {Z}}})$, and assume that one of the following is true
$:$
(1)
$G = {\rm SL}(n, {{\mathbb {R}}})$, $\dim M < n-1$ (respectively $\leq n-1$) and $n \geq 3;$(2)
$G = {\rm Sp}(2n, {{\mathbb {R}}})$, $\dim M < n$ (respectively $\leq n$) and $n \geq 2;$(3)
$G = {\rm SO}(n, n)$, $\dim M < n$ (respectively $\leq n$) and $n \geq 4;$(4)
$G = {\rm SO}(n, n+1)$, $\dim M < n-1$ (respectively $\leq n-1$) and $n \geq 3$.
Then 
$\alpha$ has finite image.
When 
$\alpha$ is a 
$C^{2}$ action, the conclusion of Theorem 1 is already obtained in [Reference Brown, Fisher and HurtadoBFH16, Reference Brown, Fisher and HurtadoBFH21]. Moreover, when 
$G = {\rm Sp}(2n, {{\mathbb {R}}})$, 
${\rm SO}(n, n)$ or 
${\rm SO}(n,n+1)$, the dimension bound in Corollary A is not optimal. However, when 
$G = {\rm SL}(n, {{\mathbb {R}}})$, we have
\[ r(G) = {{\rm rank}}_{{\mathbb{R}}}G = n-1. \]
By considering the actions of 
${\rm SL}(n, {{\mathbb {R}}})$ by projective transformations on 
$\mathbb {P}({{\mathbb {R}}}^{n})$, and by the affine transformations on 
${\mathbb {T}}^{n}$, we see that Corollary A has optimal bounds for 
$G = {\rm SL}(n, {{\mathbb {R}}})$. We note the for 
$C^{0}$ action by 
${\rm SL}(n,{{\mathbb {Z}}}), (n \geq 3)$ on compact manifold with 
$\chi (M) \neq 0 \mod 3$, the finite image property of 
$\alpha$ is proved by Ye in [Reference YeYe18].
The proofs of the results in this paper follow closely the strategy in [Reference Brown, Fisher and HurtadoBFH16]. We recommend the reader to have this paper close at hand as we make many references to these works, although we also repeat some of the main arguments for reader's convenience. Below we first describe the general strategy of the proofs in [Reference Brown, Fisher and HurtadoBFH16, Reference Brown, Fisher and HurtadoBFH20, Reference Brown, Fisher and HurtadoBFH21], and then we point out the main new ideas and modifications we make here in order to obtain results in 
$C^{1}$ regularity.
3. Review of the work of Brown, Fisher and Hurtado and outline of the proof
Step 1: Uniform subexponential growth
We fix a finite set of symmetric generators for 
$\Gamma$, denoted by 
$S =\{\gamma _i\}$. For any 
$\gamma \in \Gamma$, we let 
$\ell (\gamma )$ denote the word-length distance from 
$\gamma$ to the identity relative to 
$S$. In other words, 
$\ell (\gamma )$ is the smallest integer 
$k$ such that 
$\gamma$ may be represented by a product 
$\zeta _1 \cdots \zeta _k$ where 
$\zeta _j \in S$ for each 
$1 \leq j \leq k$.
We first recall the following notion.
Definition 1 Let 
$\alpha : \Gamma \to \operatorname {Diff}^{1}(M)$ be an action of 
$\Gamma$ on a compact manifold 
$M$ by 
$C^{1}$ diffeomorphisms. We fix a background 
$C^{\infty }$ Riemannian metric on 
$M$. We say that 
$\alpha$ has uniform subexponential growth of derivatives if for every 
$\varepsilon > 0$ there is a constant 
$C_\varepsilon > 0$ such that for all 
$\gamma \in \Gamma$ we have
\[ \| D\alpha(\gamma)\| \leq C_{\varepsilon} e^{\varepsilon \ell( \gamma)}. \]
It is clear that the above definition is independent of the choice of the metric on 
$M$ or the generating set 
$S$.
The main result of Step 1 is the following.
Proposition 1 Let 
$M$ be a compact manifold, and let 
$G$ be a connected, almost-simple real Lie group with finite center and whose real rank is at least 
$2$. Let 
$\Gamma < G$ be a lattice. Let 
$\alpha : \Gamma \to \operatorname {Diff}^{1}(M)$ be a group homomorphism. Assume 
$\Gamma$ is an uniform lattice or 
$\Gamma = {\rm SL}(n, {{\mathbb {Z}}})$, and assume either that 
$\dim M < {{\rm rank}}_{{\mathbb {R}}}(G)$, or that 
$\dim M \leq {{\rm rank}}_{{\mathbb {R}}}(G)$ and 
$\alpha (\Gamma ) \subset \operatorname {Diff}^{1}(M, {\rm vol})$. Then 
$\alpha$ has uniform subexponential growth of derivatives.
We prove Proposition 1 following the same scheme in [Reference Brown, Fisher and HurtadoBFH16]. As in [Reference Brown, Fisher and HurtadoBFH16], we define the suspension space 
$M^{\alpha }$ as the quotient of 
$G \times M$ by 
$\Gamma$-action 
$(g,x) \mapsto (g\gamma, \alpha (\gamma ^{-1})x)$. We recall that 
$M^{\alpha }$ is a fiber bundle over 
$G / \Gamma$ with fibers modeled on 
$M$. Moreover 
$M^{\alpha }$ is equipped with a left 
$G$-action, denoted by 
$\tilde \alpha$, by diffeomorphisms which preserves the foliation into fibers. We present the construction of 
$M^{\alpha }$ and its further properties in § 4.1.
As the 
$G$-action preserves the foliation into fibers of 
$M^{\alpha }$, we may consider the restriction of 
$D{{\tilde {\alpha }}}$ to the subbundle 
$E^{\rm F} :={\rm Ker}(D\pi )$ tangent to the fibers of 
$M^{\alpha }$. Let 
$A$ be the maximal split torus of 
$G$, and let 
$\mu$ be an 
$A$-ergodic 
$A$-invariant measure on 
$M^{\alpha }$. We can associate to 
$\mu$ and the derivative 
$A$-cocycle 
$D{{\tilde {\alpha }}}|_{E^{\rm F}}$ a set of fiberwise Lyapunov functionals 
$\lambda ^{\rm F}_i : Lie(A) \to {{\mathbb {R}}}$, 
$1 \leq i \leq k$ by the higher-rank Oseledec's theorem (see, e.g. [Reference Brown, Rodriguez Hertz and WangBRW16b, Part I, Theorem 2.4]). We refer the reader to [Reference Brown, Fisher and HurtadoBFH16, Proposition 3.3] for the definition and properties of Lyapunov functionals. The maximal fiberwise Lyapunov exponent for 
$a \in A$ with respect to an 
$a$-invariant probability measure 
$\mu$ is defined as
\[ \lambda^{\rm F}_+(a,\mu) = \inf_{n \to \infty}\frac{1}{n} \int \log \| D{{\tilde{\alpha}}}(a^{n}) |_{E^{\rm F}(x)}\| \,d\mu(x). \]We have the following.
Proposition 2 Suppose that 
$\Gamma$ is an uniform lattice or 
$\Gamma = {\rm SL}(n, {{\mathbb {Z}}})$, and 
$\alpha$ fails to have uniform subexponential growth of derivatives. There exists an 
$s \in A$ and an 
$A$-invariant Borel probability measure 
$\mu$ on 
$M^{\alpha }$ with 
$\lambda ^{\rm F}_{+}(s, \mu ) > 0$ such that 
$\pi _*\mu$ is the Haar measure on 
$G / \Gamma$.
When 
$\Gamma$ is an uniform lattice, the above proposition is just [Reference Brown, Fisher and HurtadoBFH16, Proposition 3.7]. When 
$\Gamma = {\rm SL}(n, {{\mathbb {Z}}})$, the above proposition is proved in [Reference Brown, Fisher and HurtadoBFH20] even though it is not explicitly stated as a single proposition. Indeed, we can define the measure 
$\mu$ in Proposition 2 by [Reference Brown, Fisher and HurtadoBFH20, Proposition 5.10] as a limit of a sequence 
$\mu _n$; and by [Reference Brown, Fisher and HurtadoBFH20, Proposition 5.6] and the two paragraphs below it, we see that 
$\pi _* \mu$ is the Haar measure on 
$G/\Gamma$.
This is the only place where we have used the hypothesis that 
$\Gamma$ is an uniform lattice or 
$\Gamma = {\rm SL}(n, {{\mathbb {Z}}})$. In a recent paper of Brown, Fisher and Hurtado [Reference Brown, Fisher and HurtadoBFH21, Proposition 8.1], they have generalized Proposition 2 to any lattice in 
$G$. Admitting their results, all of the results in the present paper hold for arbitrary lattices.
To complete the proof of Proposition 1, it remains to show the following.
Proposition 3 Let 
$\mu$ be an 
$A$-invariant Borel probability measure on 
$M^{\alpha }$ such that 
$\pi _*\mu$ is the Haar measure on 
$G / \Gamma$. If either we have that 
${{\rm rank}}_{{\mathbb {R}}}G > \dim M$, or we have that 
${{\rm rank}}_{{\mathbb {R}}}G \geq \dim M$ and 
$\alpha (\Gamma ) \subset \operatorname {Diff}^{1}(M, {\rm vol})$, then 
$\mu$ is 
$G$-invariant.
Let 
$a\in G$ be a 
${{\mathbb {R}}}$-semisimple element. The unstable, respectively stable, subgroup for 
$a$ are respectively,
\begin{align*} H^{u} &:= \Bigl\{g\Bigm| \lim_{n \to - \infty}a^{n}ga^{-n} = e \Bigr\}, \\
H^{s} &:= \Bigl\{g\Bigm| \lim_{n \to + \infty}a^{n}ga^{-n} = e \Bigr\}. \end{align*}Proposition 4 Let 
$a \in A$ be an 
${{\mathbb {R}}}$-semisimple element. Suppose 
$\mu$ is an 
$a$-invariant 
$a$-ergodic probability measure on 
$M^{\alpha }$ such that:
(1)
$\pi _*\mu$ is the Haar measure on $G /\Gamma$; and(2) all fiberwise Lyapunovv exponents of
$D\tilde \alpha (a)$ are non-positive.
Then 
$\mu$ is 
$H^{u}$-invariant.
The proof of Proposition 4 will be given in § 4. We are ready to deduce Proposition 3 from Proposition 4.
Proof of Proposition 3 We can assume without loss of generality that 
$\mu$ is 
$A$-ergodic, otherwise we may replace 
$\mu$ by any one of its 
$A$-ergodic components. This is because any 
$A$-ergodic component of 
$\mu$ projects to some 
$A$-ergodic component of 
$\pi _*\mu$; while by hypothesis 
$\pi _*\mu$ is the Haar measure on 
$G/\Gamma$ which is itself 
$A$-ergodic by Moore's ergodicity theorem (see for instance [Reference MooreMoo66] or [Reference ZimmerZim84, Theorem 2.2.6]). This allows us to define fiberwise Lyapunov functionals. We denote by 
$\lambda ^{\rm F}_1, \ldots, \lambda ^{\rm F}_k$ the total collection of distinct fiberwise Lyapunov functionals. We have that 
$k \leq \dim M$. Moreover, notice that when 
$\alpha (\Gamma ) \subset \operatorname {Diff}^{1}(M, {\rm vol})$, the sum of all Lyapunov functionals (considered with multiplicities) is zero. Then under the condition of the proposition, we can pick an arbitrary element 
$a \in (\bigcap _{i=1}^{k} \exp ( {\rm Ker}(\lambda ^{\rm F}_i))) \setminus \{ e \}$ such that
\[ \lambda{}^{\rm F}_{+}(a, \mu) = \lambda{}^{\rm F}_{+}(a^{-1}, \mu) = 0. \]
Then all 
$a$-ergodic components of 
$\mu$ have vanishing fiberwise Lyapunov exponents. By Proposition 4, we deduce that 
$\mu$ is 
$H^{u}$-invariant. By symmetry, we also have that 
$\mu$ is 
$H^{s}$-invariant. As 
$G$ is almost-simple, 
$G$ is generated by 
$H^{u}$ and 
$H^{s}$. Consequently, 
$\mu$ is 
$G$-invariant.
Proof of Proposition 1 Assume that 
$\alpha$ fails to have uniform subexponential growth of derivatives. Then by Proposition 2, there is a 
$s \in A$ and an 
$A$-invariant measure 
$\mu$ such that 
$\lambda ^{\rm F}_{+}(s, \mu ) \!> 0$ and 
$\pi _*\mu$ is the Haar measure on 
$G / \Gamma$. By Proposition 3, we deduce that 
$\mu$ is 
$G$-invariant. Recall that 
$n(G) > {{\rm rank}}_{{\mathbb {R}}}G$ where 
$n(G)$ denotes the minimal dimension of a non-trivial real representation of the Lie algebra of 
$G$. By Zimmer's cocycle superrigidity theorem (we use the version by Fisher and Margulis in [Reference Fisher and MargulisFM03, Theorem 1.4]; we refer the reader to [Reference ZimmerZim80, Reference ZimmerZim84, Reference Zimmer and Witte MorrisZW08] for some earlier results), the 
$G$-action preserves a measurable metric on 
$E^{\rm F}$. This contradicts that 
$\lambda ^{\rm F}_+(s, \mu ) > 0$. Thus 
$\alpha$ must has uniform subexponential growth of derivatives.
Step 2: Strong property $(T)$ and averaging
In this step, we follow [Reference Brown, Fisher and HurtadoBFH16] to construct a 
$\Gamma$-invariant continuous distance by using the strong property 
$(T)$ of 
$\Gamma$ proved by Lafforgue, de Laat and de la Salle in [Reference LafforgueLaf08, Reference de Laat and de la SalledLdlS15, Reference de la SalledlS19]. The main result of this step is the following proposition whose proof will be given in § 5.
Proposition 5 If 
$\alpha$ has uniform subexponential growth of derivatives, then there exists a distance 
$\bar {d}: M \times M \to [0, \infty )$ that is invariant by the 
$\Gamma$-action 
$\alpha$. Moreover, for any 
$\beta \in (0,1)$, the set 
$\alpha (\Gamma )$ is precompact in 
${\rm Hol}$-
${\rm Homeo}^{\beta }(M)$, the space of 
$\beta$-bi-Hölder homeomorphisms of 
$M$ with respect to the background Riemannian distance.
Proposition 5 replaces [Reference Brown, Fisher and HurtadoBFH16, Theorem 2.9]. In [Reference Brown, Fisher and HurtadoBFH16], the authors study a 
$C^{2}$ action of 
$\Gamma$, and the induced 
$\Gamma$-action on 
$W^{1,p}(S^{2}(T^{*}M))$, the Sobolev space of all the sections 
$\varphi$ of the bundle of symmetric two forms 
$S^{2}(T^{*}M)$ such that both 
$\varphi$ and 
$D\varphi$ are 
$L^{p}$ with respect to the Lebesgue measure. Then the strong property 
$(T)$ and the uniform subexponential growth of derivatives give us the 
$\Gamma$-invariant section in 
$W^{1,p}(S^{2}(T^{*}M))$ which is continuous if 
$p$ is sufficiently large. The above method can be adapted to the case where the action is 
$C^{1 + \epsilon }$ for any 
$\epsilon > 0$.
In our case, 
$\alpha$ is only 
$C^{1}$, and consequently 
$\alpha$ does not induce a 
$\Gamma$-action on 
$W^{1,p}(S^{2}(T^{*}M))$. We consider instead the induced 
$\Gamma$-action on 
$L^{p}(S^{2}(T^{*}M))$, and obtain a 
$L^{p}$ 
$\alpha$-invariant section of 
$S^{2}(T^{*}M)$. We use the exponential convergence inherited from the strong property 
$(T)$ and Cauchy inequality to bound the Sobolev norms of the 
$\Gamma$-action.
To make use of Proposition 5, we also need the solution of the Hilbert–Smith conjecture for sufficiently Hölder actions proved in [Reference Repovs̆ and S̆c̆epinRS̆97, Reference MaleshichMal97]. We recall the statement here.
Lemma 1 For any 
$\beta \in ({\dim M}/({\dim M +1}), 1)$ the following is true. Let 
$H$ be a compact topological group which admits a faithful action on 
$M$ by 
$\beta$-Hölder homeomorphisms; then 
$H$ is a Lie group.
Corollary 2 Let 
$G, \Gamma, \mu, \alpha$ be as in Theorem 1. Assume either that 
$\dim M < {{\rm rank}}_{{\mathbb {R}}}(G),$ or that 
$\dim M \leq {{\rm rank}}_{{\mathbb {R}}}(G)$ and 
$\alpha (\Gamma ) \subset \operatorname {Diff}^{1}(M, {\rm vol})$. Then 
$\alpha$ factors through a compact Lie group. That is
$,$ there exist
$:$ a compact Lie group 
$H;$ an injective group homomorphism 
$\iota : H \to {\rm Homeo}(M);$ and a group homomorphism 
$\phi : \Gamma \to H$ such that 
$\alpha =\iota \circ \phi$.
Proof. By Proposition 1, the action 
$\alpha$ has uniform subexponential growth of derivatives. We fix any 
$\beta \in ({\dim M}/({\dim M+1}), 1)$. By Proposition 5, the closure of 
$\alpha (\Gamma )$ in 
${\rm Hol}$-
${\rm Homeo}^{\beta }(M)$, denoted by 
$K_0$, is a compact topological subgroup of 
${\rm Homeo}(M)$. By Lemma 1, we see that 
$K_0$ is a compact Lie group.
Step 3: Margulis superrigidity with compact codomain
After Steps 1 and 2, we can apply precisely the same method as in [Reference Brown, Fisher and HurtadoBFH16] to show the finite image property. We refer the reader to [Reference Brown, Fisher and HurtadoBFH16, § 7] for details.
Proof of Theorem 1 The proof is essentially contained in [Reference Brown, Fisher and HurtadoBFH16, § 7]. We reproduce it below for the convenience of the reader.
Let 
$H$ be the compact Lie group given by Corollary B, and let 
$\iota : H \to {\rm Homeo}(M)$ and 
$\phi : \Gamma \to H$ be the associated group homomorphisms. Assume that 
$\alpha = \iota \circ \phi$ has infinite image. Then by Margulis’ arithmeticity theorem and superrigidity theorem, each almost simple factor of 
$H$ is a compact form of 
$G$. Since 
$\iota : H \to {\rm Homeo}(M)$ is injective, there is some 
$x\in M$ such that 
$\iota (H) x$ contains a compacta homeomorphic to 
$K/C$ where 
$K$ is a compact form of 
$G$ and 
$C$ is a closed proper subgroup of 
$K$. This is impossible since by hypothesis 
$\dim (K/C) \geq d(G) > \dim M$.
4. Proof of Proposition 4
4.1 Suspension space
In this subsection, we recall the suspension construction and the induced 
$G$-action in [Reference Brown, Rodriguez Hertz and WangBRW16a, S 2].
Let 
$\alpha$ be a 
$\Gamma$-action on 
$M$ by 
$C^{1}$ diffeomorphisms, i.e. 
$\alpha (gh) = \alpha (g)\alpha (h)$. We consider the right action by 
$\Gamma$ on 
$G \times M$ defined as
\[ (g,x)\cdot \gamma = (g\gamma, \alpha(\gamma^{-1})(x)),\quad \forall \gamma \in \Gamma \]
and the left 
$G$-action
\[ a \cdot (g,x) =(ag, x), \quad \forall a \in G. \]
Define the quotient manifold 
$M^{\alpha } := (G \times M)/ \Gamma$. Since the left 
$G$-action commutes with the right 
$\Gamma$-action, the left 
$G$-action descends to a left 
$G$-action on 
$M^{\alpha }$, denoted by 
${{\tilde {\alpha }}}$. Since 
$\alpha$ is a 
$C^{1}$ action, 
$M^{\alpha }$ is naturally equipped with a 
$C^{1}$ manifold structure. The action 
${{\tilde {\alpha }}}$ is given by 
$C^{1}$ diffeomorphisms of 
$M^{\alpha }$. Moreover, denote by 
$\pi : M^{\alpha } \to G/\Gamma$ the projection induced by 
$G \times M \to G$, then 
$M^{\alpha }$ is a 
$C^{1}$ fiber bundle over 
$G/\Gamma$ induced by 
$\pi$ with fibers diffeomorphic to 
$M$.
With a slight abuse of notation, we use 
$d(\cdot, \cdot )$ to denote both the right-invariant metric on 
$G$, and the quotient metric on 
$G/\Gamma$. We denote by 
$\nu$ the normalized left Haar measure on 
$G/\Gamma$.
By the construction in [Reference Brown, Fisher and HurtadoBFH20, § 2.2] (see also [Reference Brown, Rodriguez Hertz and WangBRW16a, § 2.1] for the details), there exists a 
$C^{1}$ Riemannian metric 
$\langle \cdot, \cdot \rangle$ on 
$G \times M$ with the following properties:
(1)
$\langle \cdot, \cdot \rangle$ is invariant under the right $\Gamma$-action;(2) for each
$(g,x) \in G \times M$, under the canonical identification of the $G$-orbit of $(g,x)$ with $G$, the restriction of $\langle \cdot, \cdot \rangle$ to the $G$-orbit of $(g,x)$ coincides with $d_{G}$;(3) there exist a Siegel fundamental set
$D \subset G$ for the right $\Gamma$-action (see [Reference MargulisMar91, VIII.1] for the definition) containing the identity $e \in G$, and a constant $C_1 > 1$ such that for any $g_1,g_2 \in D$, the map $(g_1,x) \mapsto (g_2,x)$ distorts the restrictions of $\langle \cdot, \cdot \rangle$ to $\{g_1\} \times M$ and $\{g_2\} \times M$ by at most $C_1$.
We use 
$\langle \cdot, \cdot \rangle _g$ to denote the restriction of 
$\langle \cdot, \cdot \rangle$ to 
$\{g\} \times M$, and view it as a metric on 
$M$. By item (1) above, we can equip 
$M^{\alpha }$ with the quotient metric of 
$\langle \cdot, \cdot \rangle$.
We fix 
$\{\gamma _i\}$, a finite symmetric generating set for 
$\Gamma$. Let 
$\ell$ denote the word-length distance on 
$\Gamma$ relative to 
$\{\gamma _i\}$. Given a fundamental domain 
$F_D \subset D$ for the right 
$\Gamma$-action on 
$G$, i.e. 
$G = F_D\Gamma$ and 
$F_D\gamma \cap F_D = \emptyset$ for 
$\forall \gamma \in \Gamma \setminus \{e\}$, the return cocycle 
$\beta : G \times G/\Gamma \to \Gamma$ associated to 
$F_D$ is defined as follows. For any 
$g \in G$, 
$x \in G/\Gamma$, we set 
$\beta (g,x)$ to be the unique element 
$\gamma \in \Gamma$ such that 
$g\tilde {x} \in F_D \gamma$, where 
$\tilde {x}$ is the lift of 
$x$ in 
$F_D$. The following are from [Reference Brown, Fisher and HurtadoBFH20] whose proofs rely on [Reference Lubotzky, Mozes and RaghunathanLMR00].
Lemma 2 If 
$F_D \subset D$ is a fundamental domain for the right 
$\Gamma$-action on 
$G$ such that 
$e \in F_D,$ then there is a constant 
$C > 0$ such that for any 
$g \in G,$ any 
$x \in G/\Gamma,$
\[ \ell( \beta(g,x)) < C d(g,e) + Cd(x, \Gamma) + C. \]Lemma 3 There is a constant 
$C > 0$ such that the following is true. For any 
$g \in G$, any 
$x \in G / \Gamma$, any 
$p \in \pi ^{-1}(x)$ we have
\[ \log \| D_p{{\tilde{\alpha}}}(g)\| < C d(g,e) + Cd(x, \Gamma) + C. \]4.2 Smooth cocycle
Let 
$a$ be as in Proposition 4. In various statements about typical points in 
$G /\Gamma$ in this rest of this section, we will always refer to the Haar measure 
$\nu$.
We first recall several basic definitions from measure theory following [Reference Cornfield, Fomin and SinaiCFS92, Appendix 1]. A partition of a Lebesgue space 
$(Y, \mathcal {Y}, \mu _Y)$ (for the definition, see [Reference Cornfield, Fomin and SinaiCFS92, Appendix 1, Definition 4]) is a family 
$\xi = \{ C \}$ of non-empty disjoint measurable subsets 
$C$ such that 
$\bigcup _{C \in \xi } C = M$. A subset 
$A \in {\mathcal Y}$ is said measurable with respect to 
$\xi$ if 
$A$ is a union of elements of 
$\xi$. The partition 
$\xi$ is said to be measurable if there exists a countable collection of sets 
$\{ B_i \mid i \in I \}$ which are measurable with respect to 
$\xi$ such that for any 
$C_1, C_2 \in \xi$ we can find an 
$i \in I$ such that either 
$C_1 \subset B_i, C_2 \not \subset B_i$ or 
$C_2 \subset B_i$, 
$C_1 \not \subset B_i$. To any measurable partition 
$\xi$ we can assign a complete 
$\sigma$-algebra 
$\mathcal {B}_\xi \subset \mathcal {Y}$ consisting of the sets 
$A \in \mathcal {Y}$ which coincide modulo 
$\mu _Y$-null sets with one of the sets which is measurable with respect to 
$\xi$. In fact such correspondence is bijective (see [Reference Cornfield, Fomin and SinaiCFS92, Appendix 1, § 3]).
Following [Reference Ledrappier and StrelcynLS82] and [Reference Margulis and TomanovMT94, § 9.3], we may find a measurable partition 
$\xi$ of 
$G / \Gamma$ with the following properties.
(1)
$\xi$ is subordinate to the partition of $G / \Gamma$ into $H^{u}$ orbits: for a.e. $x \in G / \Gamma$:
(a) the atom
$\xi (x)$ is contained in the orbit $H^{u}\cdot x$;(b) the atom
$\xi (x)$ is precompact in the orbit $H^{u}\cdot x$;(c) the atom
$\xi (x)$ contains a neighborhood of $x$ in the orbit $H^{u}\cdot x$.
(2)
$\xi$ is $a$-decreasing, i.e. $a(\xi ) \leq \xi$.
We also require that 
$\xi$ satisfies the following additional property.
(3) There is a compact set
$W \subset H^{u}$ such that for a.e. $x$,
\[ \xi(x) \subset W \cdot x. \]
To build a partition 
$\xi$ satisfying 
$(1)$–
$(3)$, we first let 
$\xi _0$ be a partition satisfying 
$(1)$ and 
$(2)$. Select a 
$\xi _0$-measurable subset 
$S \subset G / \Gamma$ with positive 
$\nu$-measure such that the diameter of 
$\xi _0(x)$ is uniformly bounded in the 
$H^{u} \cdot x$-orbit for all 
$x \in S$. It is well known that 
$a$ is ergodic with respect to the Haar measure 
$\nu$. Thus for a.e. 
$x \in G / \Gamma$, the following number is well defined:
\[ n_x = \inf \{ n \in {{\mathbb{N}}} \mid a^{n} \cdot x \in S \}. \]We set
\[ \xi(x) = a^{-n_x} \xi_0(a^{n_x}\cdot x). \]
Then 
$\xi$ still satisfies 
$(1)$ and 
$(2)$. Since 
${\rm Ad}(a^{-1})$ is a contraction restricted to the Lie algebra of 
$H^{u}$, 
$\xi$ also satisfies 
$(3)$.
Since 
$\xi$ is measurable, we may apply [Reference Avila and VianaAV10, Lemma 4.6] to find a measurable selection: there is a measurable map 
$\psi : G/\Gamma \to G/\Gamma$ such that 
$\psi$ is constant on every atom of 
$\xi$, and 
$\psi (x) \in \xi (x)$ for 
$\nu$-a.e. 
$x$. Recall our choice of a Siegel fundamental set 
$D \subset G$ and fix a fundamental domain 
$F_D \subset D$ such that 
$e \in F_D$. Let 
$\bar {\psi } : G / \Gamma \to G$ be the map that assigns 
$x \in G / \Gamma$ the unique 
$g \in F_D$ with 
$\psi (x) = g\Gamma$. Note that 
$\bar \psi$ is 
$\xi$-measurable.
Since 
$H^{u}$ is horospherical for 
$a$, for a.e. 
$x \in G / \Gamma$ the map 
$H^{u} \to G /\Gamma$, 
$h \mapsto h \cdot x$ is injective. Indeed, for a 
$\mu$-typical 
$x \in G/\Gamma$, there is a sequence 
$\{t_m\}_{m \geq 0}$ of positive numbers that tends to infinity such that 
$\{ a^{-t_m} \cdot x \}_{m \geq 0}$ is precompact. Then 
$h \mapsto h \cdot x$ must be injective on 
$H^{u}$ since each 
$H^{u}$-orbit is contracted by the backward iterates of 
$a$, and 
$G \to G/\Gamma$ is a local homeomorphism. For any such 
$x$, we let 
$W_x$ be the inverse image of 
$\xi (x)$ under the map 
$H^{u} \to G/\Gamma$, 
$h \mapsto h \cdot \psi (x)$; and let 
$\xi _1(x) = W_x \bar \psi (x)$. Notice that by definition 
$\pi (\xi _1(x)) = \xi (x)$, and 
$\xi _1(x) \cap F_D \neq \emptyset$.
As 
$F_D$ is a fundamental domain contained in 
$D$, we can choose a Borel trivialization associated to 
$F_D$, denoted by
\[ \iota: M^{\alpha} \to F_D \times M, \]
where for each 
$x \in G / \Gamma$, we identify 
$\iota |_{\pi ^{-1}(x)}$ with a diffeomorphism 
$\iota _x: \pi ^{-1}(x) \to M$. Moreover, by the construction of the metric 
$\langle \cdot, \cdot \rangle$ on 
$D \times M$, we may assume that 
$\|\iota _x\|_{C^{1}}$ is uniformly bounded over all 
$x \in G/\Gamma$.
Given a typical 
$x \in G / \Gamma$, let 
$u_x \in H^{u}$ be such that 
$x = u_x \cdot \psi (x)$. Set 
$g_x : \pi ^{-1}(x) \to \pi ^{-1}(\psi (x))$ to be
\[ g_x(y) = {{\tilde{\alpha}}}(u_x^{-1})( y). \] Given 
$x \in G / \Gamma$, set 
$F_x : M \to M$ to be
\begin{equation} F_x(y) = \iota_{\psi(a^{-1} \cdot x)}( g_{a^{-1} \cdot \psi(x)}({{\tilde{\alpha}}}(a^{-1})( \iota_{\psi(x)}^{-1}(y)))). \end{equation} Let 
$F: G / \Gamma \times M \to G/ \Gamma \times M$ be the measurable map
\begin{equation} F(x,y) = (a^{-1}\cdot x, F_x(y) ). \end{equation}
Using 
$\{g_x\}$, we define a measurable map 
$\Phi : M^{\alpha } \to G /\Gamma \times M$ as follows:
\begin{equation} \Phi(y) = (\pi(y), \iota_{\psi(\pi(y))} g_{\pi(y)}(y)). \end{equation}
Let 
$\mu$ be the 
$a$-ergodic 
$a$-invariant measure in Proposition 4, let 
$\mu ^{*} = \Phi _* \mu$.
Claim 1 
$\Phi$ is a Borel isomorphism. Moreover, for 
$\mu$-a.e. 
$x \in G/\Gamma$, 
$\Phi$ is a 
$C^{1}$ diffeomorphism from 
$\pi ^{-1}(x)$ to 
$M$, and we have
\[ F \cdot \Phi = \Phi \cdot {{\tilde{\alpha}}}(a^{-1}). \]Proof. We set 
$x = \pi (z)$. Then we have
\[ \pi(a^{-1}\cdot z) = a^{-1}\cdot \pi(z) = a^{-1} \cdot x. \]Then
\[ F \Phi(z) = (a^{-1}\cdot x, \iota_{\psi(a^{-1}\cdot x)} g_{a^{-1}\cdot \psi(x)}({{\tilde{\alpha}}}(a^{-1}) ( g_x(z)))) \]and
\[ \Phi( {{\tilde{\alpha}}}(a^{-1})( z)) = (a^{-1}\cdot x, \iota_{\psi(a^{-1}\cdot x)} g_{a^{-1}\cdot x}({{\tilde{\alpha}}}(a^{-1})( z))). \]Then by definition, it suffices to show that
\[ a u_{a^{-1}\cdot x} = u_x a u_{a^{-1}\cdot \psi(x)}. \]By definition,
\[ a u_{a^{-1}\cdot x}\cdot \psi(a^{-1}\cdot x) = a \cdot a^{-1}\cdot x = x. \]
We also notice that 
$a^{-1}\cdot \psi (x) \in a^{-1} \cdot \xi (x) \subset \xi (a^{-1}\cdot x)$. Thus
\[ \psi(a^{-1}\cdot \psi(x)) = \psi(a^{-1}\cdot x). \]Then
\[ u_x a u_{a^{-1}\cdot \psi(x)}\cdot \psi(a^{-1}\cdot x) = u_x a \cdot a^{-1}\cdot \psi(x) = x. \]This completes the proof.
Let 
$\{ \mu ^{*}_x \}$ be the disintegration of 
$\mu ^{*}$ with respect to the partition of 
$G / \Gamma \times M$ into fibers. The following properties follow immediately from the above constructions and observations.
Proposition 6 We have the following.
(1) For a.e.
$x \in G / \Gamma$ and every $x' \in \xi (x),$ $F_x = F_{x'};$ in particular, $x \mapsto F_x$ is $\xi$-measurable.(2) The function
$x \mapsto \log \| F_x^{-1}\|_{C^{1}}$ is in $L^{1}(G / \Gamma, \nu )$.(3)
$\Phi$ is a measurable conjugacy between the dynamics of $a^{-1}$ on $M^{\alpha }$ and of $F$ on $G/ \Gamma \times M$.(4) The fiberwise Lyapunov exponents for
$Da$ with respect to $\mu$ are all non-positive if, and only if, the fiberwise Lyapunov exponents of $F$ with respect to $\mu ^{*}$ are all non-negative.(5)
$\mu$ is $H^{u}$-invariant if and only if the map $x \mapsto \mu ^{*}_x$ is $\xi$-measurable.
Proof. Part 
$(1)$ follows immediately from the construction. Part 
$(3)$ is given by Claim 1. Part 
$(4)$ follows from part (3) and our hypothesis on 
$a$ in Proposition 4: all fiberwise Lyapunov exponents of 
$D\tilde \alpha (a)$ are non-positive.
To show part (2), we first notice that by (4.1) for 
$\nu$-a.e. 
$x \in G/\Gamma$, we have
\[ \| F_{x}^{-1} \|_{C^{1}} \leq \| \tilde\alpha(u_{\alpha^{-1}\cdot \psi(x)})|_{\pi^{-1}(\psi(a^{-1} \cdot x))} \|_{C^{1}}\!\| \tilde\alpha(a)|_{\pi^{-1}(a^{-1} \cdot \psi(x))}\|_{C^{1}}. \]By Lemma 3, we have
\begin{align*} \log \| \tilde\alpha(a)|_{\pi^{-1}(a^{-1} \cdot \psi(x))}\|_{C^{1}} &\leq C d(a, e) + C d(a^{-1} \cdot \psi(x), x_0) + C, \\ \log\| \tilde\alpha(u_{\alpha^{-1}\cdot \psi(x)})|_{\pi^{-1}(\psi(a^{-1} \cdot x))}\|_{C^{1}} &\leq C \sup_{b \in W}d(b, e) + C d( \psi(a^{-1} \cdot x), x_0) + C. \end{align*}
Note that there are 
$u_1, u_2 \in W$ such that
\[ a^{-1} \cdot \psi(x) = a^{-1} u_1^{-1} x, \quad \psi(a^{-1} \cdot x) =u_2^{-1} a^{-1} x. \]Then we have
\[ d(a^{-1} \cdot \psi(x), x_0), d( \psi(a^{-1} \cdot x), x_0) \leq C d(x, x_0) + C' \]
for some 
$C$ depending only on 
$G, \Gamma$, and some 
$C'$ depending only on 
$W$ and 
$a$. We now briefly explain why we have that
\begin{equation} (x \mapsto d(x, x_0)) \in L^{1}(G/\Gamma, \nu). \end{equation}
Let 
$\rho : G \to {\rm SL}(N, {{\mathbb {R}}})$ be an embedding given in [Reference MargulisMar91, Chap. VIII, § 1]. By [Reference Lubotzky, Mozes and RaghunathanLMR00, 3.5 (*)], we have
\[ d(g \Gamma, ag \Gamma) \leq d(g, ag) \leq C(1 + \log \| \rho(a) \| ). \]
We remind the reader that the first 
$d$ denotes the quotient metric on 
$X$, and the second 
$d$ denotes the right-invariant metric on 
$G$. We deduce (4.4) by [Reference MargulisMar91, Chap. VIII, § 1, Proposition 1.2]. Then part (2) follows suit.
The ‘only if’ part of part 
$(5)$ follows by definition. We assume that 
$x \mapsto \mu ^{*}_x$ is 
$\xi$-measurable. Then for 
$\mu$-a.e. 
$x$, for any 
$h \in H^{u}$ such that 
$h(\pi (x)) \in \xi (\pi (x))$, we have 
${{\tilde {\alpha }}}(h)_*\mu _{\pi (x)} = \mu _{h(\pi (x))}$ where 
$\{\mu _z\}_{z \in G/\Gamma }$ is the disintegration of 
$\mu$ along the fibers. Moreover by Claim 1, we see that 
$x \mapsto \mu ^{*}_x$ is 
$a^{n}(\xi )$-measurable for any 
$n \geq 1$. We can use the above argument for 
$a^{n}(\xi )$ instead of 
$\xi$ (for all 
$n\geq 1$) to show that 
$\mu$ is 
$H^{u}$-invariant.
4.3 Avila and Viana's invariance principle
We will use a variant of [Reference Avila and VianaAV10, Theorem B] to conclude the proof of Proposition 4. Let us first briefly recall the setting in [Reference Avila and VianaAV10].
Let 
$(\hat X, \hat {\mathcal {B}}, \hat \mu )$ be a probability space, and let 
$\hat f: \hat X \to \hat X$ be an invertible 
$\hat \mu$-preserving measurable transformation. Let 
$N$ be a compact Riemannian manifold. We set 
$\hat{\mathcal {E}} = \hat X \times N$, and denote by 
$\hat P: \hat{\mathcal {E}} \to \hat X$ the projection to the first coordinate. We say that a 
$\hat {\mathcal {B}} \otimes \mathcal {B}_N$-measurable transformation 
$\hat F: \hat{\mathcal {E}} \to \hat{\mathcal {E}}$ is a smooth cocycle over 
$\hat f$ if 
$\hat F$ is of form 
$\hat F(\hat x, \hat y) = (\hat f(\hat x), \hat F_{\hat x}(\hat y))$, where 
$\hat F_{\hat x}$ is a diffeomorphism of 
$N$ for each 
$\hat x$. We also assume the following:
\begin{equation} \int |\log ( \sup_{\hat y} \| D \hat F_{\hat x}(\hat y)^{-1} \|) | \,d \hat\mu(\hat x) < \infty. \end{equation}
In the following, for any integer 
$k$, for any 
$\hat x \in {\hat X}$ we define
\[ \hat F^{k}_{\hat x} = \begin{cases} {\hat F}_{{\hat f}^{k-1}({\hat x})} \cdots {\hat F}_{\hat x} & k \geq 0, \\ ({\hat F}_{{\hat f}^{-k}({\hat x})})^{-1} \cdots ({\hat F}_{{\hat f}^{-1}({\hat x})})^{-1} & k < 0. \end{cases} \]
We warn the reader not to confuse the above notation with 
$(\hat F_{\hat x})^{k}$.
In this case, for any 
$\hat F$-invariant probability measure 
$\hat m$ on 
$\hat{\mathcal {E}}$ that projects to 
$\hat \mu$ under 
$\hat P$, the minimal Lyapunov exponent is a well-defined quantity at 
$\hat m$-a.e. 
$(\hat x, \hat y)$ by the following formula:
\[ \lambda_-(\hat F, \hat x, \hat y) = \lim_{n \to \infty} \frac{1}{n} \log \| D\hat F_{\hat x}^{n}(\hat y)^{-1} \|^{-1}. \]The following theorem, whose proof is deferred to Appendix A, is a variant of [Reference Avila and VianaAV10, Theorem B].
Theorem 2 Let 
$\hat m$ be an 
$\hat F$-invariant measure on 
$\hat{\mathcal {E}}$ which projects down to 
$\hat \mu$. Let 
$\mathcal {B}_0 \subset \hat {\mathcal {B}}$ be a 
$\sigma$-algebra which generates 
$\hat {\mathcal {B}} \mod 0$ under 
$\hat f$. Assume that both 
$\hat f$ and 
$\hat x \mapsto \hat F_{\hat x}$ are 
$\mathcal {B}_0$-measurable 
$\mod 0$, and 
$\lambda _-(\hat F, \hat x, \hat y) \geq 0$ for 
$\hat m$-a.e. 
$(\hat x, \hat y)$, then the disintegration 
$\hat x \mapsto \hat m_{\hat x}$ of the measure 
$\hat m$ is 
$\mathcal {B}_0$-measurable modulo 
$0$.
4.4 Completing the proof
We can now finish the proof of Proposition 4.
Proof of Proposition 4 By Proposition 6, the hypothesis of Theorem 2 is satisfied with 
$(\hat X, N, \mathcal {B}_0, \hat \mu, \hat m, \hat f, \hat F)$ being 
$(G/\Gamma, M, \mathcal {B}_{\xi }, \mu, \nu, a^{-1}, F)$. Here 
$\mathcal {B}_\xi$ denotes the complete 
$\sigma$-algebra generated by the partition 
$\xi$. Then by Theorem 2, the map 
$x \mapsto \mu ^{*}_x$ is 
$\xi$-measurable. Proposition 4 then follows from Proposition 6(5).
5. Proof of Proposition 5
Recall that we fixed a finite set of symmetric generators 
$\{\gamma _i\}$ for 
$\Gamma$. The word distance 
$\ell$ on 
$\Gamma$ is defined in § 4.1.
Proof of Proposition 5 We let 
$\| \cdot \|_g$ denote the background Riemannian metric 
$g$ on 
$TM$, and let 
${{\rm Vol}}_g$ denote the volume form induced by 
$\| \cdot \|_g$. There is a 
$C^{\infty }$ Riemannian metric on 
$S^{2}(T^{*}M)$ associated to 
$\| \cdot \|_g$. We denote by 
$L^{p}(M, {{\rm Vol}}_g, S^{2}(T^{*}M))$ the space of 
$L^{p}$ sections of the tensor bundle 
$S^{2}(T^{*}M)$ with respect to 
${{\rm Vol}}_g$.
Since 
$\alpha$ has uniform subexponential growth of derivatives, by the strong property (T) of the lattice 
$\Gamma$ (proved in [Reference LafforgueLaf08, Reference de Laat and de la SalledLdlS15, Reference de la SalledlS19]), we can adapt the argument in [Reference Brown, Fisher and HurtadoBFH16] to show that there exist:Footnote 4
(1) constants
$C'''_p, \sigma _p > 0$ for every $1 \leq p < \infty$;(2)
$\bar {g} \in L^{p}(M, {{\rm Vol}}_g, S^{2}(T^{*}M))$ for all $1 \leq p < \infty$, which is non-degenerate, i.e. $\| v\|_{\bar {g}} > 0$ for ${{\rm Vol}}_g$-a.e. $x \in M$, and every non-zero $v \in T_xM$;(3) a sequence of probability measures on
$\Gamma$, denoted by $\{\omega _n\}_n$, satisfying $supp(\omega _n) \subset B_{word}(e, n) \subset \Gamma$ for every $n$, where $B_{word}(e, n)$ denotes the radius $n$ open ball in $\Gamma$ centered at $e$ with respect to the word distance,
such that, setting 
$g_n = \int \alpha (\gamma )^{*}g\, d\omega _n(\gamma )$, then we have
\begin{equation} \| g_n - \bar{g}\|_{L^{p}} < C'''_p e^{-n\sigma_p},\quad \forall 1 \leq p < \infty. \end{equation}
As a consequence, if we denote by 
${{\rm Vol}}_{\bar {g}}$ the measurable volume form induced by 
$\| \cdot \|_{\bar {g}}$, then the measure 
$d{{\rm Vol}}_{\bar {g}}$ is absolutely continuous with respect to 
$d{{\rm Vol}}_g$, and the density function 
${d{{\rm Vol}}_{\bar {g}}}/{d{{\rm Vol}}_g}$ has full support.
We define Lebesgue measurable functions 
$\bar {R}, \underline {R} : M \to {{\mathbb {R}}}_+$ as follows. Set
\[ \bar{R}(x) = \sup_{v \in T_xM, \| v\|_g = 1} \| v\|_{\bar{g}}, \quad \underline{R}(x) = \inf_{v \in T_xM, \| v\|_g = 1} \| v\|_{\bar{g}}. \]
We can see directly that for 
$d{{\rm Vol}}_g$-a.e. 
$x \in M$,
\[ \frac{d{{\rm Vol}}_{\bar{g}}}{d{{\rm Vol}}_g}(x) < \overline{R}^{\dim M}(x), \quad \frac{d{{\rm Vol}}_g}{d{{\rm Vol}}_{\bar{g}}}(x) < \underline{R}^{-\dim M}(x). \]We have the following lemma.
Lemma 4 For every 
$1 \leq p < \infty$, there is 
$C_p > 0$ such that
\[ \int \underline{R}^{-p} \,d{{\rm Vol}}_g < C_p, \quad \int \overline{R}^{p} \,d{{\rm Vol}}_g < C_p. \]Proof. The second inequality follows immediately from the fact that 
$\bar {g} \in L^{p}(M, {{\rm Vol}}_g, S^{2}(T^{*}M))$. It remains to prove the first inequality.
We define for every 
$n \geq 1$,
\[ \underline{R}_n(x) = \inf_{v \in T_xM, \| v\|_{g} = 1}\| v\|_{g_n}, \quad \forall x \in M, \]and
\[ \Omega_n = \{x \mid \underline{R}(x) \geq \tfrac{1}{2}\underline{R}_n(x)\}. \]
For the convenience of the notation, we set 
$\Omega _0 = \emptyset$. It is clear that 
$\bigcup _n \Omega _n$ is a 
$d{{\rm Vol}}_g$-conull subset of 
$M$.
By the uniform subexponential growth of derivatives, for every 
$\varepsilon > 0$ there is 
$C''_{\varepsilon } > 0$ such that
\begin{equation} \sup_{x \in M} (\underline{R}_{n}(x)^{-1}) < C''_{\varepsilon}e^{n\varepsilon}, \quad \forall n \geq 1. \end{equation}
By (5.1) and (5.2), for every 
$\varepsilon > 0$ we have
\begin{align*} {{\rm Vol}}_g(\Omega_n^{c}) &\leq {{\rm Vol}}_g( \{ x \mid |\underline{R}(x) - \underline{R}_n(x)| > \tfrac{1}{2} \underline{R}_n(x) \} )\\ &\leq 2\sup_{x \in M} (\underline{R}_{n}(x)^{-1})\int |\underline{R}(x) - \underline{R}_n(x)|\, d{{\rm Vol}}_g(x)\\ &\leq 2C''_{\varepsilon}C'''_1 e^{n\varepsilon-n\sigma_1}. \end{align*}
Then for each 
$1 \leq p < \infty$, we take 
$\varepsilon = \sigma _1/(10p)$, and we obtain
\begin{align*} \int \underline{R}(x)^{-p} \,d{{\rm Vol}}_g(x) &\leq 2^{p} \sum_{n= 0}^{\infty} \int_{\Omega_{n+1} \setminus \Omega_n} \underline{R}_{n+1}(x)^{-p} \,d{{\rm Vol}}_g(x)\\ &\leq 2^{p} \sum_{n= 0}^{\infty} \sup_x( \underline{R}_{n+1}(x)^{-p}) {{\rm Vol}}_g(\Omega_n^{c})\\ &\leq 2^{p+1} (C''_{\varepsilon})^{p+1}C'''_1 \sum_{n=0}^{\infty} e^{(n+1)p\varepsilon - n(\sigma_1 - \varepsilon)} := C_p < \infty.\tag* \end{align*}Lemma 5 For every 
$1 \leq p < \infty$, there exists 
$D_p > 0$ such that for every 
$\gamma \in \Gamma$,
\[ \int_M \| D_x\alpha(\gamma)\|_{g}^{p} \,d{{\rm Vol}}_g(x) \leq D_p. \]Proof. Take an arbitrary 
$\gamma \in \Gamma$, and set 
$F = \alpha (\gamma )$. We recall that 
$F$ preserves 
$\bar {g}$. That is, for 
$d{{\rm Vol}}_g$-a.e. 
$x$, for every 
$v \in T_xM$, we have 
$\| v\|_{\bar {g}} = \| D_xF(v)\|_{\bar {g}}$. Hence the measure 
$d{{\rm Vol}}_{\bar {g}}$ is 
$F$-invariant.
Notice that for 
$d{{\rm Vol}}_g$-a.e. 
$x \in M$,
\begin{align*} \| D_xF\|_{g} &= \sup_{v \in T_xM, \| v\|_g = 1}\| D_xF(v)\|_{g} \\ &= \sup_{v \in T_xM, \| v\|_g = 1}\| D_xF(v)\|_{\bar{g}}\frac{\| D_xF( v)\|_{g} }{\| D_xF(v)\|_{\bar{g}} }\\ &= \sup_{v \in T_xM, \| v\|_g = 1}\| v\|_{\bar{g}}\frac{\| D_xF( v)\|_{g} }{\| D_xF(v)\|_{\bar{g}} }\\ &\leq \bar{R}(x) \underline{R}(F(x))^{-1}. \end{align*}Then by Cauchy's inequality,
\[ \int \| D_xF\|_g^{p} \,d{{\rm Vol}}_g(x) \leq \biggl(\int \bar{R}(x)^{2p} \,d{{\rm Vol}}_g(x) \biggr)^{1/2} \biggl(\int \underline{R}(F(x))^{-2p} \,d{{\rm Vol}}_g(x) \biggr)^{1/2}. \]Also
\begin{align*} \int \underline{R}(F(x))^{-2p} \,d{{\rm Vol}}_g(x) &= \int \underline{R}(F(x))^{-2p} \frac{d{{\rm Vol}}_g}{d{{\rm Vol}}_{\bar{g}}}(x) \,d{{\rm Vol}}_{\bar{g}}(x)\\ &\leq \biggl( \int \underline{R}(F(x))^{-4p} \,d{{\rm Vol}}_{\bar{g}}(x) \biggr)^{1/2} \biggl( \int \biggl(\frac{d{{\rm Vol}}_g}{d{{\rm Vol}}_{\bar{g}}}(x) \biggr)^{2} \,d{{\rm Vol}}_{\bar{g}}(x) \biggr)^{1/2}\\ &\leq\biggl( \int \underline{R}(F(x))^{-4p} \,d{{\rm Vol}}_{\bar{g}}(x) \biggr)^{1/2} \biggl( \int \frac{d{{\rm Vol}}_g}{d{{\rm Vol}}_{\bar{g}}}(x) \,d{{\rm Vol}}_{g}(x) \biggr)^{1/2}\\ &\leq\biggl( \int \underline{R}(x)^{-4p} \,d{{\rm Vol}}_{\bar{g}}(x) \biggr)^{1/2} \biggl( \int \underline{R}^{-\dim M}(x) \,d{{\rm Vol}}_{g}(x) \biggr)^{1/2} \end{align*}and
\begin{align*} \int \underline{R}(x)^{-4p} \,d{{\rm Vol}}_{\bar{g}}(x) &= \int \underline{R}(x)^{-4p} \frac{d{{\rm Vol}}_{\bar{g}}}{d{{\rm Vol}}_g}(x) \,d{{\rm Vol}}_{g}(x)\\ &\leq \biggl( \int \underline{R}(x)^{-8p} \,d{{\rm Vol}}_{g}(x) \biggr)^{1/2} \biggl( \int \biggl(\frac{d{{\rm Vol}}_{\bar{g}}}{d{{\rm Vol}}_g}(x) \biggr)^{2} \,d{{\rm Vol}}_{g}(x) \biggr)^{1/2}\\ &\leq \biggl( \int \underline{R}(x)^{-8p} \,d{{\rm Vol}}_{g}(x) \biggr)^{1/2} \biggl( \int \overline{R}^{2\dim M}(x) \,d{{\rm Vol}}_{g}(x) \biggr)^{1/2}. \end{align*}By Lemma 4,
\[ \int \| D_xF(v)\|_g^{p} \,d{{\rm Vol}}_g(x) \leq C_{2p}^{1/2} C_{8p}^{1/8} C_{2\dim M}^{1/8} C_{\dim M}^{1/4}. \]
Since 
$\gamma$ is chosen arbitrarily, we can conclude the proof by taking 
$D_p$ to be the right hand side of the last inequality.
We fix an embedding 
$\iota : M \rightarrow {{\mathbb {R}}}^{N}$ for some integer 
$N$. Let 
$\pi _i: {{\mathbb {R}}}^{N} \to {{\mathbb {R}}}$ be the projection to the 
$i$th coordinate. We have seen that for every 
$1 \leq p < \infty$, there exists a constant 
$C'_p > 0$ such that for every 
$1 \leq i \leq N$, for every 
$\gamma \in \Gamma$,
\[ \int |D_x(\pi_i \iota \alpha(\gamma))|^{p} \,d{{\rm Vol}}_g(x) < C'_p. \]
Take 
$p > \dim M/(1-\beta )$. Then by Sobolev's embedding theorem, we can see that the set 
$\{\alpha (\gamma ) \mid \gamma \in \Gamma \}$ is pre-compact in 
${\rm Hol}$-
${\rm Homeo}^{\beta }(M)$. We know that any pre-compact subset of 
${\rm Hol}$-
${\rm Homeo}^{\beta }(M)$ is equicontinuous in 
${\rm Homeo}(M)$. Thus the closure of 
$\alpha (\Gamma )$ in 
${\rm Homeo}(M)$ is a compact topological group 
$K_0$, and we can directly verify by definition that 
$K_0 \subset {\rm Hol}$-
${\rm Homeo}^{\beta }(M)$. It is then direct to construct a 
$\Gamma$-invariant continuous distance on 
$M$ by averaging.
Acknowledgements
Zhang thanks the support and hospitality of IAS and KTH since part of this work was done during his postdoc there.
Appendix A
We now give the proof of Theorem 2 in this appendix. We recall the construction in [Reference Avila and VianaAV10, § 3]. There is a Lebesgue space 
$(X, \mathcal {B}, \mu )$ obtained by identifying any two points of 
$\hat X$ which are not distinguished by any element of 
$\mathcal {B}$; and a projection 
$\pi : \hat X \to X$ such that 
$\mathcal {B} = \pi _* \mathcal {B}_0$ and 
$\mu = \pi _* \hat \mu$. Since 
$\hat f$ is 
$\mathcal {B}_0$-measurable modulo 
$0$, there exists a 
$\mathcal {B}$-measurable modulo 
$0$ transformation 
$f: X \to X$ such that 
$\pi \circ \hat f = f \circ \pi$. Let 
$\mathcal {E} = X \times N$ and 
$P : \mathcal {E} \to X$ the canonical projection. Since 
$\hat F$ is 
$\mathcal {B}_0$-measurable modulo 
$0$, we may write 
$F_{\pi (\hat x)} = \hat F_{\hat x}$ for some 
$\mathcal {B}$-measurable modulo 
$0$ fiber bundle morphism 
$F: \mathcal {E} \to \mathcal {E}$ over 
$f$. The measure 
$m = (\pi \times id)_* \hat m$ is 
$F$-invariant and projects down to 
$\mu$. Denote by 
$\{ m_x \}_{x \in X}$ the measure disintegration of 
$m$ corresponding to the partition of 
$\mathcal {E}$ into the fibers. By the 
$F$-invariance of 
$m$ we deduce that for 
$\mu$-a.e. 
$x \in X$,
\begin{equation} m_{f(x)} = \int (F_{x'})_* m_{x'} \,d\mu^{f^{-1}(\mathcal{B}_0)}_x(x'). \end{equation} For any integer 
$l \geq 0$, we define 
$J_l : \mathcal {E} \to [0, \infty )$ by considering the Lebesgue decomposition of 
$(F_x^{-l})_* m_{f^{l}(x)}$ relative to 
$m_x$ :
\[ (F_x^{-l})_* m_{f^{l}(x)} = J_l(x, \cdot) m_x + \eta^{(l)}_x. \]
We abbreviate 
$J_1$ as 
$J$.
Define
\[ h(F, m) = \int -\log J \,dm. \]
Following the proof of [Reference Avila and VianaAV10, Theorem B], we will show that 
$m(\{ J = 0\}) = 0$ and in addition the following is true.
Proposition A.1 We have
\[ 0 \leq h(F, m) \leq - \dim N \int \min\{ 0, \lambda_-(\hat F) \} \,d\hat m. \]The statement of Proposition A.1 is the same as [Reference Avila and VianaAV10, Proposition 3.1], except that we are now assuming (4.5), while in [Reference Avila and VianaAV10] the authors assume that 
$\log \| D \hat F_{\hat x}(\hat y)^{-1} \|$, 
$\log \| D \hat H_{\hat x}(\hat y) \|$ and 
$\log \| D \hat H_{\hat x}(\hat y)^{-1} \|$ are all uniformly bounded, and the dependences of 
$D\hat F_{\hat x}(\hat y)$, 
$D\hat H_{\hat x}(\hat y)$ on 
$\hat x, \hat y$ are uniformly continuous. Thus we will need to make some adjustments to the proof in [Reference Avila and VianaAV10] (see also [Reference LedrappierLed86]).
Under the hypothesis of Theorem 2, we can conclude by Proposition A.1 that 
$h(F, m)$ vanishes. Once we know that 
$h( F, m)$ vanishes, we can apply [Reference Avila and VianaAV10, Proposition 3.2] to conclude the proof of Theorem 2. We recall the statement below.
Proposition A.2 If 
$h(F, m) = 0$ then 
$\hat x \mapsto \hat m_{\hat x}$ is 
$\mathcal {B}_0$-measurable modulo 
$0$.
The proof of Proposition A.2 is rather general and the condition (4.5) suffices. Now it suffices to give the proof of Proposition A.1. The proof here follows essentially the scheme in [Reference LedrappierLed86].
Proof of Proposition A.1 By the same argument in [Reference Avila and VianaAV10, § 3.2], we may assume without loss of generality that 
$\hat m$ is ergodic for 
$\hat F$. In this case, 
$\min (0, \lambda _-(\hat F))$ is a constant 
$\hat m$-almost everywhere, and is denoted by 
$-\lambda \leq 0$.
For any integer 
$k$, for any 
$(x, \xi ) \in X \times N$ we define
\[ F^{k}_x = \begin{cases} F_{f^{k-1}(x)} \cdots F_x & k \geq 0, \\ F_{f^{-k}(x)}^{-1} \cdots F_{f^{-1}(x)}^{-1} & k < 0, \end{cases} \]and
\[ L_k(x, \xi) = \| D_\xi F^{-k}_{f^{k}(x)} \|, \quad C_k(x) = \sup_{\xi \in N} L_k(x, \xi), \quad \tilde C_k(x, \xi) = C_k(x). \]Notice that we have
\begin{equation} 0 \leq \log \tilde C_k(x, \xi) \leq \sum_{i=0}^{k-1} \log \tilde C_1(F^{i}(x,\xi)). \end{equation} Given 
$(x, \xi ) \in X \times N$, we denote by 
$B(\xi, \delta )$ the ball in 
$N$ centered at 
$\xi$ of radius 
$\delta > 0$ and write
\[ B((x, \xi), \delta) = \{ x \} \times B(\xi, \delta). \]
For each integer 
$l \geq 0$, we write
\[ J_l(x, \xi; \delta) = \frac{(F_{f^{l}(x)}^{-l})_* m_{f^{l}(x)}(B(\xi, \delta))}{m_x(B(\xi, \delta))} \]and
\[ J^{*}_l(x, \xi) = \max_{\delta > 0} J_l(x, \xi; \delta). \]
It is clear that 
$J_l \geq 0$ and 
$J^{*}_l \geq 1$.
We fix some 
$\epsilon > 0$. Then there is 
$\beta _1 = \beta _1(\epsilon ) > 0$ so that for any set 
$A \subset X \times N$ with 
$m(A) < \beta _1$, we have
\begin{equation} \int_A \log \tilde C_1 \,d\mu < \epsilon. \end{equation}
Fix some integer 
$l > 0$ such that the measurable set 
$\Lambda _1 \subset X \times N$ defined by
\[ \Lambda_1 = \{ (x, \xi) \mid L_l(x, \xi) \leq e^{(\lambda + \epsilon)l} \} \]
satisfies 
$m(\Lambda _1) > 1 - \beta _1/2$. Then there is a subset 
$\Lambda \subset \Lambda _1$ with
\begin{equation} m(\Lambda) > 1 - \beta_1 \end{equation}
such that the derivatives 
$D_{\xi }F_x$ are uniformly continuous in 
$\xi$ over all 
$x \in \Lambda$, and for some 
$\delta _1 = \delta _1(\epsilon, l, \Lambda ) > 0$, for any 
$x \in \Lambda$, for any 
$\delta \in (0, \delta _1(\epsilon ))$ we have
\begin{equation} F_{x}^{-l}(B(\xi, \delta)) \subset B(F_{x}^{-l}(\xi), e^{(\lambda + 2\epsilon)l}\delta). \end{equation}
We denote by 
$E_l$ the collection of ergodic component of 
$m$ for 
$F^{l}$. Since 
$\mu$ is 
$F$-ergodic, we deduce that 
$E_l$ is finite and 
$F$ induces a cyclic permutation of 
$E_l$. Moreover, for 
$m$-almost every 
$(x,\xi )$ we denote by 
$m_{(x,\xi )}$ the ergodic component at 
$(x,\xi )$.
By [Reference LedrappierLed86, Proposition 5], we know that
\[ \log J^{*}_l \in L^{1}(\mathcal{E}, m). \]More precisely, we have the following.
Lemma A.1 For 
$m$-almost every 
$(x, \xi )$, we have
\[ J_l(x, \xi) = \prod_{i=0}^{l-1} J(F^{i}(x, \xi)). \]
Consequently, for any 
$m' \in E_l$, we have
\begin{equation} l h(F, m) = - \int \log J_l(x, \xi) \,dm'(x, \xi). \end{equation}Proof. It is clear that the first equality holds when 
$l=1$. For any 
$l > 1$, we have
\begin{align*} (F^{-l}_{f^{l}(x)})_* m_{f^{l}(x)} &= (F^{-1}_x)_*(F^{-l+1}_{f^{l}(x)})_* m_{f^{l}(x)} \\ &= (F^{-1}_x)_*(J_{l-1}(F(x, \xi))m_{f(x)} + \eta^{(l-1)}_{f(x)}) \\ &= J_{l-1}(F(x, \xi)) \cdot J(x,\xi) m_x + J_{l-1}(F(x, \xi)) \eta_x + (F^{-1}_x)_*\eta^{(l-1)}_{f(x)}. \end{align*}
Here 
$\eta ^{(l-1)}_{f(x)}$ is the singular component of 
$(F^{-l+1}_{f^{l}(x)})_* m_{f^{l}(x)}$ with respect to 
$m_{f(x)}$. By definition, 
$\eta _x$ is singular with respect to 
$m_x$. Moreover, 
$(F^{-1}_x)_*\eta ^{(l-1)}_{f(x)}$ is also singular with respect to 
$m_{x}$ for 
$m$-a.e 
$x$. Otherwise, we would know that 
$\eta ^{(l-1)}_{f(x)}$ is not singular with respect to 
$(F_x)_*m_{x}$; then by (A.1) we would know that 
$\eta ^{(l-1)}_{f(x)}$ is not singular with respect to 
$m_{f(x)}$, which is a contradiction. Consequently, we see that
\[ J_l = J_{l-1} \circ F \cdot J. \]
We then conclude the proof of the first equality by induction. The equality (A.6) in the lemma is an immediate consequence of the first equality, and the fact that 
$m = ({1}/{l})\sum _{i=0}^{l-1} (F^{i})_*m'$ for any 
$m' \in E_l$.
We choose some 
$\beta _2 = \beta _2(\epsilon, l) > 0$ such that for 
$m' \in E_l$, and for every 
$A \subset X \times N$ with 
$m'(A) < \beta _2$, we have
\begin{equation} \int_A (\log J^{*}_l + \log J_l) \,dm' < \epsilon. \end{equation}We define
\[ Z = \{ (x, \xi) \mid J_l(x, \xi) = 0 \}, \quad G = Z^{c} = \{ (x, \xi) \mid J_l(x,\xi) > 0 \}. \]
We fix a large constant 
$D > 0$. Given a constant 
$\delta > 0$, we define
\begin{align*} G(\delta) &= \{ (x, \xi) \in G \mid \log J_l(x, \xi; \delta') \leq \log J_l(x, \xi) + \epsilon, \forall \delta' \in (0, \delta) \},\\ Z(\delta) &= \{ (x, \xi) \mid \log J_l(x, \xi; \delta') \leq -D, \forall \delta' \in (0, \delta) \}. \end{align*}
We fix some 
$\delta _2 = \delta _2(\epsilon, l) > 0$ sufficiently small so that for every 
$m' \in E_l$,
\begin{equation} m'(G \setminus G(\delta_2)) < \beta_2. \end{equation} We take an arbitrary 
$\delta _0 \in (0, \min (\delta _1, \delta _2))$. Given 
$(x, \xi ) \in X \times N$, define 
$\delta _l(x, \xi ; 0) = \delta _0$ and for 
$k \geq 1$ recursively define
\[ \delta_l(x, \xi; k) = \begin{cases} e^{(-\lambda - 2\epsilon)l} \delta_l(x, \xi; k-1) & F^{kl}(x, \xi) \in \Lambda,\\ [\tilde C_l(F^{kl}(x, \xi))]^{-1} \delta_l(x, \xi; k-1) & F^{kl}(x, \xi) \notin \Lambda. \end{cases} \]Observe that we have
\[ \delta_l(x, \xi; k+1) \leq \delta_l(x, \xi; k) \leq \delta_0, \quad \forall k \geq 0 \]
and by (A.5) and the definition of 
$\tilde C$ we deduce that
\[ F^{-l}(B(F^{(k+1)l}(x, \xi), \delta_l(x, \xi; k+1))) \subset B(F^{kl}(x, \xi), \delta_l(x, \xi; k)), \quad k \geq 0. \]Lemma A.2 For 
$m$-almost every 
$(x, \xi ),$ we have
\[ \liminf_{n \to \infty} \frac{1}{n} \log \delta_l(x, \xi ; n) \geq (- \lambda - 3 \epsilon) l. \]Proof. By definition, we have
\[ \log \delta_l(x, \xi ; n) = \log \delta_0 + \sum_{k=0}^{n-1} (l(-\lambda - 2\epsilon)1_{F^{kl}(x,\xi) \in \Lambda} - \log \tilde C_l(F^{kl}(x,\xi))1_{F^{kl}(x,\xi) \notin \Lambda}). \]Then by pointwise ergodic theorem, we have
\[ \liminf_{n \to \infty} \frac{1}{n} \log \delta_l(x, \xi ; n) \geq - (\lambda + 2\epsilon)l m_{(x,\xi)}(\Lambda) - \int_{\Lambda^{c}} \log \tilde C_l \,dm_{(x, \xi)}. \]By (A.2), (A.3) and (A.4), we obtain
\begin{align*} \int_{\Lambda^{c}} \log \tilde C_l \,dm_{(x, \xi)} &\leq \sum_{i=0}^{l-1} \int_{\Lambda^{c}} \log \tilde C_1 \circ F^{i} \,dm_{(x, \xi)} \\ &\leq \sum_{i=0}^{l-1} \int_{\Lambda^{c}} \log \tilde C_1 d(F^{i})_*m_{(x, \xi)} \\ &= l \int_{\Lambda^{c}} \log \tilde C_1 \,dm \leq l \epsilon. \end{align*}
The equality follows from 
$m = \frac {1}{l} \sum _{i=0}^{l-1}(F^{i})_*m_{(x, \xi )}$ for 
$m$-almost every 
$(x, \xi )$. Then
\[ \liminf_{n \to \infty} \frac{1}{n} \log \delta_l(x, \xi ; n) \geq - (\lambda + 3\epsilon) l. \] Recall that, by [Reference LedrappierLed86, Proposition 5], for any Borel probability measure 
$\nu$ on 
$N$, we have
\[ \limsup_{r \to 0} \frac{\log \nu(B(\xi, r))}{\log r} \leq \dim N, \quad \text{$\nu$-a.e. $\xi$}. \]
Thus we may pick a subset 
$\Omega _1 \subset X \times N$ such that
\[ \limsup_{r \to 0} \frac{ \log \inf_{(x, \xi) \in \Omega_1} m_{x}(B(\xi, r))}{\log r} \leq \dim M \]
and 
$m'(\Omega _1) > 0$ for every 
$m' \in E_l$. Then for 
$m$-almost every 
$(x, \xi )$, there is an infinite sequence of 
$n$ such that 
$F^{nl}(x, \xi ) \in \Omega _1$. Then for all sufficiently large 
$n$ in such sequence we have
\begin{align} \frac{1}{n}\log m_{f^{nl}(x)}(B(F^{nl}(x, \xi), \delta_l(x, \xi; n))) &\geq \frac{1}{n} \log \delta_l(x, \xi; n) (\dim N + \epsilon) \nonumber\\ &\geq ( - \lambda - 4\epsilon)(\dim N + \epsilon)l. \end{align}On the other hand, we have
\begin{align*} & m_{f^{nl}(x)}(B(F^{nl}(x, \xi), \delta_l(x, \xi; n))) \\ &\quad= m_x(B(x, \delta_l(x, \xi; 0))) \prod_{j=0}^{n-1} \frac{m_{f^{(j+1)l}(x)}(B(F^{(j+1)l}(x, \xi), \delta_l(x, \xi; j+1) ))}{m_{f^{jl}(x)}(B(F^{jl}(x, \xi), \delta_l(x, \xi; j)) )}\\ &\quad\leq \prod_{j=0}^{n-1} \frac{m_{f^{(j+1)l}(x)}(B(F^{(j+1)l}(x, \xi), \delta_l(x, \xi; j+1) ))}{m_{f^{jl}(x)}(B(F^{jl}(x, \xi), \delta_l(x, \xi; j)))}\\ &\quad\leq \prod_{j=0}^{n-1} \frac{m_{f^{(j+1)l}(x)}(F^{l}(B(F^{jl}(x, \xi), \delta_l(x, \xi; j) )))}{m_{f^{jl}(x)}(B(F^{jl}(x, \xi), \delta_l(x, \xi; j)))}\\ &\quad= \prod_{j=0}^{n-1} J_l(F^{jl}(x, \xi), \delta_l(x, \xi; j)). \end{align*}
Take an arbitrary 
$m' \in E_l$. Then for 
$m'$-almost every 
$(x, \xi )$ we have
\begin{align*} & \limsup \frac{1}{n} \log m_{f^{nl}(x)}(B(F^{nl}(x, \xi), \delta_l(x, \xi; n)))\\ &\quad\leq \limsup \frac{1}{n} \sum_{j=0}^{n-1} \log J_{l}(F^{jl}(x, \xi), \delta_l(x,\xi; j))\\ &\quad\leq \int_{Z(\delta_0)} \log J_l \,dm' + \int \log J^{*}_l \,dm' \\ &\quad\leq - D m'(Z(\delta_0)) + l\int \log J^{*} \,dm. \end{align*}
The last inequality follows from Lemma A.1 and the definition of 
$Z(\delta _0)$. Combining the above inequality with (A.9), we conclude that
\[ m'(Z(\delta_0)) \leq \frac{l(\int \log J^{*} \,dm + (\lambda + 4\epsilon)(\dim N + \epsilon))}{D}. \]
Since the above holds for any 
$\delta _0$ sufficiently small and for any 
$m' \in E_l$, we deduce that
\[ m(Z) \leq \limsup_{\delta_0 \to 0} m(Z(\delta_0)) \leq \frac{l(\int \log J^{*} \,dm + (\lambda + 4\epsilon)(\dim N + \epsilon))}{D}. \]
By letting 
$D$ tend to infinity, we conclude that 
$m(Z) = 0$, and consequently 
$m(G) = 1$.
Now notice that for 
$m'$-almost every 
$(x, \xi )$ we have
\begin{align*} & \limsup \frac{1}{n} \log m_{f^{nl}(x)}(B(F^{nl}(x, \xi), \delta_l(x, \xi; n))) \\ &\quad\leq \limsup \frac{1}{n} \sum_{j=0}^{n-1} \log J_{l}(F^{jl}(x, \xi), \delta_l(x,\xi; j)) \\ &\quad\leq \int_{G(\delta_0)} ( \log J_l + l \epsilon ) \,dm' + \int_{G(\delta_0)^{c}} \log J^{*}_l \,dm' \\ &\quad\leq \int \log J_l \,dm' + 5l \epsilon. \end{align*}
The last inequality follows from (A.8), 
$G(\delta _2) \subset G(\delta _0)$ and (A.7). Combining the above inequality with (A.9) and (A.6) in Lemma A.1, we obtain
\[ h(F, m) - 5 \epsilon \leq (\lambda + 4\epsilon)(\dim N + \epsilon). \]
Since 
$\epsilon$ is arbitrary, we conclude the proof of Proposition A.1.
