---
abstract: 'We prove that almost every finite collection of matrices in $GL_d( \mathbb{R} )$ and $SL_d({{\mathbb R}})$ with positive entries is Diophantine. Next we restrict ourselves to the case $d=2$. A finite set of $SL_2({{\mathbb R}})$ matrices induces a (generalized) iterated function system on the projective line $\RP^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.'
address:
- 'Yuki Takahashi, Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel'
- 'Boris Solomyak, Department of Mathematics, Bar-Ilan University, Ramat Gan, Israel'
author:
- BORIS SOLOMYAK
- YUKI TAKAHASHI
bibliography:
- 'bib.bib'
title: 'Diophantine property of matrices and attractors of projective iterated function systems in $\RP^1$'
---

Introduction and main results
=============================

Diophantine property of matrices
--------------------------------

Recently there has been interest in Diophantine properties in non-Abelian groups. The following is a variant of [@GJS1999 Definition 4.2].

Let $\Ak = \{A_i\}_{i\in \Lam}$ be a finite subset of a topological group $G$ equipped with a metric $\varrho$. Write $A_\bi = A_{i_1}\cdots A_{i_n}$ for $\bi = i_1\ldots i_n$. We say that the set $\Ak$ is [*Diophantine*]{} if there exists a constant $c>0$ such that for every $n\in \N$, we have $$\label{Dioph1}
\bi,\bj\in \Lam^n,\ A_\bi\ne A_\bj \implies \varrho(A_\bi,A_\bj) > c^n.$$ The set $\Ak$ is [*strongly Diophantine*]{} if there exists $c>0$ such that for all $n\in \N$, $$\label{Dioph2}
\bi,\bj\in \Lam^n,\ \bi\ne \bj \implies \varrho(A_\bi,A_\bj) > c^n.$$

Clearly, $\Ak$ is strongly Diophantine if and only if it is Diophantine and generates a free semigroup. Gamburd, Jacobson, and Sarnak [@GJS1999 Definition 4.2] gave a definition of a Diophantine set, which is equivalent to ours, except that they always consider symmetric sets (that is, $g\in \Ak\ \Rightarrow\ g^{-1}\in \Ak$). Diophantine-type questions in groups arise in connection with spectral gap estimates, see [@GJS1999; @Bourgain2014].

See [@ABRS2015; @ABRS2018] for a recent discussion of Diophantine properties in groups and related problems. In [@ABRS2018] a Lie group $G$ is called Diophantine, if almost every $k$ elements of $G$, chosen independently at random according to the Haar measure, together with their inverses, form a Diophantine set in $G$. Gamburd et al. [@GJS1999] conjectured that $SU_2({{\mathbb R}})$ is Diophantine. More generally, it is conjectured that semi-simple Lie groups are Diophantine. Kaloshin and Rodnianski [@KR2001] proved a weaker Diophantine-type property: for a.e. $(A,B) \in SO_3({{\mathbb R}})\times SO_3({{\mathbb R}})$, there exists $c>0$ such that for any $n{\geqslant}1$ and any two distinct words $W_1, W_2$ over the set $\Ak=\{A,B,A^{-1},B^{-1}\}$ of length $n$, $$\|W_1-W_2\|{\geqslant}c^{n^2}.$$ It is mentioned in [@KR2001] that their method is general, and applies to $SU_2({{\mathbb R}})$ as well, and also to $m$-tuples of matrices for any $m{\geqslant}2$.

Next we state our first result. For any collection of linearly independent vectors $v_1,\ldots,v_{d}$ in ${{\mathbb R}}^{d}$ consider the simplicial cone $$\label{cone}
\Sig=\Sig_{v_1,\ldots,v_{d}} = \{x_1 v_1 + \cdots + x_{d} v_{d}:\ x_1,\ldots,x_{d}{\geqslant}0\}.$$ If a matrix $A\in GL_{d}({{\mathbb R}})$ satisfies $$A({\Sig}{\smallsetminus}\{0\}) \subset \Sig^\circ,$$ we say that $\Sig$ is [*strictly invariant*]{} for $A$. Given a cone $\Sig=\Sig_{v_1,\ldots,v_{d}}$, denote by $\Xk_{\Sig,m}$ (respectively, $\Yk_{\Sig,m}$) the set of all $GL_{d}({{\mathbb R}})$ (respectively, $SL_{d}({{\mathbb R}})$) $m$-tuples of matrices for which $\Sig$ is strictly invariant. We consider $\Xk_{\Sig,m}$ as an open subset of ${{\mathbb R}}^{d^2m}$ and $\Yk_{\Sig,m}$ as a $(d^2-1)m$-dimensional manifold.

\[main\_thm\] Let $\Sig=\Sig_{v_1,\ldots,v_{d}}$ be a simplicial cone in ${{\mathbb R}}^{d}$ and $m{\geqslant}2$.

[(i)]{} For a.e. $\mathcal{A} \in \mathcal{X}_{\Sig, m}$, the $m$-tuple $\mathcal{A}$ is strongly Diophantine. In particular, a.e. $m$-tuple of positive $GL_{d}({{\mathbb R}})$ matrices is strongly Diophantine.

[(ii)]{} For a.e. $\mathcal{A} \in \mathcal{Y}_{\Sig, m}$, the $m$-tuple $\mathcal{A}$ is strongly Diophantine. In particular, a.e. $m$-tuple of positive $SL_{d}({{\mathbb R}})$ matrices is strongly Diophantine.

*1. Unfortunately, our results do no cover any example of a symmetric set, since the strict invariance property cannot hold for a matrix $A$ and $A^{-1}$ simultaneously.*

2\. Every $m$-tuple of matrices with algebraic entries is Diophantine (but not necessarily strongly Diophantine), see, e.g., [@GJS1999 Prop.4.3].

3\. It is well-known that Diophantine numbers in ${{\mathbb R}}$ form a set of full measure, which is, however, meagre in Baire category sense (its complement contains a dense $G_\delta$ set). Baire category genericity of non-Diophantine $m$-tuples in $SU_2({{\mathbb R}})$ has been pointed out in [@GJS1999]. In $G=SL_d({{\mathbb R}})$ the situation is different, since there are, for example, open sets of $m$-tuples in $G\times G$ which satisfy (\[Dioph2\]). For instance, if ${{\mathbb R}}^d_+$ is mapped by $A,B$ into closed cones that are disjoint, except at the origin, then (\[Dioph2\]) holds for $\{A,B\}$. On the other hand, there are open sets in $(SL_d({{\mathbb R}}))^m$ in which non-Diophantine pairs are dense. For instance, the set of elliptic matrices in $SL_2({{\mathbb R}})$ is open, and a standard argument shows that a generic $m$-tuple that contains an elliptic matrix is not Diophantine.

The scheme of the proof of Theorem \[main\_thm\] is as follows. We consider the induced action of the matrices on the projective space, and show that, given a non-degenerate family of $m$-tuples strictly preserving an open set, depending on a parameter real-analytically, for all parameters outside an exceptional set of zero Hausdorff dimension, the induced iterated function system (IFS) satisfies a version of the “exponential separation condition”. This property implies the strong Diophantine condition for the matrices. We then locally foliate the space of $m$-tuples of matrices and apply Fubini’s Theorem. The result on the zero-Hausdorff dimensional set of exceptions uses the notion of [*order-$k$ transversality*]{}, which is a modified version of that which appeared in the work of Hochman [@Hochman2014; @Hochman2015]. The strict open set preservation property is needed to ensure that the induced IFS is contracting (uniformly hyperbolic).

Projective IFS and linear cocycles
----------------------------------

Let $\Ak = \{A_i\}_{i\in \Lam}$ be a finite collection of $SL_d({{\mathbb R}})$ matrices. The linear action of $SL_d({{\mathbb R}})$ on ${{\mathbb R}}^{d}$ induces an action on the projective space $\RP^{d-1}$, and thus $\Ak$ defines an IFS $\Phi_\Ak=\{\varphi_A\}_{A\in \Ak}$ on $\RP^{d-1}$, called a [*(real) projective IFS*]{}. Such IFS were studied by Barnsley and Vince [@BV2012], and by De Leo [@Leo2015a; @Leo2015b]. Following [@BV2012], we say that the IFS $\Phi_\Ak$ has an attractor $K$ if for every nonempty compact set $B$ in a neighborhood of $K$, we have $\lim_{k\to \infty}\Phi_\Ak^k(B) = K$ in the Hausdorff metric, where $\Phi_\Ak(B) = \bigcup_{A\in \Ak} \varphi_A(B)$. Let $\Ak = \{A_i\}_{i\in \Lam}$ be a finite collection of $GL_d({{\mathbb R}})$ matrices and $\Phi_\Ak$ the associated IFS on $\RP^{d-1}$.

An alternative, but closely related viewpoint, is to consider the linear cocycle $A: \Lam^{{\mathbb Z}}\to SL_d({{\mathbb R}})$ over the shift on $\Lam^{{\mathbb Z}}$, defined by $A(\bi) = A_{i_1}$. Strict contractivity of the projective IFS turns out to be equivalent to [*uniform hyperbolicity*]{} of the cocycle [@BG2009]. Here we restrict ourselves to the case of $d=2$, which was investigated in great detail by Avila, Bochi, and Yoccoz [@ABY2010]. There is a natural identification between $[0, \pi)$ and the projective space $\RP^1$. Below we use this identification freely, and whenever necessary we view $[0, \pi)$ as $\mathbb{R} / \pi \mathbb{Z}$. For $A\in GL_2({{\mathbb R}})$ denote the action of $A$ on $[0, \pi)\cong \RP^1$ by the symbol $\varphi_A$.

Denote by $d_{\Prob}$ the metric on $\RP^1$ induced from the identification with ${{\mathbb R}}/\pi{{\mathbb Z}}$. Below we work with $m$-tuples of $SL_2({{\mathbb R}})$-matrices, since the action of $GL_2({{\mathbb R}})$ factors through the $SL_2({{\mathbb R}})$ action in the obvious way, via $A\mapsto (\det A)^{-1}A$.

A [*multicone*]{} is a proper nonempty open subset $U$ of $\RP^1$, having finitely many connected components with disjoint closures.

In the following theorem we extracted the results relevant for us from [@ABY2010; @BV2012] (note that [@BV2012] considers real projective IFS of any dimension).

\[th-unihyp\] Let $\Ak = \{A_i\}_{i\in \Lam}$ be a family of $SL_2({{\mathbb R}})$ matrices and let $\Phi_\Ak$ be the associated IFS on $\RP^1$. The following are equivalent:

[(i)]{} the IFS $\Phi_\Ak$ has an attractor $K \ne \RP^1$;

[(ii)]{} the associated linear cocycle over $\Lam^{{\mathbb Z}}$ is uniformly hyperbolic;

[(iii)]{} there is a multicone $U$, such that $\Phi_\Ak(\ov{U}) \subset U$;

[(iv)]{} there is nonempty open set $V\subset \RP^1$ such that $\Phi_\Ak$ is contractive on $\ov{V}$, with respect to a metric equivalent to $d_{\Prob}$.

Following [@ABY2010], we will call a multicone satisfying $\Phi_\Ak(\ov{U})\subset U$, a [*strictly invariant multicone*]{} for the family of matrices and for the IFS. There are examples, see [@ABY2010], which show that one may need a multicone having $k$ components, for any given $k{\geqslant}2$, even for a pair of $SL_2({{\mathbb R}})$ matrices $\{A_1,A_2\}$.

Our next result concerns the dimension of the attractor. Following De Leo [@Leo2015a], consider the $\zeta$-function $$\zeta_\Ak(t) = \sum_{n{\geqslant}1} \sum_{\bi \in \Lam^n} \|A_\bi\|^{-t},$$ and define the [*critical exponent*]{} of $\Ak$ by \[critic\] s\_= \_[t0]{} {t: \_(t) = }.

\[thm-attr\] Let $\Ak=  \{A_i\}_{i\in \Lam}$ be a finite set of $SL_2({{\mathbb R}})$ matrices which has a strictly invariant multicone (or satisfies any of the equivalent conditions from Theorem \[th-unihyp\]), and let $K$ be the attractor of the associated IFS $\Phi_\Ak$ on $\RP^1$. Assume that at least two of the maps $\varphi_{A_i}$ have distinct attracting fixed points. If $\Ak$ is strongly Diophantine, then $\dim_H(K) = \min\{1,\half s_\Ak\}$, where $s_\Ak$ is the critical exponent (\[critic\]).

In the special case when the IFS $\Phi_\Ak$ satisfies the Open Set Condition, this result is due to De Leo [@Leo2015a Th.4]. Recall that the strong Diophantine condition holds, in particular, when $\Ak$ generates a free semigroup and all the entries of $A_i$ are algebraic.

\[rem-DeLeo\] [*It is further shown in [@Leo2015a] that for $\Ak$ hyperbolic (and in some parabolic cases), $$s_\Ak = \lim_{r\to \infty} \frac{N_{\Ak}(r)}{\log r},$$ where $N_{\Ak}(r)$ is the number of elements of norm ${\leqslant}r$ of the semigroup generated by $\Ak$. An analogy is pointed out with the classical results on Kleinian and Fuchsian groups, see, e.g., [@Sullivan1984].* ]{}

Let $\Phi=\Phi_\Ak$. An alternative way to express the dimension, and one we actually use in the proof, is in terms of [*Bowen’s pressure formula*]{} \[Bowen1\] P\_(s) = 0,where $P_\Phi(\cdot)$ is the pressure function associated with the IFS $\Phi$. Throughout the paper we use the notation $$\varphi_\bi = \varphi_{i_1}\ldots \varphi_{i_n}.$$ The pressure is defined by \[Bowen0\] P\_(t) = \_[n]{} \_[\^n]{} \_’\^t, where $\|\cdot\|$ is the supremum norm on $\ov{U}$. As will be clear from the Bounded Distortion Property, the definition of $P_\Phi(t)$ does not depend on the choice of strictly invariant multicone $U$, and moreover, \[critic2\] 2s = s\_.

It is a classical result, going back to Bowen [@Bowen1979] and Ruelle [@Ruelle1982], see also [@Falconer_Tech], that if $\{\varphi_i\}_{i\in \Lam}$ is a hyperbolic IFS on ${{\mathbb R}}$ of smoothness $C^{1+{{\varepsilon}}}$, satisfying the Open Set Condition, then the dimension of the attractor $K$ is given by the Bowen’s equation. In the case that the maps $\varphi_i$ are affine, $s>0$ is the unique solution of $$\sum_{i\in \Lam} r_i^s = 1,$$ where $r_i \in (0,1)$ is the contraction ratio of $\varphi_i$. For an IFS with overlaps this is not necessarily true. In [@SSU1], Simon, Solomyak, and Urbański showed that for a one-parameter family of nonlinear IFS with overlaps (hyperbolic and some parabolic) satisfying the [*order-1 transversality condition*]{}, for Lebesgue-a.e. parameter the dimension of the attractor is given by $$\label{dimul}
\dim_H (K) = \min\{1,s\},$$ where $s$ is from (\[Bowen1\]) and the pressure is given by (\[Bowen0\]).

\[def-sep\] Let ${{\mathcal F}}= \{f_i\}_{i\in \Lam}$ be an IFS on a metric space $(X,\varrho)$, that is, $f_i:X\to X$. We say that ${{\mathcal F}}$ satisfies the *exponential separation condition* on a set $J\subset \Xk$ if there exists $c > 0$ such that for all $n\in \N$ we have $$\label{exp_sep}
\sup_{x \in J} \varrho(f_\bi(x), f_\bj(x) ) > c^{n},\ \ \mbox{for all}\  \bi,\bj\in \Lam^{n}\ \ \mbox{with}\ \ i_1\ne j_1\ \ \mbox{and}\ \ f_\bi\not \equiv f_\bj.$$ If, in addition, the semigroup generated by ${{\mathcal F}}$ is free, that is, $f_\bi \equiv f_\bj \ \Longleftrightarrow\ \bi=\bj$, we say that ${{\mathcal F}}$ satisfies the [*strong exponential separation condition*]{}. If these properties hold for infinitely many $n$, then we say that ${{\mathcal F}}$ satisfies the [*(strong) exponential separation condition on $J$ along a subsequence*]{}.

It is rather straightforward to show that the (strong) Diophantine condition for an $m$-tuple in $SL_2({{\mathbb R}})$ matrices is equivalent to the (strong) exponential separation condition for the associated projective IFS (see Lemma \[lem-Dioph\] below).

In [@Hochman2014 Cor. 1.2], Hochman proved (\[dimul\]) for an affine IFS $\mathcal{F} = \{ f_i \}_{i \in \Lambda}$ satisfying the exponential separation condition on $J=\{0\}$ along a subsequence. Thus our Theorem \[thm-attr\] is, in a sense, a generalization of Hochman’s result to the case of contractive projective IFS.

[*In fact, Hochman [@Hochman2014] used the condition (\[exp\_sep\]) without the requirement $i_1\ne j_1$. However, for an IFS $\{f_i\}_{i\in \Lam}$ on an interval $J\subset {{\mathbb R}}$, such that $$\inf_{x\in J, i\in \Lam}|f_i'(x)|{\geqslant}r_{\min}>0,$$ requiring $i_1\ne j_1$ in (\[exp\_sep\]) does not weaken the exponential separation condition — it only affects the constant $c$. This follows from the estimate $$|f_\bi(x) -f_\bj(x)| =|f_{(\bi\wedge \bj)\bu}(x) - f_{(\bi\wedge \bj)\bv}(x) |{\geqslant}r_{\min}^n |f_\bu(x) - f_\bv(x)|, \ \ \bi,\bj\in \Lam^n,$$ where $\bi\wedge \bj$ is the common initial segment of $\bi$ and $\bj$, so that $u_1 \ne v_1$.* ]{}

IFS of linear fractional transformations
----------------------------------------

It is well-known that the action of $GL_2({{\mathbb R}})$ on $\RP^1$ can be expressed in terms of linear fractional transformations. For $$A = 
\begin{pmatrix}
a & b \\
c & d \\
\end{pmatrix}
\in GL_2(\mathbb{R}),$$ let $f_A(x) = (ax + b)/(cx + d)$, and define $\psi : [0, \pi) \to \mathbb{R^*}$ by $\psi (\theta) = \cos\theta/\sin\theta$, where ${{\mathbb R}}^* = {{\mathbb R}}\cup \{\infty\}$. It is easy to see that the following diagram commutes:

\(m) \[matrix of math nodes, row sep=3em, column sep=4em, minimum width=2em\] [ \[0, ) & [\[0, )]{}\
&\
]{}; (m-1-1) edge node \[left\] [$\psi$]{} (m-2-1) edge node \[above\] [$\varphi_A$]{} (m-1-2) (m-2-1.east|-m-2-2) edge node \[above\] [$f_A$]{} (m-2-2) (m-1-2) edge node \[left\] [$\psi$]{} (m-2-2);

Observe that $\psi$ is smooth, and on any compact subset of $(0,\pi)$ the derivatives of $\psi$ and $\psi^{-1}$ are bounded. The following is then an immediate corollary of Theorem \[thm-attr\].

\[cor-ifs\] Let $\mathcal{F} = \{ f_i \}_{ i \in \Lambda }$ be a finite collection of linear fractional transformations with real coefficients. Assume that there exists $U\subset {{\mathbb R}}$, a finite union of bounded open intervals with disjoint closures, such that $f_i( \overline{U} ) \subset U$ for all $i \in \Lambda$. If $\mathcal{F}$ satisfies the strong exponential separation condition on $\ov{U}$, then we have $
\dim_H (K) = \min\{ 1, s \}, 
$ where $s > 0$ is the unique zero of the pressure function $P_{{\mathcal F}}$.

Furstenberg measure
-------------------

Let $\mathcal{A} = \{ A_i \}_{i \in \Lambda}$ be a finite collection of $SL_2(\mathbb{R})$ matrices, and let $p = ( p_i )_{i \in \Lambda}$ be a probability vector. Assume that $p_i > 0$ for all $i \in \Lambda$ (we always assume this for any probability vector). We consider the finitely supported probability measure $\mu$ on $SL_2({{\mathbb R}})$: \[def-mu\] = \_[i]{} p\_i \_[A\_i]{}. Our standing assumption is that $\mathcal{A}$ generates an unbounded and totally irreducible subgroup (i.e., does not preserve any finite set in $\RP^1$). Then there exists a unique probability measure $\nu$ on $\RP^1$ satisfying $\mu\cdot\nu = \nu$, that is, $$\label{Furst1}
\nu = \sum_{i \in \Lambda} p_i A_i \nu,$$ where $A_i \nu$ is the push-forward of $\nu$ under the action of $A_i$, see [@Furstenberg1963]. The measure $\nu$ is the [*stationary measure*]{}, or the [*Furstenberg measure*]{}, for the random matrix product $A_{i_n}\cdots A_{i_1}$ where the matrices are chosen i.i.d. from $\Ak$ according to the probability vector $p$.

The properties of the Furstenberg measure for $SL_2({{\mathbb R}})$ random matrix products, such as absolute continuity, singularity, Hausdorff dimension, etc., were studied by many authors, including [@Ledrappier1983; @BL1985]. In [@Pincus1994; @Lyons2000; @SSU2] this investigation was linked with the study of IFS consisting of linear fractional transformations. The reader is referred to [@HS2017] for a discussion of more recent applications. We will recall the main result of [@HS2017], since it will be the main tool in proving Theorem \[thm-attr\]. Let $\chi_{\mathcal{A}, p}$ be the *Lyapunov exponent*, which is the almost sure value of the limit $$\label{Lyap1}
\lim_{n \to \infty} \frac{1}{n} \log \|  A_{i_1 \cdots i_n } \|,$$ where $i_1, i_2, \cdots \in \Lambda$ is a sequence chosen randomly according to the probability vector $p = ( p_i )_{ i \in \Lambda}$. The Lyapunov exponent is usually defined as the almost sure value of the limit $$\label{Lyap2}
\lim_{n \to \infty} \frac{1}{n} \log \|  A_{i_n \cdots i_1 } \|,$$ but it is easy to see that (\[Lyap1\]) and (\[Lyap2\]) define the same value (e.g., by Egorov’s Theorem). Under the standing assumptions, the limit exists almost surely and is positive [@Furstenberg1963]. The Hausdorff dimension of a measure $\nu$ is defined by $$\dim_H(\nu) = \inf\{\dim_H (E):\ \nu(E^c)=0\}.$$ For a probability vector $p = ( p_i )_{i \in \Lambda}$, we denote the *entropy* $H(p)$ by $$H(p) = - \sum_{i \in \Lambda} p_i \log p_i.$$

\[thmHS\] Let $\mathcal{A} = \{ A_i \}_{i \in \Lambda}$ be a finite collection of $SL_2(\mathbb{R})$ matrices. Assume that $\mathcal{A}$ is strongly Diophantine and generates an unbounded and totally irreducible subgroup. Let $p = ( p_i )_{i \in \Lambda}$ be a probability vector, and let $\nu$ be the associated Furstenberg measure. Then we have $$\label{eq-Fursten}
\dim_H (\nu) = \min \Big\{ 1, \frac{H( p )}{ 2 \chi_{ \mathcal{A}, p } } \Big\}.$$

Theorem \[main\_thm\] implies, in particular, that the dimension formula (\[eq-Fursten\]) holds for the Furstenberg measure associated with a.e. finite family of positive matrices (independent of the probability vector).

Next we address the question: what is the Hausdorff dimension of the support of the Furstenberg measure? Sometimes, the support is all of $\RP^1$, in which case the answer is trivially one. The definition (\[Furst1\]) implies that the support is invariant under the IFS $\Phi$ induced by $\Ak$. Thus, Theorem \[thm-attr\] has the following immediate corollary:

Let $\Ak=  \{A_i\}_{i\in \Lam}$ be a Diophantine set of $SL_2({{\mathbb R}})$ matrices which has a strictly invariant multicone, $\mu$ a finitely supported measure defined by (\[def-mu\]), and $\nu$ the associated Furstenberg measure. Then $\dim_H({\mathrm{supp} \,}\nu) = \min\{1,\half s_\Ak\}$, where $s_\Ak$ is the critical exponent of $\Ak$.

Denote by $\Hk_m$ the set of $m$-tuples in $SL_2({{\mathbb R}})$ which have a strictly invariant multicone. Avila (see [@Yoccoz2004 Prop.6]) proved that the interior of the complement of $\Hk_m$ in $(SL_2({{\mathbb R}}))^m$ is ${{\mathcal E}}_m$, where ${{\mathcal E}}_m$ is the set of $m$-tuples which generate a semigroup containing an elliptic matrix. Observe that if an elliptic matrix is conjugate to an irrational rotation, then certainly the invariant set (support of the Furstenberg measure) is all of $\RP^1$. On the other hand, if it is conjugate to a rational rotation, then the semigroup generated by $\Ak$ contains the identity and the strong Diophantine property fails. We expect that our methods can be extended to cover strongly Diophantine families on the boundary of $\Hk_m$, which include parabolic systems.

Structure of the paper
----------------------

The rest of the paper is organized as follows. In the next section we prove Theorem \[main\_thm\]. In Section 3 we consider projective IFS and prove Theorem \[thm-attr\]. Finally, in Section 4 we include proofs of some standard technical results for the reader’s convenience.

Diophantine property of $GL_{d+1}(\mathbb{R})$ and $SL_{d+1}(\mathbb{R})$ matrices
==================================================================================

For notational reasons it is convenient to consider $GL_{d+1}({{\mathbb R}})$ instead of $GL_d({{\mathbb R}})$.

$GL_{d+1}({{\mathbb R}})$ actions
---------------------------------

Let $A\in GL_{d+1}({{\mathbb R}})$ be a matrix that strictly preserves a cone $\Sig=\Sig_{v_1,\ldots,v_{d+1}}\subset {{\mathbb R}}^{d+1}$. Without loss of generality, we can assume that $\Sig{\smallsetminus}\{0\}$ is contained in the halfspace $\{x\in {{\mathbb R}}^{d+1}: x_{d+1}>0\}$. It is convenient to represent the induced action of $A$ on $\RP^{d}$ on the affine hyperplane $\{x\in {{\mathbb R}}^{d+1}: x_{d+1}=1\}$, and consider the corresponding action on ${{\mathbb R}}^{d}$. To be precise, for $x= (x_1,\ldots,x_{d})\in {{\mathbb R}}^{d}$, we consider $(x,1) =  (x_1,\ldots,x_{d},1)\in {{\mathbb R}}^{d+1}$ and let $$f_A(x) = {\rm P}_{d}\Bigl(\frac{A(x,1)}{A(x,1)_{d+1}}\Bigr),\ \ \mbox{when}\ \ A(x,1)_{d+1}\ne 0,$$ where ${\rm P}_{d}$ is the projection onto the first $d$ coordinates. The components of $f_A$ are rational functions, which are, of course, real-analytic on their domain. Consider $$\ov{V} := {\rm P}_{d}(\Sig \cap \{x\in {{\mathbb R}}^{d+1}: x_{d+1}=1\}).$$ By assumption, $f_A$ is well-defined on $\ov{V}$, and we have $f_A(\ov{V})\subset V$.

We will also consider the action of $A$ on the unit sphere, given by $$\varphi_A(x):=A\cdot x =\frac{Ax}{\|x\|},$$ for a unit vector $x\in \Sp^{d}$. Consider also $\ov{U}$, the intersection of $\Sig$ with the upper hemisphere. We have $\varphi_A(\ov{U}) \subset U$. Lines through the origin provide a 1-to-1 correspondence between $\ov{U}$ and $\ov{V}$, which is bi-Lipschitz in view of the assumption $\Sig {\smallsetminus}\{0\} \subset \{x\in {{\mathbb R}}^{d+1}: x_{d+1}>0\}$.

It is well-known [@Birk1957] (see also [@BV2012 Section 9]) that strictly preserving a cone implies that $\varphi_A$ is a strict contraction in the Hilbert metric on $\ov{U}$, which is by-Lipschitz with the round metric. We thus obtain the following:

\[lem-Hilbert\] Suppose that the finite family $\Ak = \{A_i\}_{i\in \Lam} \subset GL_d({{\mathbb R}})$ strictly preserves a simplicial cone $\Sig =\Sig_{v_1,\ldots,v_{d+1}}\subset \{x\in {{\mathbb R}}^{d+1}: x_{d+1}>0\}\cup \{0\}$. Then the associated IFS ${{\mathcal F}}_\Ak = \{f_A\}_{A\in \Ak}$ is real-analytic and uniformly hyperbolic on $\ov{V}\subset {{\mathbb R}}^{d}$, in the sense that there exist $C>0$ and $\gam\in (0,1)$ such that $$\max_{x\in \ov{V}} \|f'_\bi(x)\| {\leqslant}C\gam^n,\ \ \mbox{for all}\ \bi\in \Lam^n,$$ where $f_\bi = f_{A_{i_1}}\circ \cdots f_{A_{i_n}}$ and $\|f_\bi'(x)\|$ is the operator norm of the differential at the point $x$.

From exponential separation to the Diophantine property
-------------------------------------------------------

Recall the strong exponential separation condition (Definition \[def-sep\]).

\[prop-Dioph2\] Let $\Ak$ be a finite family of $GL_{d+1}({{\mathbb R}})$ matrices, and let $\Phi_\Ak$ be the induced IFS on $\Sp^d$. If $\Phi_\Ak$ satisfies the strong exponential separation on a nonempty set, then $\Ak$ is strongly Diophantine.

Let $C_1 = \max_{i\in \Lam}\{1,\|A_i\|\}$ and $C_2 = \max_{i\in \Lam} \{1,\|A_i^{-1}\|\}$. Suppose that $\bi\ne \bj$ in $\Lam^n$. Let us write $$\bi = (\bi\wedge \bj) \bu,\ \ \bj = (\bi\wedge \bj) \bv,$$ where $\bi\wedge \bj$ is the common initial segment of $\bi$ and $\bj$, so that $\bu = u_1\ldots u_k,\ \bv = v_1\ldots v_k$ for some $k{\leqslant}n$, with $u_1 \ne v_1$. We have $$\label{tiu1}
\|A_\bi- A_\bj\| {\geqslant}\|A_{\bi\wedge\bj}^{-1}\|^{-1} \|A_\bu - A_\bv\| {\geqslant}C_2^{-n} \|A_\bu - A_\bv\|.$$

\[lem-claim\] For any $A, B \in GL_{d+1}(\mathbb{R})$ and any unit vector $x\in {{\mathbb R}}^{d+1}$, we have $$\| A \cdot x - B \cdot x \| {\leqslant}\| A^{-1} \| \bigl( 1 + \| B \| \|B^{-1}\|\bigr)\cdot \| A - B \|.$$

We have $$\begin{aligned}
\| A \cdot x - B \cdot x \| &= \left\| \frac{Ax}{\| Ax \|} - \frac{Bx}{\| Bx \|} \right\| \\
&{\leqslant}\left\| \frac{Ax}{ \| Ax \|} - \frac{ Bx }{ \| Ax \| } \right\| + 
\left\| \frac{Bx}{ \| Ax \| } - \frac{ Bx }{ \| Bx \| } \right\| =: R_1 + R_2. 
\end{aligned}$$ Since $$1 = \| A^{-1} (Ax) \| {\leqslant}\| A^{-1} \| \| Ax \|,$$ we have $\| Ax \|^{-1} {\leqslant}\| A ^{-1}\|$. Therefore, $$R_1 {\leqslant}\| A - B \| \cdot \| A^{-1} \|.$$ Similarly, $$\begin{aligned}
R_2 &{\leqslant}\| B \| \cdot \big| \| Ax \| - \| Bx \| \big| \cdot \| Ax \|^{-1} \| Bx \|^{-1} \\
&{\leqslant}\| B \| \cdot \| A - B \| \cdot \| A^{-1} \| \| B^{-1} \|, \\
\end{aligned}$$ and the desired estimate follows.

Applying the lemma to $A_\bu$ and $A_\bv$ yields, in view of $\|A_\bw\|{\leqslant}C_1^n$, $\|A_\bw^{-1}\|{\leqslant}C_2^n$ for any $\bw \in \Lam^k$, $k{\leqslant}n$: $$\label{eq-tiu2}
\|A_\bu - A_\bv\| {\geqslant}2^{-n} C_1^{-n} C_2^{-2n} \|A_\bu\cdot x- A_\bv \cdot x\|.$$

Now we continue with the proof of the lemma. By assumption, $\Phi_\Ak$ satisfies the exponential separation condition on a nonempty set. Let $c\in (0,1)$ be the constant from the definition (\[exp\_sep\]). It follows that there exists $x\in \Sp^d$ such that $$\|A_\bu\cdot x - A_\bv\cdot x\|{\geqslant}c^k{\geqslant}c^n.$$ Combining this inequality with (\[eq-tiu2\]) and (\[tiu1\]) yields $$\|A_\bi - A_\bj\|{\geqslant}2^{-n} C_1^{-n} C_2^{-3n}c^n,$$ confirming the strong Diophantine property.

Dimension of exceptions for one-parameter real-analytic families
----------------------------------------------------------------

We consider a one-parameter family of real-analytic IFS on a compact subset of ${{\mathbb R}}^d$, and show that under some mild assumptions it satisfies the exponential separation condition outside of a Hausdorff dimension zero set. This section is based on [@Hochman2014 Section 5.4] and [@Hochman2015 Section 6.6], but we had to make a substantial number of modifications in the definitions and proofs.

Let $\mathcal{J}$ be a compact interval in ${{\mathbb R}}$ and $V$ a bounded open set in ${{\mathbb R}}^d$. Suppose that for each $i \in \Lambda$ we are given a real-analytic function $$f_i:\,\ov{V}\times \Jk\to V.$$ This means that it is real-analytic on some neighborhood of $\ov{V}\times \Jk$. We will sometimes write this function as $$f_{i,t}(x) = f_i(x,t),\ \ x\in \ov{V},\ \ t\in \Jk.$$ Denote $\mathcal{F}_t = \{ f_{i, t} \}_{ i \in \Lambda }$. This is a real-analytic IFS on $\ov{V}$, depending on the parameter $t\in \Jk$ real-analytically. For $\bi = i_1\ldots i_n$ we write $f_{\bi,t} = f_{i_1,t} \circ \cdots \circ f_{i_n,t}$.

Further, assume that this family of IFS is uniformly hyperbolic in the following sense: there exist $C>0$ and $0 < \gam<1$, such that \[gluk1\] f’\_[,t]{}(x) C\^n,     \^n,  x, t. Here in the left-hand side is the norm of differential with respect to $x\in {{\mathbb R}}^d$. Fix $x_0 \in V$. For any finite sequence $\bi \in\Lambda^n$ we define $$F_\bi(t) = f_{\bi,t}(x_0).$$ Of course, this depends on $x_0$, but we suppress it from notation. For $\bi\in \Lam^\N$ we have $$\label{conver}
\Pi_t(\bi) = F_\bi(t) :=  \lim_{n\to \infty} F_{\bi|_n}(t),$$ where $\Pi_t:\Lam^\N\to {{\mathbb R}}^d$ is the natural projection corresponding to the IFS ${{\mathcal F}}_t$ and $\bi|_n = i_1\ldots i_n$. Notice that this limit is well-defined, independent of $x_0$, and is uniform in $t\in \Jk$, by uniform hyperbolicity (\[gluk1\]).

\[lem-analytic\] The function $F_\bi(\cdot)$ is real-analytic on $\Jk$, for any $\bi\in \Lam^\N$. Moreover, $F_{\bi|_n}(\cdot) \to F_\bi(\cdot)$ uniformly on $\Jk$ for all $\bi\in \Lam^\N$, together with derivatives of all orders.

By assumption, for every $\bi \in \Lam^n$, the function $F_\bi$ extends to a holomorphic function in a complex neighborhood of $\Jk$, and we are going to prove that for all $\bi \in \Lam^\N$ the sequence $F_{\bi|_n}$ converges to $F_\bi$ on a sufficiently small neighborhood uniformly. In order to achieve this, note that since $f_i(x,t):\,\ov{V}\times \Jk \to V$ is real-analytic, it can be extended to a holomorphic (complex-analytic) function $\wtil{f}_i(z,\tau)$, defined on a neighborhood of $\ov{V}\times \Jk$ in ${{\mathbb C}}^d\times {{\mathbb C}}$. Denote by $[\ov{V}]^\delta$ the $\delta$-neighborhood of $\ov{V}$ in ${{\mathbb C}}^d$ and let $\wtil{f}_{i,\tau} = \wtil{f}_i(\cdot,\tau)$. Choose $\ell \in\N$ so that $C\gam^\ell < 1/2$. Then $\|f'_{\bi,t}(x)\|<1/2$ for $\bi\in \Lam^\ell$, and $x\in \ov{V}$. By continuity, there exists $\delta>0$ such that $\wtil{f}_{\bi,t}$, with $\bi\in \Lam^\ell$, is holomorphic on $[\ov{V}]^\delta$ and $$\label{niki1}
\|{\wtil{f}_{\bi,t}}'(z)\|< 1/2,\ \mbox{for all}\ \bi\in \Lam^\ell,\ z\in [\ov{V}]^\delta,\ t\in \Jk.$$ Here in the left-hand side is the norm of the differential with respect to $z\in {{\mathbb C}}^d$. Thus, each $\wtil{f}_{\bi,t}$, with $\bi\in \Lam^\ell$, is a strict contraction on $[\ov{V}]^\delta$, and since $\wtil{f}_{\bi,t}({\ov{V}}) = f_{\bi,t}(\ov{V})\subset V$, we obtain that $[\ov{V}]^\delta$ is mapped into its interior by $\wtil{f}_{\bi,t}$, for $t\in \Jk$. Then the same must be true for all $\tau$ in a sufficiently small complex neighborhood of $\Jk$, which we denote by $\Ok$. We can find a constant $L>0$ such that $$\label{niki2}
\|\wtil{f}_{\bj,\tau}'(z)\|{\leqslant}L,\ \mbox{for all}\ \bj \ \mbox{such that}\ \ |\bj| {\leqslant}\ell-1,\ z\in [\ov{V}]^\delta,\ \tau\in \Ok,$$ since there are finitely many holomorphic functions involved.

Now, it follows from (\[niki1\]) and (\[niki2\]) that the function $\wtil{f}_{\bj,t}$, for [*all*]{} $\bj\in \bigcup_{n=1}^\infty \Lam^n$ and $t\in \Jk$, is well-defined and holomorphic in ${{\mathcal W}}:= [\ov{V}]^{\delta/L}$, and moreover, it maps ${{\mathcal W}}$ into $[V]^\delta$. In addition, $\wtil{{{\mathcal F}}}^\ell_\tau = \{\wtil{f}_{\bi,\tau}\}_{\bi \in \Lam^\ell}$ is a strictly contracting IFS on ${{\mathcal W}}$, depending on $\tau\in \Ok$ holomorphically. It follows that the finite iterates $\wtil{f}_{\bi|_{n},\tau}$ converge to $\Pi_\tau(\bi)$, the natural projection for $\wtil{{{\mathcal F}}}_\tau$, as $n\to \infty$, uniformly for $\tau\in \Ok$. The uniform limit of holomorphic functions in an open set in ${{\mathbb C}}$ is holomorphic, and since $F_\bi(t) = \Pi_t(\bi)$ is the restriction of a holomorphic map to an interval on the real line, it is real-analytic. The uniform convergence of holomorphic functions implies uniform convergence of their derivatives as well.

Next, for $\bi,\bj \in \bigcup_{n=1}^\infty \Lam^n \cup \Lam^{\N}$, let $$\Delta_{\bi, \bj}(t) = F_\bi(t) - F_\bj(t) \in {{\mathbb R}}^d.$$ For any ${{\varepsilon}}> 0$, let $$E_{{{\varepsilon}}} = \bigcap_{N=1}^{\infty} \bigcup_{n > N} 
\Big( \bigcup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1}  \, ( \Delta_{\bi, \bj} )^{-1} B_{{{\varepsilon}}^n} \Big)$$ and $$\label{def-E}
E = \bigcap_{{{\varepsilon}}> 0} E_{{{\varepsilon}}},$$ where $B_{{{\varepsilon}}^n} = \{x\in {{\mathbb R}}^d:\ \|x\|{\leqslant}{{\varepsilon}}^n\}$. It is easy to see that if $t \notin E$ then $\mathcal{F}_t$ satisfies the strong exponential separation condition.

[*In [@Hochman2014; @Hochman2015] Hochman considered the case where $\mathcal{F}_t$ is an affine IFS. He defined the sets $E'_{{{\varepsilon}}}$ and $E'$ as follows: $$E'_{{{\varepsilon}}} = \bigcup_{N=1}^{\infty} \bigcap_{n > N} 
\Big( \bigcup_{\bi, \bj \in \Lambda^n, \bi \neq \bj} ( \Delta_{\bi, \bj} )^{-1} B_{{{\varepsilon}}^n} \Big)$$ and $$\label{def-Etag}
E' = \bigcap_{{{\varepsilon}}> 0} E'_{{{\varepsilon}}}.$$ If $t \notin E'$ then $\mathcal{F}_t$ satisfies the strong exponential separation condition [*along a subsequence*]{}.* ]{}

For a family of IFS ${{\mathcal F}}_t,\ t\in \Jk$, as above, and for $\bi,\bj\in \Lam^\N$ let $\Delta_{\bi,\bj}(t) = F_\bi(t) - F_\bj(t)$. We say that the family is [*non-degenerate*]{} if $$\label{nondegen}
\Delta_{\bi, \bj}(\cdot) \equiv 0 \iff \bi = \bj \ \ \mbox{for all $\bi, \bj \in \Lambda^{\mathbb{N}}$.}$$

We next prove the following:

\[thm\_important\] Suppose that the family of IFS ${{\mathcal F}}_t,\ t\in \Jk$, is non-degenerate. Then the set $E$ from (\[def-E\]) has Hausdorff dimension zero, and therefore, ${{\mathcal F}}_t$ satisfies the strong exponential separation condition on $\Jk$, outside of a set of zero Hausdorff dimension.

\[cor\_sep\] For a family of IFS ${{\mathcal F}}_t,\ t\in \Jk$, as above, assume that there exists $t_0 \in \mathcal{J}$ such that the sets $\{ f_{i, t_0}( \overline{V} ) \}_{i \in \Lambda}$ are pairwise disjoint. Then (\[nondegen\]) holds, and hence the set $E$ from (\[def-E\]) has Hausdorff dimension zero.

Hochman [@Hochman2014; @Hochman2015] proved, for a non-degenerate family of affine IFS, with a real-analytic dependence on parameter, that the set $E'$ from (\[def-Etag\]) has packing dimension zero.

For any smooth function $F : \mathcal{J} \to {{\mathbb R}}^d$, denote $F^{(p)}(t) = \frac{d^p}{dt^p} F(t)$.

The family $\{ \mathcal{F}_t \}_{t \in \mathcal{J} }$ is said to be *transverse of order $k$* if there exists $c > 0$ such that for all $n \in \mathbb{N}$ and $\bi, \bj \in \Lambda^{n}$, with $i_1 \neq j_1$, we have $$\forall t \in \mathcal{J} \ \  \exists p \in \{ 0, \cdots, k \} \text{\, s.t.  } 
\| \Delta^{(p)}_{\bi, \bj}(t) \| > c.$$ Here the norm $\|\cdot\|$ is simply the Euclidean norm in ${{\mathbb R}}^d$.

[*The above definition is different from [@Hochman2014] and it simplifies the proof of Theorem \[thm\_important\].* ]{}

\[delta\_prop\] Suppose that the non-degeneracy condition (\[nondegen\]) holds. Then $\{ \mathcal{F}_t \}_{t \in \mathcal{J}}$ is transverse of order $k$ for some $k \in \mathbb{N}$.

Suppose that for all $k\in \N$ the family $\{ \mathcal{F}_t \}_{t \in \mathcal{J}}$ is not transverse of order $k$. Then by assumption, for $\{c_k\}$ with $c_k < 1/k$, we can choose $n(k)$, $\bi^{(k)}, \bj^{(k)} \in \Lambda^{n(k)}$ with $i^{(k)}_1 \neq j^{(k)}_1$ and a point $t_k \in \mathcal{J}$ such that $$\| \Delta^{(p)}_{ \bi^{(k)}, \bj^{(k)} } (t_k) \| < c_k$$ for $0 {\leqslant}p {\leqslant}k$. Passing to a subsequence $\{ k_{l} \}$, we can assume that $t_{k_{l}} \to t_0\in \Jk$, $\bi^{( k_{l} )} \to \bi \in \Lambda^{\mathbb{N}}$ and $\bj^{( k_{l} )} \to \bj \in \Lambda^{\mathbb{N}}$, with $i_1\ne j_1$. Arguing as in the proof of Lemma \[lem-analytic\], the complex extension of $\Delta_{\bi^{( k_{l} )}, \bj^{( k_{l} )}}$ converges to the complex extension of $\Delta_{\bi, \bj}$ uniformly on a complex neighborhood of $\Jk$, and hence the same holds for $p$-th derivatives. Thus for all $p {\geqslant}0$, we have $$\| \Delta^{(p)}_{\bi, \bj} (t_0) \| = 
\lim_{l \to \infty} 
\| \Delta^{(p)}_{ \bi^{(k_{l}),} \bj^{(k_{l})} } (t_{k_{l}}) \| = 0.$$ Since $\Delta_{\bi, \bj}$ is real-analytic, the vanishing of its derivatives implies $\Delta_{\bi, \bj} \equiv 0$ on $\Jk$, contradicting (\[nondegen\]), since $\bi\ne \bj$ by construction.

For a $C^k$-smooth function $F : \mathcal{V} \to \mathbb{C}$, write $$\| F \|_{ \mathcal{V}, k } = \max_{ p \in \{0, \cdots, k\} } \sup_{t \in \mathcal{V}} | F^{(p)}(t) |,\ \ \|F\|_{\Vk} = \|F\|_{\Vk,0},$$ and similarly for vector-functions.

\[key\_lem\] Let $k \in \mathbb{N}$ and let $F : \mathcal{J} \to \mathbb{R}$ be a $k$ times continuously differentiable function. Let $M = \| F \|_{\mathcal{J}, k}$, and let $0 < c < 1$ be such that for every $t \in \mathcal{J}$ there is $p \in \{ 0, \cdots, k\}$ with $| F^{(p)}(t) | > c$. Then there exists $C=C_{c,M,|J|}{\geqslant}1$ such that for every $0 < \rho < (c/2)^{2^k}$, the set $F^{-1} (-\rho, \rho) \cap \mathcal{J}$ can be covered by $C^k$ intervals of length ${\leqslant}2( \rho/c )^{1/2^k} $ each.

\[lem-conc\] If $\{ \mathcal{F}_t \}_{t \in \mathcal{J} }$ is transverse of order $k {\geqslant}1$ on the compact interval $\mathcal{J}$, then the set $E$ from (\[def-E\]) has Hausdorff dimension zero.

Extending the real-analytic functions to the complex plane, as in Lemma \[lem-analytic\], since $$\sup_n \sup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} \| \Delta_{\bi, \bj} \|_{\Ok} < \infty$$ on a neighborhood $\Ok$ of $\Jk$, and $\Delta_{\bi, \bj}(\cdot)$ is holomorphic on $\Ok$ for all $\bi, \bj \in \Lambda^n$, we have $$\label{normal}
M:=\sup_{n} \sup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} \| \Delta_{\bi, \bj} \|_{ \mathcal{J} , k} < \infty.$$ Let $$\label{En}
E_{{{\varepsilon}}, n} = \bigcup_{\bi, \bj \in \Lambda^n, i_1 \neq j_1} ( \Delta_{\bi, \bj} )^{-1}( B_{{{\varepsilon}}^n} ).$$ Then $$\label{E}
E_{{{\varepsilon}}} = \bigcap_{N = 1}^{\infty} \bigcup_{n > N} E_{{{\varepsilon}}, n}.$$ Let $\bi, \bj \in \Lambda^n$, with $i_1\ne j_1$, and assume that $\|\Delta_{\bi,\bj}(t)\|< {{\varepsilon}}^n$. By Lemma \[key\_lem\] applied to a component of $\Delta_{\bi,\bj}$, for ${{\varepsilon}}$ sufficiently small, the set $$( \Delta_{\bi, \bj} )^{-1} ( B_{{{\varepsilon}}^n} )$$ may be covered by $C^k$ intervals of length ${\leqslant}2( {{\varepsilon}}^n \cdot c^{-1} )^{1/2^k}$. It follows that the set $E_{{{\varepsilon}},n}$ from (\[En\]) may be covered by $O( | \Lambda |^{2n} \cdot C^k )$ intervals of length ${\leqslant}( {{\varepsilon}}^n \cdot c^{-1} )^{1/2^k}$. Fix $s>0$ and write $\Hk^s$ for the $s$-dimensional Hausdorff measure. We obtain from (\[E\]) that $$\Hk^s(E_{{\varepsilon}}) {\leqslant}O(1)\cdot \sum_{n{\geqslant}1} |\Lam|^{2n} C^k {\bigl({{\varepsilon}}^n \cdot c^{-1}\bigr)}^{s/2^k} < \infty$$ for ${{\varepsilon}}$ sufficiently small. It follows that $\Hk^s(E)=0$.

This is now immediate from Lemmas \[delta\_prop\] and \[lem-conc\].

Proof of Theorem \[main\_thm\]
------------------------------

The next lemma follows by an application of Fubini’s Theorem.

\[Fubini\] Let $F \subset \mathbb{R}^n$ and let $v \in \mathbb{R}^n$ be a nonzero vector. Assume that for every $x_0 \in \mathbb{R}^n$, the set $\{ x_0 + t v : t \in \mathbb{R} \} \cap F$ has 1-dimensional Lebesgue measure $0$. Then the set $F$ has $n$-dimensional Lebesgue measure $0$.

\(i) Let $\Sig=\Sig_{v_1,\ldots,v_{d+1}}$ be a simplicial cone in ${{\mathbb R}}^{d+1}$. Let $\Uk \subset \Xk_{\Sig,m}$ be a small open set in $(GL_{d+1}({{\mathbb R}}))^m$ of $m$-tuples of matrices for which $\Sig$ is strictly invariant. Choose vectors $w_i \in {{\mathbb R}}^{d+1} \ (i \in \Lambda)$, with distinct directions, in such a way that \[w-cond\] w\_i \_[ A\_i v\_1, …,A\_i v\_[d+1]{} ]{}     (A\_i)\_[i ]{} . This is possible when $\Uk$ is sufficiently small. Let $(A_i)_{i \in \Lambda} \in \Uk$, and for each $t {\geqslant}0$ and $i\in \Lam$ let $A_{i,t}$ be such that $$A_{i, t} v_j = A_i v_j + t w_i,\ j=1,\ldots,d+1.$$ Condition (\[w-cond\]) guarantees that $\{A_{i,t}v_j\}_{i=1}^{d+1}$ is linearly independent, and hence $A_{i,t}\in GL_{d+1}({{\mathbb R}})$ for all $t>0$. This is a consequence of the following elementary claim.

[**Claim.**]{} [*Let $y_1,\ldots,y_{d+1}\in {{\mathbb R}}^{d+1}$ be linearly independent, and suppose that $w = \sum_{k=1}^{d+1} a_k y_k$ for some $a_k{\geqslant}0$. Then the family $\{y_1 + w,\ldots, y_{d+1} + w\}$ is linearly independent as well.*]{}

We have $$\sum_{j=1}^{d+1} c_j \Bigl(y_j + \sum_{k=1}^{d+1} a_k y_k\Bigr) = 0\ \ \Longrightarrow \ \ \sum_{j=1}^{d+1} \Bigl(c_j + a_j \sum_{k=1}^{d+1} c_k \Bigr) y_j = 0,$$ hence $c_j + a_j \sum_{k=1}^{d+1} c_k = 0$ for all $j$. If $\sum_{k=1}^{d+1} c_k\ne 0$, we obtain a contradiction, in view of $a_j{\geqslant}0,\ j=1,\ldots, d+1$; thus $c_j=0,\ j=1,\ldots, d+1$, as claimed.

Let $\Ak_t = \{A_{i,t}\}_{i\in \Lam}$ be the family of matrices defined above, for $t{\geqslant}0$, and let ${{\mathcal F}}_t={{\mathcal F}}_{\Ak_t}$ be the corresponding one-parameter family of IFS on the set $\ov{V} \subset {{\mathbb R}}^d$ obtained by projection of $\Sig \cap \{x\in {{\mathbb R}}^{d+1}: x_d=1\}$ onto ${{\mathbb R}}^d$. Notice that the cone $\Sig$ is strictly preserved by all $\Ak_t,\ t{\geqslant}0$, by construction, hence by Lemma \[lem-Hilbert\], these IFS are all uniformly hyperbolic. It is easy to see that the IFS and their dependence on $t$ is real-analytic, since the IFS are given by rational functions. Condition (\[gluk1\]) holds for $t\in [0,M]$, for any $M<\infty$, by uniform hyperbolicity and compactness. Finally, observe that, given ${{\varepsilon}}>0$, for $t$ sufficiently large, we have $$f_{i,t}(\ov{V}) \subset {\rm P}_d\bigl(\Sig_{{\varepsilon}}(w_i)\cap \{x_{d+1}=1\}\bigr),$$ where $\Sig_{{\varepsilon}}(w_i)$ is the cone of vectors ${{\varepsilon}}$-close to $w_i$ in direction. By construction, $w_i$ are all distinct, hence Corollary \[cor\_sep\] applies. We obtain that for all $t\in [0,\infty)$ outside a set of Hausdorff dimension zero, the IFS ${{\mathcal F}}_t$ satisfies the exponential separation condition, and then Proposition \[prop-Dioph2\] implies that the $m$-tuple of matrices $(A_{i,t})_{i\in \Lam}$ is Diophantine for all $t$ outside of a zero-dimensional set, so certainly for Lebesgue-a.e. $t$. Now Lemma \[Fubini\] yields the desired claim.

\(ii) We consider $(SL_{d+1}({{\mathbb R}}))^m$ as a codimension-$m$ submanifold of $(GL_{d+1}({{\mathbb R}}))^m \subset {{\mathbb R}}^{(d+1)^2m}$. In the proof of part (i) we showed that for a.e. $(A_i)_{i\in \Lam} \in  \mathcal{X}_{\Sig, m}$, the induced IFS on a subset of ${{\mathbb R}}^d$ satisfies the strong exponential separation condition. Suppose that there is a positive measure subset ${{\mathcal E}}\subset \Yk_{\Sig,m}$ for which the strong Diophantine condition is violated. Then for every $(A_i)_{i\in \Lam}\in {{\mathcal E}}$, the induced IFS $\Phi$ does not have strong exponential separation, by another application of Proposition \[prop-Dioph2\]. However, $(A_i)_{i\in \Lam}\in \Yk_{\Sig,m}$ and $(c_i A_i)_{i\in \Lam}\in \Xk_{\Sig,m}$, for any $c_i>0$, induce the same IFS on the projective space, and we get a set of positive measure in $\Xk_{\Sig,m}$ for which the strong Diophantine condition does not hold. This is a contradiction, and the theorem is proved completely.

Dimension of the attractor
==========================

Let $A \in SL_2(\mathbb{R})$. It is easy to see that $A^{*} A$ has eigenvalues $\| A \|^2$, $\| A \|^{-2}$. Let $(\cos t_A, \sin t_A)^{\mathrm{t}}$ be the unit eigenvector corresponding to the eigenvalue $\| A \|^{-2}$, where $t_A \in [0, \pi)$. We recall some basic properties of the map $\varphi_A$. For more details see sections 2.2, 2.3 and 2.4 in [@HS2017]. The following simple lemma is [@HS2017 Section 2.4].

\[trivial0\] Let $A \in SL_2(\mathbb{R})$. Then the induced map $\varphi_{A}$ expands by at most $\| A \|^2$ and contracts by at most $\| A \|^{-2}$. Furthermore, for any ${{\varepsilon}}> 0$ there exists $C_{{\varepsilon}}> 1$ such that $\| A \|^{-2} {\leqslant}| \varphi_{A}'(x) | < C_{{\varepsilon}}\| A \|^{-2}$ for all $x \in [0, \pi) {\smallsetminus}(t_A - {{\varepsilon}}, t_A + {{\varepsilon}})$.

The following lemma is now immediate.

\[trivial\] Let $U \subsetneq (0, \pi)$ be an open set. Then, for every ${{\varepsilon}}> 0$ there exists $C_{{\varepsilon}}> 1$ such that for any $A \in SL_2( \mathbb{R} )$ with $(t_A-{{\varepsilon}},t_A+{{\varepsilon}}) \subset U$, we have $$\pi - C_{{\varepsilon}}\| A \|^{-2} < | \varphi_A( U ) | < \pi.  $$

Lemma \[trivial\] implies the following:

\[trivial3\] Let $U \subsetneq (0, \pi)$ be an open set. Then, for every ${{\varepsilon}}> 0$ there exists $M =M({{\varepsilon}})> 0$ such that the following holds: for any $A \in SL_2( \mathbb{R} )$ that satisfies $\varphi_A( U ) \subset U$ and $\| A \| > M$, we have $(t_A-{{\varepsilon}},t_A+{{\varepsilon}}) \not\subset U$.

Let $\mathcal{A} = \{A_i \}_{i \in \Lambda}$ be a finite collection of $SL_2(\mathbb{R})$ matrices and let $\Phi = \{\varphi_A\}_{A\in \Ak}$ be the corresponding IFS on $[0,\pi)\cong\RP^1$. Recall the notation: $$\Phi(E) = \bigcup_{A\in \Ak} \varphi_A(E).$$ Assume that there is a strictly invariant multicone $U\subset [0,\pi)$, that is, a nonempty open set having finitely many connected components with disjoint closures, such that $\ov{U}\ne \RP^1$ and $\Phi(\ov{U})\subset U$. By Theorem \[th-unihyp\], the associated cocycle is uniformly hyperbolic, which implies that there exist $c>0$ and $\lam>1$ such that $$\label{eq-hyp}
\|A_\bi\| {\geqslant}c\lam^n\ \ \mbox{for all}\ \bi\in \Lam^n,\ n\in \N,$$ see [@Yoccoz2004] and [@ABY2010 Theorem 2.2]. Fix ${{\varepsilon}}>0$ such that the $(2{{\varepsilon}})$-neighborhood of $\Phi(\ov{U})$ is contained in $U$, and let $M=M({{\varepsilon}})$ from Lemma \[trivial3\]. By (\[eq-hyp\]), there exists $n_0\in \N$ such that $\|A_\bi\|>M$ for $\bi\in \Lam^n$, $n{\geqslant}n_0$. Lemma \[trivial3\] implies that $(t_{A_\bi}-{{\varepsilon}}, t_{A_\bi}+{{\varepsilon}}) \not \subset U$, hence $$(t_{A_\bi}-{{\varepsilon}}, t_{A_\bi}+{{\varepsilon}}) \cap \Phi(\ov{U}) = \emptyset,\ \ \mbox{for all}\ \bi\in \Lam^n,\ n{\geqslant}n_0.$$ Hence, by Lemmas \[trivial0\] and \[trivial\] we obtain $$\label{eq-BDP}
\|A_\bi\|^{-2} {\leqslant}|\varphi'_{\bi}(x)| {\leqslant}C_{{\varepsilon}}\|A_\bi\|^{-2},\ \ \mbox{for all}\ \ x\in \ov{U},\ \bi\in \Lam^n,\ n{\geqslant}n_0.$$

Thus we obtain

\[lem-contract\] [(i)]{} The Bounded Distortion Property holds for $\Phi$ on $U$: there exists $C'>1$ such that $$\label{eq-bdp}
\frac{1}{C'} {\leqslant}\frac{|\varphi_{\bi}'(x)|}{|\varphi_{\bi}'(y)|} {\leqslant}C'\ \ \mbox{for all}\ \ x,y\in \ov{U}, \ \bi\in \Lam^n,\ n\in \N.$$

[(ii)]{} The IFS $\Phi^k$ is contractive on $U$ in the metric $d_{\Prob}$ for sufficiently large $k$. More precisely, there exists $C''>0$ such that $$\label{eq-HYP}
\|\varphi'_\bi\|_{\ov{U}} {\leqslant}C'' \lam^{-2n},$$ where $\lam>1$ is from (\[eq-hyp\]).

[(iii)]{} We have $$s = s_\Ak/2,$$ where $s$ is the solution of the Bowen’s equation $P_\Phi(s)=0$, with the pressure given by (\[Bowen0\]) and $s_\Ak$ is the critical exponent, given by (\[critic\]).

Let $p = ( p_i )_{i \in \Lambda}$ be a probability vector, and let $x_0 \in U$. Let $\chi_{\Phi, p}$ be the almost sure value of the limit $$\label{lim}
\lim_{n \to \infty} - \frac{1}{n} \log |  ( \varphi_{ i_1 \cdots i_n } )' (x_0)  |,$$ where $i_1, i_2, \cdots \in \Lambda$ is a sequence chosen randomly according to the probability vector $p = ( p_i )_{ i \in \Lambda}$. The equations (\[eq-BDP\]) and (\[Lyap1\]) imply

\[lem-Lyap\] We have $\chi_{ \Phi, p } = 2 \chi_{ \mathcal{A}, p }$.

By the Birkhoff Ergodic Theorem, applied to the shift transformation on $\Lam^\N$ with the measure $\mu=p^\N$, in view of the bounded distortion (\[eq-bdp\]), we have

$$\label{eq-Lyap3}
\chi_{\Phi,p} = \lim_{n \to \infty} - \frac{1}{n} \log \left|  ( \varphi_{ i_1 \cdots i_n } )^{'} ( \Pi (\bi)) \right|=-\int_{\Lam^\N} \log\bigl|\varphi_{i_1}'(\Pi(\bi))\bigr|\,d\mu(\bi),$$

where $\Pi:\Lam^\N\to \RP^1$ is the natural projection corresponding to $\Phi$.

\[lem-Dioph\] [(i)]{} Let $\Ak$ be a finite set of matrices in $GL_2({{\mathbb R}})$, and let $\Phi$ be the IFS induced by $\Ak$ on the projective line $\RP^1$. If $\Phi$ satisfies the strong exponential separation condition on a nonempty set, then $\mathcal{A}$ is strongly Diophantine.

[(ii)]{} Let $\Ak$ be a finite set of matrices in $SL_2({{\mathbb R}})$, and let $\Phi$ be the IFS induced by $\Ak$ on the projective line $\RP^1$. Then $\Phi$ satisfies the strong exponential separation condition on a set containing at least three points if and only if $\mathcal{A}$ is strongly Diophantine.

\(i) This is a special case of Proposition \[prop-Dioph2\], since exponential separation for the induced action on a subset of $\RP^1$ is equivalent to that for the induced action on a subset of the circle.

\(ii) One direction, that the strong exponential separation for $\Phi$ implies the strong Diophantine property for $\Ak$, follows from (i). For the converse, we refer to [@HS2017 Lemma 2.5], which says that $SL_2({{\mathbb R}})$ is quantitatively separated by the action on three points of $\RP^1$.

Recall that $\A$ is a finite set of $SL_2({{\mathbb R}})$ matrices satisfying the strong Diophantine condition and having a strictly invariant multicone $U$, and $\Phi=\Phi_\Ak$ is the associated IFS on $U$. Then $\Phi$ has a compact attractor $K$, and our goal is to show that $\dim_H(K) =s$, where $P_\Phi(s)=0$ and $P_\Phi$ is given by (\[Bowen0\]). It is known that $$\label{dim_ineq}
\dim_H (K) {\leqslant}s,$$ see the appendix for a short proof. Let us show the opposite inequality.

Let $d_n > 0$ be the solution of the equation $$\sum_{\bi\in \Lam^n} | U_\bi |^{d_n} = 1,$$ where $U_\bi =\varphi_\bi(U)$ and $|\cdot|$ denotes the Lebesgue measure on $[0,\pi)\cong\RP^1$. It is not hard to see that $$\label{dim_lim}
\lim_{n \to \infty} d_n = s.$$ For convenience of the reader, we include the proof in the appendix, following [@SiSo1999].

Let $p^{(n)} = ( p^{(n)}_\bi )_{\bi \in \Lam^n}$ be the probability vector such that $p^{(n)}_\bi = | U_\bi |^{d_n}$. Let $\eta^{(n)}$ be the invariant probability measure for the IFS $\Phi^n$ on $U$, corresponding to $p^{(n)}$. Since $\eta^{(n)}$ is supported on $K$, we have $\dim \eta^{(n)} {\leqslant}\dim_{H} (K)$.

We claim that $\Ak$ satisfies the assumptions of Theorem \[thmHS\]. Indeed, the existence of a strictly invariant multicone is known to imply that all the matrices in $\Ak$ are hyperbolic, hence the group generated by $\Ak$ is unbounded. Further, we assumed that not all attracting fixed points of $\Ak$ are the same, hence this group is totally irreducible. Thus the Furstenberg measure for $(\Ak^n,p^{(n)})$ is unique, and it coincides with $\eta^{(n)}$. Since $\Ak$ is Diophantine, we have that $\Ak^n$ is Diophantine as well. Now, by Theorem \[thmHS\] and Lemma \[lem-Lyap\] we have $$\frac{ H( p^{(n)} ) }{ \chi_{ \Phi^n, p^{(n)} } } {\leqslant}\dim_{H} (K).$$

We claim that there exists $C > 0$ such that $$\label{claim2}
\chi_{\Phi^n, p^{(n)} } {\leqslant}-\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log | U_\bi | + C\ \ \mbox{for all}\ \ n\in \N.$$ Indeed, by (\[eq-Lyap3\]), we have $$\chi_{\Phi^n, p^{(n)} } {\leqslant}\sum_{\bi \in\Lam^n} \mu([\bi])\cdot \log\bigl(\min_{x\in \ov{U}}|\varphi_\bi'(x)|\bigr)^{-1},$$ where $[\bi]$ is the cylinder set of sequences starting with $\bi$. Now $\mu([\bi])=|U|^{d_n}$, and $$\min_{x\in \ov{U}}|\varphi_\bi'(x)|{\geqslant}\frac{|U_\bi|}{C'|U|},$$ by the Bounded Distortion Property (\[eq-bdp\]). Therefore, $$\begin{aligned}
\chi_{ \Phi^n, p^{(n)} }  
& {\leqslant}\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log \frac{ C' | U | }{ | U_\bi | } \\
&= -\sum_{\bi \in \Lambda^n} | U_\bi |^{d_n} \log | U_\bi | + \log C' | U |. 
\end{aligned}$$ By (\[claim2\]), we have $$\begin{aligned}
\frac{ H( p^{(n)} ) }{ \chi_{ \Phi^n, p^{(n)} } } &> 
\frac{ -\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log | U_\bi |^{d_n} }
{ -\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log | U_\bi | + C } \\
&= d_n \left( 1 + \frac{C}{ -\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log | U_\bi | } \right)^{-1}. 
\end{aligned}$$ Since $\lim_{n \to \infty} d_n = s$ and $\lim_{n \to \infty} -\sum_{ \bi \in \Lambda^n } | U_\bi |^{d_n} \log | U_\bi | = \infty$, we obtain $s {\leqslant}\dim_{H} (K)$, as desired. Finally, $s=s_\Ak$ by Lemma \[lem-contract\](iii).

Appendix: the proof of (\[dim\_ineq\]) and (\[dim\_lim\]) {#appendix}
=========================================================

Proof of (\[dim\_lim\]) [@SiSo1999]
-----------------------------------

We have a projective IFS $\Phi = \{\varphi_i\}_{i\in \Lam}$ on a strictly invariant multicone $U$. Observe that $$P_\Phi(t) = \lim_{n\to \infty} \frac{1}{n} \log \sum_{\bi \in \Lam^n} \|\varphi_\bi'\|^t = \lim_{n \to \infty} \frac{1}{n} \log \sum_{\bi \in \Lambda^n} \left| U_{\bi} \right|^t,$$ by the Bounded Distortion Property (\[eq-bdp\]). Let $$Q_n = \frac{1}{n} \log \sum_{\bi \in \Lambda^n} \left| U_{\bi} \right|^s.$$ Since $P_\Phi(s ) = 0$, we have $\lim_{n \to \infty} Q_n = 0$. Let $r_1>0$ be such that $r_1 {\leqslant}|\varphi_i'(x)|$ for all $i \in \Lambda$ and $x \in \ov{U}$. Recall (\[eq-HYP\]), which says that $${\|\varphi'_\bi\|}_{\ov{U}} {\leqslant}C'' \lam^{-2n},$$ for $C''>0$ and $\lam>1$. Then $r^n_1 |U| {\leqslant}|U_\bi| < C''\lam^{-2n} |U|$ for $\bi\in\Lam^n$, and hence we have $$( r^n_1 |U| )^{s-d_n} \cdot |U_\bi|^{d_n} < |U_\bi|^s <  C'' ( \lam^{-2n} |U| )^{s-d_n} \cdot |U_\bi|^{d_n}.$$ In view of $\sum_{\bi\in \Lambda^n} |U_\bi|^{d_n} = 1$, we have $$\frac{1}{n} \log ( r^n_1 |U| )^{s-d_n} < Q_n < \frac{1}{n} \bigl[\log C'' + (s-d_n)(\log|U| - 2n\log\lam)\bigr],$$ and it follows that $$\frac{Q_n -(\log C'')/n}{ -2\log \lam+ ( \log |U| ) / n } < s - d_n < \frac{Q_n}{ \log r_1 + ( \log |U| ) / n },$$ which implies $d_n\to s$, as desired.

Proof of (\[dim\_ineq\])
------------------------

Fix ${{\varepsilon}}> 0$. Then for sufficiently large $n$ we have $d_n < s + {{\varepsilon}}/2$. Thus $$\sum_{\bi \in \Lambda^n} | U_\bi |^{s + {{\varepsilon}}} < \sum_{\bi \in \Lambda^n} | U_\bi |^{d_n + {{\varepsilon}}/2} 
< ( r_2^n |U| )^{{{\varepsilon}}/2} \to 0, \text{\ as } n \to \infty.$$ Therefore, the $(s + {{\varepsilon}})$-dimensional Hausdorff measure of $K$ is zero. By the definition of the Hausdorff dimension, this proves (\[dim\_ineq\]).

Acknowledgements {#acknowledgements .unnumbered}
================

The authors would like to acknowledge Balazs Barańy for helpful comments and for telling us about the papers [@ABY2010; @BG2009].
