Diophantine property of matrices and attractors of projective iterated function systems in $\mathbb{RP}^1$

TL;DR

This paper demonstrates that most finite sets of matrices with positive entries are Diophantine and explores the dimension of attractors of projective iterated function systems in the case of $SL_2( )$ matrices, linking hyperbolicity and Diophantine properties.

Contribution

It establishes the Diophantine property for almost all finite matrix collections and relates it to the dimension of attractors in projective iterated function systems for $d=2$.

Findings

Almost every finite collection of matrices in $GL_d( )$ and $SL_d( )$ with positive entries is Diophantine.

Under hyperbolicity and Diophantine assumptions, the attractor's dimension equals the minimum of 1 and the critical exponent.

The work connects Diophantine properties with fractal dimensions of attractors in projective dynamics.

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 $\mathbb{RP}^1$. Assuming uniform hyperbolicity and the Diophantine property, we show that the dimension of the attractor equals the minimum of 1 and the critical exponent.

Equations

$$\mathbf{i},\mathbf{j}\in {\Lambda }^n,A_{\mathbf{i}}\ne A_{\mathbf{j}}⟹\varrho (A_{\mathbf{i}},A_{\mathbf{j}})>c^n.$$

$$\mathbf{i},\mathbf{j}\in {\Lambda }^n,\mathbf{i}\ne \mathbf{j}⟹\varrho (A_{\mathbf{i}},A_{\mathbf{j}})>c^n.$$

$$\Vert W_1-W_2\Vert ⩾c^{n^2}.$$

$$\Sigma ={\Sigma }_{v_1,\dots ,v_d}=\{x_1{v}_1+\cdots +x_d{v}_d:x_1,\dots ,x_d⩾0\}.$$

$$A(\Sigma \setminus \{0\})\subset {\Sigma }^{\circ },$$

$$\Vert A_{\mathbf{i}}\Vert ⩾c{\lambda }^n\text{mbox}forall\mathbf{i}\in {\Lambda }^n,n\in \mathbb{N}.$$

$${\zeta }_{\mathcal{A}}(t)=\sum _{n⩾1}\sum _{\mathbf{i}\in {\Lambda }^n}\Vert A_{\mathbf{i}}{\Vert }^{-t},$$

$$s_{\mathcal{A}}=\underset{t⩾0}{\sup }\{t:{\zeta }_{\mathcal{A}}(t)=\infty \}.$$

$$s_{\mathcal{A}}=\underset{r\to \infty }{\lim }\frac{\log N_{\mathcal{A}}(r)}{\log r},$$

$$P_{\Phi }(s)=0,$$

$${\phi }_{\mathbf{i}}={\phi }_{i_1}\dots {\phi }_{i_n},\text{mbox}where{\phi }_i={\phi }_{A_i}.$$

$$P_{\Phi }(t)=\underset{n\to \infty }{\lim }\frac{1}{n}\log \sum _{\mathbf{i}\in {\Lambda }^n}\Vert {\phi }_{\mathbf{i}}^{\prime }{\Vert }^t,$$

$$2s=s_{\mathcal{A}}.$$

$$\sum _{i\in \Lambda }{r}_i^s=1,$$

$${\dim }_H(K)=\min \{1,s\},$$

$$\underset{x\in X^{\prime }}{\sup }\varrho (f_{\mathbf{i}}(x),f_{\mathbf{j}}(x))>c^n,\text{mbox}forall\mathbf{i},\mathbf{j}\in {\Lambda }^n\text{mbox}with{f}_{\mathbf{i}}\not\equiv f_{\mathbf{j}}.$$

$$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in G{L}_2(\mathbb{R}),$$

$$\mu =\sum _{i\in \Lambda }{p}_i{\delta }_{A_i}.$$

$$\nu =\sum _{i\in \Lambda }{p}_i{A}_i\nu ,$$

$$\underset{n\to \infty }{\lim }\frac{1}{n}\log \Vert A_{i_1\cdots i_n}\Vert ,$$

$$\underset{n\to \infty }{\lim }\frac{1}{n}\log \Vert A_{i_n\cdots i_1}\Vert ,$$

$${\dim }_H(\nu )=\inf \{{\dim }_H(E):\nu (E^c)=0\}.$$

$$H(p)=-\sum _{i\in \Lambda }{p}_i\log p_i.$$

$$\dim_{H}(\nu)=\min\Big{\{}1,\frac{H(p)}{2\chi_{\mathcal{A},p}}\Big{\}}.$$

$$f_{A}(x)={\rm P}_{d}\Bigl{(}\frac{A(x,1)}{A(x,1)_{d+1}}\Bigr{)},\ \ \mbox{when}\ \ A(x,1)_{d+1}\neq 0,$$

$$\overline{V}:={\mathrm{P}}_d(\Sigma \cap \{x\in {\mathbb{R}}^{d+1}:x_{d+1}=1\}).$$

$${\phi }_A(x):=A\cdot x=\frac{Ax}{\Vert Ax\Vert },$$

$$\underset{x\in \overline{V}}{\max }\Vert f_{\mathbf{i}}^{\prime }(x)\Vert ⩽C{\gamma }^n,\text{mbox}forall\mathbf{i}\in {\Lambda }^n,$$

$$\underset{x\in X^{\prime }}{\sup }\varrho (f_{\mathbf{i}}(x),f_{\mathbf{j}}(x))>c^n,\text{mbox}forall\mathbf{i},\mathbf{j}\in {\Lambda }^n\text{mbox}with{i}_1\ne j_1.$$

$$\underset{x\in I,i\in \Lambda }{\inf }|f_i^{\prime }(x)|⩾r_{\min }>0,$$

Full text

Both authors were supported by the Israel Science Foundation grant 396/15 (PI. B. Solomyak).

1. Introduction and main results

1.1. Diophantine property of matrices

Recently there has been interest in Diophantine properties in non-Abelian groups. The following is a variant of [15, Definition 4.2].

Definition 1.1**.**

Let ${\mathcal{A}}=\{A_{i}\}_{i\in\Lambda}$ be a finite subset of a semi-simple Lie group $G$ equipped with a metric $\varrho$. Write $A_{\mathbf{i}}=A_{i_{1}}\cdots A_{i_{n}}$ for ${\mathbf{i}}=i_{1}\ldots i_{n}$. We say that the set ${\mathcal{A}}$ is Diophantine if there exists a constant $c>0$ such that for every $n\in{\mathbb{N}}$, we have

[TABLE]

The set ${\mathcal{A}}$ is strongly Diophantine if there exists $c>0$ such that for all $n\in{\mathbb{N}}$,

[TABLE]

It is enough to check (1.1) for all $n$ sufficiently large, since for a finite number of $n$’s it always holds with some constant $c>0$. Clearly, ${\mathcal{A}}$ is strongly Diophantine if and only if it is Diophantine and generates a free semigroup; thus it also suffices to check (1.2) for all $n$ sufficiently large.

Gamburd, Jakobson, and Sarnak [15, Definition 4.2] gave a definition of a Diophantine set, which is similar to ours, except that they considered words in the alphabet ${\mathcal{A}}\cup{\mathcal{A}}^{-1}={\mathcal{A}}\cup\{A_{i}^{-1}\}_{i\in\Lambda}$. Diophantine-type questions in groups arise in connection with spectral gap estimates, see [15, 8].

See [1, 2] for a recent discussion of Diophantine properties in groups and related problems. There, the definition is more general, replacing the separation in (1.1) with a negative power of the cardinality of the $n$-ball in the word metric in the group generated by ${\mathcal{A}}$. In [2] a semi-simple 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. [15] conjectured that $SU_{2}({\mathbb{R}})$ is Diophantine. More generally, it is conjectured that semi-simple Lie groups are Diophantine. Kaloshin and Rodnianski [20] 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 ${\mathcal{A}}=\{A,B,A^{-1},B^{-1}\}$ of length $n$,

[TABLE]

It is stated in [20] that their method is general, and applies to $SU_{2}({\mathbb{R}})$ as well, and also to $m$-tuples of matrices for any $m\geqslant 2$. In [2, pp. 9-10] it is mentioned that the same method works for other semi-simple Lie groups. Breuillard [10, Cor.1.11] showed that a closely related weak form of Diophantine property holds for every $m$-tuple generating a dense subgroup of $SU_{2}({\mathbb{R}})$.

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

[TABLE]

If a matrix $A\in GL_{d}({\mathbb{R}})$ satisfies

[TABLE]

we say that $\Sigma$ is strictly invariant for $A$. Given a cone $\Sigma=\Sigma_{v_{1},\ldots,v_{d}}$, denote by ${\mathcal{X}}_{\Sigma,m}$ (respectively, ${\mathcal{Y}}_{\Sigma,m}$) the set of all $GL_{d}({\mathbb{R}})$ (respectively, $SL_{d}({\mathbb{R}})$) $m$-tuples of matrices for which $\Sigma$ is strictly invariant. We consider ${\mathcal{X}}_{\Sigma,m}$ as an open subset of ${\mathbb{R}}^{d^{2}m}$ and ${\mathcal{Y}}_{\Sigma,m}$ as a $(d^{2}-1)m$-dimensional submanifold.

Theorem 1.2**.**

Let $\Sigma=\Sigma_{v_{1},\ldots,v_{d}}$ be a simplicial cone in ${\mathbb{R}}^{d}$ and $m\geqslant 2$.

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

Remark 1.3**.**

1. Unfortunately, our results do not 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., [15, 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 [15]. 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 (1.2). For instance, if ${\mathbb{R}}^{d}_{+}$ is mapped by $A,B$ into closed cones that are disjoint, except at the origin, then (1.2) holds for $\{A,B\}$. On the other hand, there are open sets in $(SL_{d}({\mathbb{R}}))^{m}$ in which non-Diophantine $m$-tuples 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 1.2 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 [16, 17]. The strict open set preservation property is needed to ensure that the induced IFS is contracting.

1.2. Projective IFS and linear cocycles

Let ${\mathcal{A}}=\{A_{i}\}_{i\in\Lambda}$ be a finite collection of $GL_{d}({\mathbb{R}})$ matrices. The linear action of $GL_{d}({\mathbb{R}})$ on ${\mathbb{R}}^{d}$ induces an action on the projective space ${\mathbb{RP}}^{d-1}$, and thus ${\mathcal{A}}$ defines an IFS $\Phi_{\mathcal{A}}=\{\varphi_{A}\}_{A\in{\mathcal{A}}}$ on ${\mathbb{RP}}^{d-1}$, called a (real) projective IFS. Such IFS were studied by Barnsley and Vince [4], and by De Leo [12, 11]. Following [4], we say that the IFS $\Phi_{\mathcal{A}}$ has an attractor $K$ if for every nonempty compact set $B$ in a neighborhood of $K$, we have $\lim_{k\to\infty}\Phi_{\mathcal{A}}^{k}(B)=K$ in the Hausdorff metric, where $\Phi_{\mathcal{A}}(B)=\bigcup_{A\in{\mathcal{A}}}\varphi_{A}(B)$. It is shown in [4] that the attractor is necessarily unique. Assume, for simplicity, that matrices in ${\mathcal{A}}$ are orientation-preserving, that is, ${\mathcal{A}}\subset GL^{+}_{d}({\mathbb{R}})=\{A\in GL_{d}({\mathbb{R}}):\det(A)>0\}$. The action of $GL^{+}_{d}({\mathbb{R}})$ factors through the $SL_{d}({\mathbb{R}})$ action, via $A\mapsto|\det A|^{-1/d}\cdot A$; thus, it is often enough to work with families of $SL_{d}({\mathbb{R}})$ matrices.

An alternative, but closely related viewpoint, is to consider the linear cocycle $A:\Lambda^{\mathbb{Z}}\to SL_{d}({\mathbb{R}})$ over the shift on $\Lambda^{\mathbb{Z}}$, defined by $A({\mathbf{i}})=A_{i_{1}}$, see [6]. Here we restrict ourselves to the case of $d=2$, which was investigated in great detail by Yoccoz [29] and Avila, Bochi, and Yoccoz [3].

Definition 1.4** (De Leo [12]).**

A finite set of $SL_{2}({\mathbb{R}})$ matrices ${\mathcal{A}}=\{A_{i}\}_{i\in\Lambda}$ is called hyperbolic if there exist $c>0$ and $\lambda>1$ such that

[TABLE]

Recall that $A\in SL_{2}({\mathbb{R}})$ is elliptic, parabolic, or hyperbolic if ${\rm trace}(A)$ is, respectively, in $(-2,2)$, $\{-2,2\}$, or ${\mathbb{R}}\smallsetminus[-2,2]$. Definition 1.4 is consistent with this, in the sense that $A\in SL_{2}({\mathbb{R}})$ is hyperbolic if and only if $\{A\}$ is a hyperbolic set of matrices. The property (1.4) was used in [29, 3] as a criterion (necessary and sufficient) for uniform hyperbolicity of the cocycle. (We do not need the original definition of a uniformly hyperbolic cocycle, referring the reader to [29].)

There is a natural identification between $[0,\pi)$ and the projective space ${\mathbb{RP}}^{1}$. 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 $[0,\pi)\cong{\mathbb{RP}}^{1}$ by the symbol $\varphi_{A}$. Denote by $d_{{\mathbb{P}}}$ the metric on ${\mathbb{RP}}^{1}$ induced from the identification with ${\mathbb{R}}/\pi{\mathbb{Z}}$.

Definition 1.5**.**

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

In the following theorem we extracted the results relevant for us from [3, 4] (note that [4] considers real projective IFS of any dimension).

Theorem 1.6** ([3, 4]).**

Let ${\mathcal{A}}=\{A_{i}\}_{i\in\Lambda}$ be a family of $SL_{2}({\mathbb{R}})$ matrices and let $\Phi_{\mathcal{A}}$ be the associated IFS on ${\mathbb{RP}}^{1}$. The following are equivalent:

(i)* the IFS $\Phi_{\mathcal{A}}$ has an attractor $K\neq{\mathbb{RP}}^{1}$;*

(ii)* the set of matrices ${\mathcal{A}}$ is hyperbolic;*

(iii)* there is a multicone $U$, such that $\Phi_{\mathcal{A}}(\overline{U})\subset U$;*

(iv)* there is a nonempty open set $V\subset{\mathbb{RP}}^{1}$ such that $\Phi_{\mathcal{A}}$ is contractive on $\overline{V}$, with respect to a metric equivalent to $d_{{\mathbb{P}}}$.*

By the classical Hutchinson’s Theorem [19], for a contractive IFS on $\overline{V}$, the attractor $K$ is the unique non-empty compact invariant subset of $\overline{V}$. A contraction $\varphi_{A}$ on $\overline{V}$ has a unique fixed point, which is called the attracting fixed point. The attracting fixed points of all $\varphi_{A}$’s belong to the attractor $K$. Thus the attractor is not a singleton if and only if at least two of the attracting fixed points are distinct. In the latter case, as is well-known, $K$ is perfect, i.e., it has no isolated points.

We will call a multicone $U$ satisfying $\Phi_{\mathcal{A}}(\overline{U})\subset U$, a strictly invariant multicone for the family of matrices and for the IFS. There are examples, see [3], which show that one may need a multicone having $k$ components, for any given $k\geqslant 2$, even for a hyperbolic pair of $SL_{2}({\mathbb{R}})$ matrices.

Our next result concerns the dimension of the attractor. Following De Leo [12], consider the $\zeta$-function

[TABLE]

and define the critical exponent of ${\mathcal{A}}$ by

[TABLE]

Theorem 1.7**.**

Let ${\mathcal{A}}=\{A_{i}\}_{i\in\Lambda}$ 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 1.6), and let $K$ be the attractor of the associated IFS $\Phi_{\mathcal{A}}$ on ${\mathbb{RP}}^{1}$. Assume that the attractor $K$ is not a singleton. If ${\mathcal{A}}$ is strongly Diophantine, then $\dim_{H}(K)=\min\{1,\frac{1}{2}s_{\mathcal{A}}\}$, where $s_{\mathcal{A}}$ is the critical exponent (1.5).

In the special case when the IFS $\Phi_{\mathcal{A}}$ satisfies the Open Set Condition, this result is due to De Leo [12, Th.4]. Recall that the strong Diophantine condition holds, in particular, when ${\mathcal{A}}$ generates a free semigroup and all the entries of $A_{i}$ are algebraic.

Remark 1.8**.**

It is further shown in [12] that for ${\mathcal{A}}$ hyperbolic (and in some parabolic cases),

[TABLE]

where $N_{{\mathcal{A}}}(r)$ is the number of elements of norm $\leqslant r$ of the semigroup generated by ${\mathcal{A}}$. An analogy is pointed out with the classical results on Kleinian and Fuchsian groups, see, e.g., [28]. **

Let $\Phi=\Phi_{\mathcal{A}}$. An alternative way to express the dimension, and one we actually use in the proof, is in terms of Bowen’s pressure formula

[TABLE]

where $P_{\Phi}(\cdot)$ is the pressure function associated with the IFS $\Phi$. Throughout the paper we use the notation

[TABLE]

The pressure is defined by

[TABLE]

where $\|\cdot\|$ is the supremum norm on $\overline{U}$. 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,

[TABLE]

It is a classical result, going back to Bowen [9] and Ruelle [24], see also [13], that if $\{\varphi_{i}\}_{i\in\Lambda}$ 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 Bowen’s equation. In the case that the maps $\varphi_{i}$ are affine, $s>0$ is the unique solution of

[TABLE]

where $r_{i}\in(0,1)$ is the contraction ratio of $\varphi_{i}$. For an IFS with overlaps this is not necessarily true. In [26], 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

[TABLE]

where $s$ is the solution of Bowen’s equation (1.6) (the solution is unique in the hyperbolic case; in the parabolic cases considered in [26] one needs to take the minimal solution). More recently, starting with [16], (a version of) the following condition appeared in the literature.

Definition 1.9**.**

Let ${\mathcal{F}}=\{f_{i}\}_{i\in\Lambda}$ 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 $X^{\prime}\subseteq X$ if there exists $c>0$ such that for all $n\in{\mathbb{N}}$ sufficiently large we have

[TABLE]

If, in addition, the semigroup generated by ${\mathcal{F}}$ is free, that is, $f_{\mathbf{i}}\equiv f_{\mathbf{j}}\ \Longleftrightarrow\ {\mathbf{i}}={\mathbf{j}}$, 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 $X^{\prime}$ along a subsequence.

We prove in Proposition 2.4 below that the (strong) exponential separation condition on a non-empty set for the projective IFS on ${\mathbb{RP}}^{d-1}$ implies the (strong) Diophantine condition for an $m$-tuple of $GL_{d}({\mathbb{R}})$ matrices.

In [16, Cor. 1.2], Hochman proved (1.9) for an affine IFS $\mathcal{F}=\{f_{i}\}_{i\in\Lambda}$ on ${\mathbb{R}}$, satisfying the strong exponential separation condition along a subsequence on the set $X^{\prime}=\{0\}$. (In fact, it follows from [16] that strong exponential separation along a subsequence on any finite subset of ${\mathbb{R}}$ implies the same conclusion.) Thus our Theorem 1.7 is, in a sense, a generalization of Hochman’s result to the case of contractive projective IFS.

1.3. IFS of linear fractional transformations

It is well-known that the action of $GL_{2}({\mathbb{R}})$ on ${\mathbb{RP}}^{1}$ can be expressed in terms of linear fractional transformations. For

[TABLE]

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:

{[0,\pi)}$${{[0,\pi)}}$${\mathbb{R^{*}}}$${\mathbb{R^{*}}}$$\psi$$\varphi_{A}$$f_{A}$$\psi

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 1.7, in view of Proposition 2.4 below.

Corollary 1.10**.**

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 $i\in\Lambda$. Let $K$ be the attractor of the IFS ${\mathcal{F}}$, and assume that $K$ is not a singleton. If $\mathcal{F}$ satisfies the strong exponential separation condition on a non-empty subset of $\overline{U}$, then $\dim_{H}(K)=\min\{1,s\},$ where $s>0$ is the unique zero of the pressure function $P_{\mathcal{F}}$.

1.4. 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 $SL_{2}({\mathbb{R}})$:

[TABLE]

Our standing assumption is that $\mathcal{A}$ generates an unbounded and totally irreducible subgroup (i.e., does not preserve any finite set in ${\mathbb{RP}}^{1}$). Then there exists a unique probability measure $\nu$ on ${\mathbb{RP}}^{1}$ satisfying $\mu\cdot\nu=\nu$, that is,

[TABLE]

where $A_{i}\nu$ is the push-forward of $\nu$ under the action of $A_{i}$, see [14]. 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 ${\mathcal{A}}$ 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 [21, 7]. In [23, 22, 27] this investigation was linked with the study of IFS consisting of linear fractional transformations. The reader is referred to [18] for a discussion of more recent applications. We will recall the main result of [18], since it will be the main tool in proving Theorem 1.7.

Let $\chi_{\mathcal{A},p}$ be the Lyapunov exponent, which is the almost sure value of the limit

[TABLE]

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

[TABLE]

but it is well-known that (1.13) and (1.14) define the same value, since $A_{i_{1}\cdots i_{n}}$ and $A_{i_{n}\cdots i_{1}}$ have the same distribution. Under the standing assumptions, the limit exists almost surely and is positive [14]. The Hausdorff dimension of a measure $\nu$ is defined by

[TABLE]

For a probability vector $p=(p_{i})_{i\in\Lambda}$, we denote the entropy $H(p)$ by

[TABLE]

Theorem 1.11** ([18]).**

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

[TABLE]

Theorem 1.2 implies, in particular, that the dimension formula (1.15) 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 ${\mathbb{RP}}^{1}$, in which case the answer is trivially one. The definition (1.12) implies that the support is invariant under the IFS $\Phi$ induced by ${\mathcal{A}}$.

Corollary 1.12**.**

Let ${\mathcal{A}}=\{A_{i}\}_{i\in\Lambda}$ be a strongly Diophantine set of $SL_{2}({\mathbb{R}})$ matrices which has a strictly invariant multicone and generates a totally irreducible subgroup. Let $\mu$ be a finitely supported measure on ${\mathcal{A}}$ defined by (1.11), and $\nu$ the associated Furstenberg measure. Then $\dim_{H}(\mathrm{supp}\,\nu)=\min\{1,\frac{1}{2}s_{\mathcal{A}}\}$, where $s_{\mathcal{A}}$ is the critical exponent of ${\mathcal{A}}$.

The corollary immediately follows from Theorem 1.7. Indeed, having a strictly invariant multicone implies that the subgroup generated by ${\mathcal{A}}$ is unbounded, and together with total irreducibility this implies that the stationary measure $\nu$ is unique. Its support is a compact set, invariant for the IFS $\Phi_{\mathcal{A}}$, hence it is the attractor, which is unique under our assumption. The attractor is not a singleton by total irreducibility.

Denote by ${\mathcal{H}}_{m}$ the set of $m$-tuples in $SL_{2}({\mathbb{R}})$ which have a strictly invariant multicone. Proposition 6, attributed to a personal communication from Avila, which appeared, with a proof, in the paper by Yoccoz [29], asserts that the interior of the complement of ${\mathcal{H}}_{m}$ in $(SL_{2}({\mathbb{R}}))^{m}$ is ${\mathcal{E}}_{m}$, 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 ${\mathbb{RP}}^{1}$. On the other hand, if it is conjugate to a rational rotation, then the semigroup generated by ${\mathcal{A}}$ 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 ${\mathcal{H}}_{m}$, which include parabolic systems.

1.5. Structure of the paper

The rest of the paper is organized as follows. In the next section we prove Theorem 1.2. In Section 3 we consider projective IFS and prove Theorem 1.7. Finally, in Section 4 we include proofs of some standard technical results for the reader’s convenience.

2. 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 $GL_{d}({\mathbb{R}})$.

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

Let $A\in GL_{d+1}({\mathbb{R}})$ be a matrix that strictly preserves a cone $\Sigma=\Sigma_{v_{1},\ldots,v_{d+1}}\subset{\mathbb{R}}^{d+1}$. Without loss of generality, by a change of coordinates, we can assume that $\Sigma\smallsetminus\{0\}$ 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 ${\mathbb{RP}}^{d}$ 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

[TABLE]

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

[TABLE]

By assumption, $f_{A}$ is well-defined on $\overline{V}$, and we have $f_{A}(\overline{V})\subset V$.

We will also consider the action of $A$ on the unit sphere, given by

[TABLE]

for a unit vector $x\in{\mathbb{S}}^{d}$. Consider $\overline{U}$, the intersection of $\Sigma$ with the upper hemisphere. We have $\varphi_{A}(\overline{U})\subset U$. Lines through the origin provide a 1-to-1 correspondence between $\overline{U}$ and $\overline{V}$, which is bi-Lipschitz in view of the assumption $\Sigma\smallsetminus\{0\}\subset\{x\in{\mathbb{R}}^{d+1}:x_{d+1}>0\}$.

It is well-known [5] (see also [4, Section 9]) that strictly preserving a cone implies that $\varphi_{A}$ is a strict contraction in the Hilbert metric on $\overline{U}$, which is by-Lipschitz with the round metric. We thus obtain the following:

Lemma 2.1**.**

Suppose that the finite family ${\mathcal{A}}=\{A_{i}\}_{i\in\Lambda}\subset GL_{d}({\mathbb{R}})$ strictly preserves a simplicial cone $\Sigma=\Sigma_{v_{1},\ldots,v_{d+1}}\subset\{x\in{\mathbb{R}}^{d+1}:x_{d+1}>0\}\cup\{0\}$. Then the associated IFS ${\mathcal{F}}_{\mathcal{A}}=\{f_{A}\}_{A\in{\mathcal{A}}}$ is real-analytic and uniformly hyperbolic on $\overline{V}\subset{\mathbb{R}}^{d}$, in the sense that there exist $C>0$ and $\gamma\in(0,1)$ such that

[TABLE]

where $f_{\mathbf{i}}=f_{A_{i_{1}}}\circ\cdots\circ f_{A_{i_{n}}}$ and $\|f_{\mathbf{i}}^{\prime}(x)\|$ is the operator norm of the differential at the point $x$.

2.2. From exponential separation to the Diophantine property

Recall the strong exponential separation condition (Definition 1.9). For technical reasons it is convenient to weaken it slightly.

Definition 2.2**.**

Let ${\mathcal{F}}=\{f_{i}\}_{i\in\Lambda}$ be an IFS on a metric space $(X,\varrho)$. We say that ${\mathcal{F}}$ satisfies the SESDC condition on $X^{\prime}\subseteq X$ if there exists $c>0$ such that for all $n\in{\mathbb{N}}$ sufficiently large we have

[TABLE]

The abbreviation “SESDC” stands for “strong exponential separation on distinct (first order) cylinders”.

Remark 2.3**.**

For an IFS $\{f_{i}\}_{i\in\Lambda}$ on an interval $I\subset{\mathbb{R}}$, such that

[TABLE]

with $r_{\min}\in(0,1)$, requiring $i_{1}\neq j_{1}$ in (2.1) does not weaken the exponential separation condition (1.10); it only affects the constant $c$. This follows from the estimate

[TABLE]

where ${\mathbf{i}}\wedge{\mathbf{j}}$ is the common initial segment of ${\mathbf{i}}$ and ${\mathbf{j}}$, so that $u_{1}\neq v_{1}$. However, in higher dimensions it is sometimes easier to check the SESDC than the strong exponential separation condition. **

Proposition 2.4**.**

Let ${\mathcal{A}}$ be a finite family of $GL_{d+1}({\mathbb{R}})$ matrices, and let $\Phi_{\mathcal{A}}$ be the induced IFS on ${\mathbb{S}}^{d}$. If $\Phi_{\mathcal{A}}$ satisfies the SESDC condition on a non-empty set, then ${\mathcal{A}}$ is strongly Diophantine.

Proof.

Let $C_{1}=\max_{i\in\Lambda}\{1,\|A_{i}\|\}$ and $C_{2}=\max_{i\in\Lambda}\{1,\|A_{i}^{-1}\|\}$. Suppose that ${\mathbf{i}}\neq{\mathbf{j}}$ in $\Lambda^{n}$. Let us write

[TABLE]

where ${\mathbf{i}}\wedge{\mathbf{j}}$ is the common initial segment of ${\mathbf{i}}$ and ${\mathbf{j}}$, so that ${\mathbf{u}}=u_{1}\ldots u_{k},\ {\mathbf{v}}=v_{1}\ldots v_{k}$ for some $k\leqslant n$, with $u_{1}\neq v_{1}$. We have

[TABLE]

Lemma 2.5**.**

For any $A,B\in GL_{d+1}(\mathbb{R})$ and any unit vector $x\in{\mathbb{R}}^{d+1}$, we have

[TABLE]

Proof.

We have

[TABLE]

Since

[TABLE]

we have $\|Ax\|^{-1}\leqslant\|A^{-1}\|$. Therefore,

[TABLE]

Similarly,

[TABLE]

and the desired estimate follows. ∎

Applying the lemma to $A_{\mathbf{u}}$ and $A_{\mathbf{v}}$ yields, in view of $\|A_{\mathbf{w}}\|\leqslant C_{1}^{n}$, $\|A_{\mathbf{w}}^{-1}\|\leqslant C_{2}^{n}$ for any ${\mathbf{w}}\in\Lambda^{k}$, $k\leqslant n$:

[TABLE]

Now we continue with the proof of the proposition. By assumption, $\Phi_{\mathcal{A}}$ satisfies the SESDC condition on a non-empty set. Let $c\in(0,1)$ be the constant from the definition (2.1). It follows that for all $n\geqslant n_{0}$ there exists $x\in{\mathbb{S}}^{d}$ such that

[TABLE]

for all $n\geqslant n_{0}$. Combining this inequality with (2.3) and (2.2) yields

[TABLE]

confirming the strong Diophantine property. ∎

2.3. 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 SESDC condition on a single point, for parameters outside of a Hausdorff dimension zero set. This section is based on [16, Section 5.4], 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 ${\mathbb{R}}^{d}$. Let $\Lambda$ be a finite set, $|\Lambda|\geqslant 2$, and suppose that for each $i\in\Lambda$ we are given a real-analytic function

[TABLE]

This means that it is real-analytic on some neighborhood of $\overline{V}\times{\mathcal{J}}$. We will sometimes write this function as

[TABLE]

Denote $\mathcal{F}_{t}=\{f_{i,t}\}_{i\in\Lambda}$. This is a real-analytic IFS on $\overline{V}$, depending on the parameter $t\in{\mathcal{J}}$ real-analytically. For ${\mathbf{i}}=i_{1}\ldots i_{n}$ we write $f_{{\mathbf{i}},t}=f_{i_{1},t}\circ\cdots\circ f_{i_{n},t}$.

Further, assume that this family of IFS is uniformly hyperbolic in the following sense: there exist $C>0$ and $0<\gamma<1$, such that

[TABLE]

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 ${\mathbf{i}}\in\Lambda^{n}$ we define

[TABLE]

Of course, this depends on $x_{0}$, but we suppress it from notation. For ${\mathbf{i}}\in\Lambda^{\mathbb{N}}$ we have

[TABLE]

where $\Pi_{t}:\Lambda^{\mathbb{N}}\to{\mathbb{R}}^{d}$ is the natural projection corresponding to the IFS ${\mathcal{F}}_{t}$ and ${\mathbf{i}}|_{n}=i_{1}\ldots i_{n}$. Notice that this limit is well-defined, independent of $x_{0}$, and is uniform in $t\in{\mathcal{J}}$, by uniform hyperbolicity (2.4).

Lemma 2.6**.**

The function $F_{\mathbf{i}}(\cdot)$ is real-analytic on ${\mathcal{J}}$, for any ${\mathbf{i}}\in\Lambda^{\mathbb{N}}$. Moreover, $F_{{\mathbf{i}}|_{n}}(\cdot)\to F_{\mathbf{i}}(\cdot)$ uniformly on ${\mathcal{J}}$ for all ${\mathbf{i}}\in\Lambda^{\mathbb{N}}$, together with derivatives of all orders.

Proof.

By assumption, for every ${\mathbf{i}}\in\Lambda^{n}$, the function $F_{\mathbf{i}}$ extends to a holomorphic function in a complex neighborhood of ${\mathcal{J}}$, and we are going to prove that for all ${\mathbf{i}}\in\Lambda^{\mathbb{N}}$ the sequence $F_{{\mathbf{i}}|_{n}}$ converges to $F_{\mathbf{i}}$ on a sufficiently small neighborhood uniformly. In order to achieve this, note that since $f_{i}(x,t):\,\overline{V}\times{\mathcal{J}}\to V$ is real-analytic, it can be extended to a holomorphic (complex-analytic) function $\widetilde{f}_{i}(z,\tau)$, defined on a neighborhood of $\overline{V}\times{\mathcal{J}}$ in ${\mathbb{C}}^{d}\times{\mathbb{C}}$. Denote by ${[\overline{V}]^{\delta}}$ the closed $\delta$-neighborhood of $\overline{V}$ in ${\mathbb{C}}^{d}$ and let $\widetilde{f}_{i,\tau}=\widetilde{f}_{i}(\cdot,\tau)$. Choose $\ell\in{\mathbb{N}}$ so that $C\gamma^{\ell}<1/2$. Then $\|f^{\prime}_{{\mathbf{i}},t}(x)\|<1/2$ for ${\mathbf{i}}\in\Lambda^{\ell}$, and $x\in\overline{V}$. By continuity, there exists $\delta>0$ such that $\widetilde{f}_{{\mathbf{i}},t}$, with ${\mathbf{i}}\in\Lambda^{\ell}$, is holomorphic on $[\overline{V}]^{\delta}$ and

[TABLE]

Here in the left-hand side is the norm of the differential with respect to $z\in{\mathbb{C}}^{d}$. Thus, each $\widetilde{f}_{{\mathbf{i}},t}$, with ${\mathbf{i}}\in\Lambda^{\ell}$, is a strict contraction on $[\overline{V}]^{\delta}$, and since $\widetilde{f}_{{\mathbf{i}},t}({\overline{V}})=f_{{\mathbf{i}},t}(\overline{V})\subset V$, we obtain that $[\overline{V}]^{\delta}$ is mapped into its interior by $\widetilde{f}_{{\mathbf{i}},t}$, for $t\in{\mathcal{J}}$. Then the same must be true for all $\tau$ in a sufficiently small complex neighborhood of ${\mathcal{J}}$, which we denote by ${\mathcal{O}}$. We can find a constant $L>1$ such that

[TABLE]

since there are finitely many holomorphic functions involved.

Now, it follows from (2.6) and (2.7) that the function $\widetilde{f}_{{\mathbf{j}},t}$, for all   ${\mathbf{j}}\in\bigcup_{n=1}^{\infty}\Lambda^{n}$ and $t\in{\mathcal{J}}$, is well-defined and holomorphic in the interior of ${\mathcal{W}}:=[\overline{V}]^{\delta/L}$, and moreover, it maps ${\mathcal{W}}$ into $[\overline{V}]^{\delta}$. In addition, $\widetilde{{\mathcal{F}}}^{\ell}_{\tau}=\{\widetilde{f}_{{\mathbf{i}},\tau}\}_{{\mathbf{i}}\in\Lambda^{\ell}}$ is a strictly contracting IFS on ${\mathcal{W}}$, depending on $\tau\in{\mathcal{O}}$ holomorphically. It follows that the finite iterates $\tau\mapsto\widetilde{f}_{{\mathbf{i}}|_{n},\tau}(x_{0})$ converge to $\Pi_{\tau}({\mathbf{i}})$, the natural projection for $\widetilde{{\mathcal{F}}}_{\tau}$, as $n\to\infty$, uniformly for $\tau\in{\mathcal{O}}$. The uniform limit of holomorphic functions in an open set in ${\mathbb{C}}$ is holomorphic, and since $F_{\mathbf{i}}(t)=\Pi_{t}({\mathbf{i}})$ 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 ${\mathbf{i}},{\mathbf{j}}\in\bigcup_{n=1}^{\infty}\Lambda^{n}$, let

[TABLE]

Recall that this depends on $x_{0}$: $\Delta_{{\mathbf{i}},{\mathbf{j}}}(t)=f_{{\mathbf{i}},t}(x_{0})-f_{{\mathbf{j}},t}(x_{0})$. For any ${\varepsilon}>0$, let

[TABLE]

and define the exceptional set $E=E(x_{0})$ by

[TABLE]

where $B_{{\varepsilon}^{n}}=\{x\in{\mathbb{R}}^{d}:\ \|x\|\leqslant{\varepsilon}^{n}\}$ is the Euclidean ball in ${\mathbb{R}}^{d}$.

Lemma 2.7**.**

If $t\notin E=E(x_{0})$, then $\mathcal{F}_{t}$ satisfies the SESDC condition on $\{x_{0}\}\subset\overline{V}$.

Proof.

Observe that $t\notin E$ implies $t\notin E_{\varepsilon}$ for some ${\varepsilon}>0$, hence $|f_{{\mathbf{i}},t}(x_{0})-f_{{\mathbf{j}},t}(x_{0})|\geqslant{\varepsilon}^{n}$, for all ${\mathbf{i}},{\mathbf{j}}\in\Lambda^{n}$ with $i_{1}\neq j_{1}$, for all $n$ sufficiently large. ∎

Remark 2.8**.**

In [16, 17] Hochman considered the case where $\mathcal{F}_{t}$ is an affine IFS. He defined the sets $E^{\prime}_{{\varepsilon}}$ and $E^{\prime}$ as follows:

[TABLE]

and

[TABLE]

If $t\notin E^{\prime}$ then $\mathcal{F}_{t}$ satisfies the strong exponential separation condition on $\{x_{0}\}$ along a subsequence. **

Denote by $[w]_{q}$ the $q$-th component of a vector $w\in{\mathbb{R}}^{d}$, $1\leqslant q\leqslant d$.

Definition 2.9**.**

We say that the family of IFS ${\mathcal{F}}_{t},\ t\in{\mathcal{J}}$, as above, is non-degenerate in $q$-th component for some $1\leqslant q\leqslant d$ if

[TABLE]

In [16] an IFS family on ${\mathbb{R}}$ is called non-degenerate if $\Delta_{{\mathbf{i}},{\mathbf{j}}}\equiv 0\iff{\mathbf{i}}={\mathbf{j}}$. This condition is equivalent to (2.10) for $d=1$ when the maps of the IFS are injective.

We next prove the following:

Theorem 2.10**.**

Suppose that the family of IFS ${\mathcal{F}}_{t},\ t\in{\mathcal{J}}$, is non-degenerate in $q$-th component for some $1\leqslant q\leqslant d$. Then the set $E$ from (2.8) has Hausdorff dimension zero, and therefore, ${\mathcal{F}}_{t}$ satisfies the SESDC condition on $\{x_{0}\}$ for all parameters $t\in{\mathcal{J}}$ outside of an exceptional set of Hausdorff dimension zero.

Hochman [16, 17] proved, for a non-degenerate family of affine IFS, with a real-analytic dependence on parameter, that the set $E^{\prime}$ from (2.9) has zero packing dimension.

Corollary 2.11**.**

For a family of IFS ${\mathcal{F}}_{t},\ t\in{\mathcal{J}}$, as above, assume that there exist $t_{0}\in\mathcal{J}$ and $q$, $1\leqslant q\leqslant d,$ such that the sets $\pi_{q}\left(f_{i,t_{0}}(\overline{V})\right)$ are pairwise disjoint for $i\in\Lambda$, where $\pi_{q}:{\mathbb{R}}^{d}\to{\mathbb{R}}$ is the orthogonal projection to the $q$-th coordinate. Then (2.10) holds, and hence the set $E$ from (2.8) has Hausdorff dimension zero.

To deduce the corollary from Theorem 2.10, it suffices to note that $[F_{\mathbf{i}}(t_{0})]_{q}\in\pi_{q}\left(f_{i_{1},t_{0}}(\overline{V})\right)$ whence $\Delta_{{\mathbf{i}},{\mathbf{j}}}(t_{0})\neq 0$ whenever $i_{1}\neq j_{1}$.

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

Definition 2.12**.**

The family $\{\mathcal{F}_{t}\}_{t\in\mathcal{J}}$ is said to be transverse of order $k$ in $q$-th component for some $q,\ 1\leqslant q\leqslant d$, if there exists $c>0$ such that for all $n\in\mathbb{N}$ and ${\mathbf{i}},{\mathbf{j}}\in\Lambda^{n}$, with $i_{1}\neq j_{1}$, we have

[TABLE]

Lemma 2.13**.**

Suppose that the non-degeneracy in $q$-th component (2.10) holds. Then $\{\mathcal{F}_{t}\}_{t\in\mathcal{J}}$ is transverse of order $k$ in $q$-th component for some $k\in\mathbb{N}$.

Proof.

Suppose that for all $k\in{\mathbb{N}}$ the family $\{\mathcal{F}_{t}\}_{t\in\mathcal{J}}$ is not transverse of order $k$ in $q$-th component . Then by assumption, for $\{c_{k}\}$ with $c_{k}<1/k$, we can choose $n(k)$, ${\mathbf{i}}^{(k)},{\mathbf{j}}^{(k)}\in\Lambda^{n(k)}$, with $i^{(k)}_{1}\neq j^{(k)}_{1}$ and a point $t_{k}\in\mathcal{J}$ such that

[TABLE]

for $0\leqslant p\leqslant k$. Since $\Lambda$ is finite, passing to a subsequence $\{k_{l}\}$, we can assume that $t_{k_{l}}\to t_{0}\in{\mathcal{J}}$, ${\mathbf{i}}^{(k_{l})}\to{\mathbf{i}}\in\Lambda^{\mathbb{N}}$ and ${\mathbf{j}}^{(k_{l})}\to{\mathbf{j}}\in\Lambda^{\mathbb{N}}$, with $i_{1}\neq j_{1}$. By Lemma 2.6, the complex extension of $\Delta_{{\mathbf{i}}^{(k_{l})},{\mathbf{j}}^{(k_{l})}}$ converges to the complex extension of $\Delta_{{\mathbf{i}},{\mathbf{j}}}$ uniformly on a complex neighborhood of ${\mathcal{J}}$, and hence the same holds for $p$-th derivatives. Thus for all $p\geqslant 0$, we have

[TABLE]

Since $[\Delta_{{\mathbf{i}},{\mathbf{j}}}]_{q}$ is real-analytic, the vanishing of its derivatives implies $[\Delta_{{\mathbf{i}},{\mathbf{j}}}]_{q}\equiv 0$ on ${\mathcal{J}}$, contradicting (2.10), since $i_{1}\neq j_{1}$ by construction. ∎

For a $C^{k}$-smooth function $F:J\to\mathbb{C}$, where $J\subset{\mathbb{R}}$ is a compact interval, write

[TABLE]

and similarly for vector-functions.

Lemma 2.14** (Lemma 5.8 in [16]).**

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

Lemma 2.15**.**

If the family of IFS $\{\mathcal{F}_{t}\}_{t\in\mathcal{J}}$ is transverse of order $k\geqslant 1$ in $q$-th component, for some $q$, $1\leqslant q\leqslant d$, on the compact interval $\mathcal{J}$, then the set $E$ from (2.8) has Hausdorff dimension zero.

Proof.

Extending the real-analytic functions to the complex plane, as in Lemma 2.6, since

[TABLE]

on a neighborhood ${\mathcal{O}}$ of ${\mathcal{J}}$, and $\Delta_{{\mathbf{i}},{\mathbf{j}}}(\cdot)$ is holomorphic on ${\mathcal{O}}$ for all ${\mathbf{i}},{\mathbf{j}}\in\Lambda^{n}$, we have

[TABLE]

Let

[TABLE]

Then

[TABLE]

Let ${\mathbf{i}},{\mathbf{j}}\in\Lambda^{n}$, with $i_{1}\neq j_{1}$. Since the family is transverse in $q$-th component, we can apply Lemma 2.14 to $[\Delta_{{\mathbf{i}},{\mathbf{j}}}]_{q}$, to obtain that for ${\varepsilon}$ sufficiently small, the set

[TABLE]

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 (2.12) may be covered by $O(|\Lambda|^{2n}\cdot C^{k})$ intervals of length $\leqslant 2({\varepsilon}^{n}\cdot c^{-1})^{1/2^{k}}$. Fix $s>0$ and choose ${\varepsilon}>0$ such that $|\Lambda|^{2}{\varepsilon}^{s/2^{k}}<1$. Writing ${\mathcal{H}}^{s}$ for the $s$-dimensional Hausdorff measure, we obtain from (2.13) that

[TABLE]

It follows that ${\mathcal{H}}^{s}(E_{\varepsilon})<\infty$ whence $\dim_{H}(E)\leqslant\dim_{H}(E_{\varepsilon})\leqslant s$, and since $s>0$ was arbitrary we obtain $\dim_{H}(E)=0$. ∎

Proof of Theorem 2.10.

This is now immediate from Lemmas 2.13 and 2.15. ∎

2.4. Proof of Theorem 1.2

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

Lemma 2.16**.**

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 $\{x_{0}+tv:t\in\mathbb{R}\}\cap F$ has 1-dimensional Lebesgue measure [math]. Then the set $F$ has $n$-dimensional Lebesgue measure [math].

Proof of Theorem 1.2.

(i) Let $\Sigma=\Sigma_{v_{1},\ldots,v_{d+1}}$ be a simplicial cone in ${\mathbb{R}}^{d+1}$. By a coordinate change, we can assume without loss of generality, that $\Sigma\smallsetminus\{0\}$ is contained in the subspace $x_{d+1}>0$. Let ${\mathcal{U}}\subset{\mathcal{X}}_{\Sigma,m}$ be a small open set in $(GL_{d+1}({\mathbb{R}}))^{m}$ of $m$-tuples of matrices for which $\Sigma$ is strictly invariant. Choose vectors $w_{i}\in{\mathbb{R}}^{d+1}\ (i\in\Lambda)$, such that $[w_{i}]_{d+1}=1$ for all $i$ and for some fixed $q$, $1\leqslant q\leqslant d$, the components $[w_{i}]_{q}$, for $i\in\Lambda$, are all distinct, in such a way that

[TABLE]

This is possible when ${\mathcal{U}}$ is sufficiently small. (In fact, there is no difficulty in ensuring that all components of $w_{i}$ are distinct since the cones have nonempty interior.) Let $(A_{i})_{i\in\Lambda}\in{\mathcal{U}}$, and for each $t\geqslant 0$ and $i\in\Lambda$ let $A_{i,t}$ be such that

[TABLE]

Condition (2.14) guarantees that $\{A_{i,t}v_{j}\}_{j=1}^{d+1}$ is linearly independent, and hence $A_{i,t}\in GL_{d+1}({\mathbb{R}})$ for all $t\geqslant 0$. 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.

Proof of the Claim..

We have

[TABLE]

hence $c_{j}+a_{j}\sum_{k=1}^{d+1}c_{k}=0$ for all $j$. If $\sum_{k=1}^{d+1}c_{k}\neq 0$, 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 ${\mathcal{A}}_{t}=\{A_{i,t}\}_{i\in\Lambda}$ be the family of matrices defined above, for $t\geqslant 0$, and let ${\mathcal{F}}_{t}={\mathcal{F}}_{{\mathcal{A}}_{t}}$ be the corresponding one-parameter family of IFS on the set $\overline{V}\subset{\mathbb{R}}^{d}$ obtained by projection of $\Sigma\cap\{x\in{\mathbb{R}}^{d+1}:x_{d+1}=1\}$ onto ${\mathbb{R}}^{d}$. Notice that the cone $\Sigma$ is strictly preserved by all ${\mathcal{A}}_{t},\ t\geqslant 0$, by construction, hence by Lemma 2.1, these IFS are all uniformly hyperbolic. Both the IFS and their dependence on $t$ are real-analytic, since the IFS are given by rational functions. Condition (2.4) 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

[TABLE]

By construction, $[w_{i}]_{q}$ are all distinct, hence Corollary 2.11 applies for ${\varepsilon}>0$ sufficiently small. We obtain that for all $t\in[0,\infty)$ outside a set of Hausdorff dimension zero, the IFS ${\mathcal{F}}_{t}$ satisfies the SESDC condition on a non-empty set, and then Proposition 2.4 implies that the $m$-tuple of matrices $(A_{i,t})_{i\in\Lambda}$ is Diophantine for all $t$ outside of a zero-dimensional set, so certainly for Lebesgue-a.e. $t$. Now Lemma 2.16 yields the desired claim.

(ii) We consider $(SL_{d+1}({\mathbb{R}}))^{m}$ as a codimension-$m$ submanifold of $(GL_{d+1}({\mathbb{R}}))^{m}\subset{\mathbb{R}}^{(d+1)^{2}m}$. In the proof of part (i) we showed that for a.e. $(A_{i})_{i\in\Lambda}\in\mathcal{X}_{\Sigma,m}$, the induced IFS on a subset of ${\mathbb{R}}^{d}$ satisfies the SESDC condition on a non-empty set. Suppose that there is a positive measure subset ${\mathcal{E}}\subset{\mathcal{Y}}_{\Sigma,m}$ for which the strong Diophantine condition is violated. Then for every $(A_{i})_{i\in\Lambda}\in{\mathcal{E}}$, the induced IFS $\Phi$ does not satisfy the SESDC on a non-empty set by Proposition 2.4. However, $(A_{i})_{i\in\Lambda}\in{\mathcal{Y}}_{\Sigma,m}$ and $(c_{i}A_{i})_{i\in\Lambda}\in{\mathcal{X}}_{\Sigma,m}$, for any $c_{i}>0$, induce the same IFS on the projective space, and we get a set of positive measure in ${\mathcal{X}}_{\Sigma,m}$ for which the SESDC condition on a non-empty set does not hold. This is a contradiction, and the theorem is proved completely. ∎

3. Dimension of the attractor

Let $A\in SL_{2}(\mathbb{R})$ be a hyperbolic matrix. Then $A^{*}A$ has distinct 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 $\varphi_{A}$, the induced action of $A$ on ${\mathbb{RP}}^{1}\cong[0,\pi)$. For more details see sections 2.2, 2.3 and 2.4 in [18]. Below we use the Euclidean metric on $[0,\pi)$ and denote by $|F|$ the Lebesgue measure of a measurable $F\subset[0,\pi)$. The following simple lemma is [18, Section 2.4].

Lemma 3.1**.**

Let $A\in SL_{2}(\mathbb{R})$. Then $\|A\|^{-2}\leqslant|\varphi_{A}^{\prime}(x)|\leqslant\|A\|^{2}$ for all $x\in[0,\pi)$. Furthermore, for any ${\varepsilon}>0$ there exists $C_{\varepsilon}>1$ such that $|\varphi_{A}^{\prime}(x)|\leqslant C_{\varepsilon}\|A\|^{-2}$ for all $x\in[0,\pi)\smallsetminus(t_{A}-{\varepsilon},t_{A}+{\varepsilon})$.

The following lemma is now immediate.

Lemma 3.2**.**

Let $U\subset[0,\pi)$ be an open set, with $|U|<\pi$. 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

[TABLE]

Let $\mathcal{A}=\{A_{i}\}_{i\in\Lambda}$ be a finite collection of $SL_{2}(\mathbb{R})$ matrices and let $\Phi=\{\varphi_{A}\}_{A\in{\mathcal{A}}}$ be the corresponding IFS on $[0,\pi)\cong{\mathbb{RP}}^{1}$. We continue to use the notation:

[TABLE]

Recall that a strictly invariant multicone $U\subset[0,\pi)$ is a nonempty open set having finitely many connected components with disjoint closures, such that $\overline{U}\neq{\mathbb{RP}}^{1}$ and $\Phi(\overline{U})\subset U$. By Theorem 1.6, the set ${\mathcal{A}}$ is hyperbolic, which means that there exist $c>0$ and $\lambda>1$ such that

[TABLE]

Lemma 3.3**.**

Let $U$ be a strictly invariant multicone for the IFS $\Phi$. Then there exists a constant $C_{1}>1$ such that

(i)* we have*

[TABLE]

(ii)* the Bounded Distortion Property holds for $\Phi$ on $U$:*

[TABLE]

(iii)* we have*

[TABLE]

where $C_{2}=C_{1}/c^{2}$, with $c>0$ and $\lambda>1$ from (3.1), and $r_{1}=\left(\max_{A\in{\mathcal{A}}}\|A\|\right)^{-2}$;

(iv)* we have*

[TABLE]

where $s$ is the unique solution of Bowen’s equation $P_{\Phi}(s)=0$, with the pressure given by (1.7) and $s_{\mathcal{A}}$ is the critical exponent, given by (1.5).

Proof.

(i) In view of Lemma 3.1, we only need to check the right inequality in (3.2). By Lemma 3.2 and strict invariance of $U$, we have for every $A_{\mathbf{i}}$, with ${\mathbf{i}}\in\Lambda^{n}$ and $n$ sufficiently large, that $t_{A_{\mathbf{i}}}\not\in U$. Since $\Phi(\overline{U})\subset U$, there exists ${\varepsilon}>0$ such that

[TABLE]

For every ${\mathbf{i}}=i_{1}\ldots i_{n}$, with $n\geqslant n_{0}$, and every $x\in\overline{U}$,

[TABLE]

This confirms (3.2) for $n\geqslant n_{0}$, with $C_{1}:=C_{\varepsilon}(\max_{A\in{\mathcal{A}}}\|A\|)^{4}$, and for the first finitely many $n$ it trivially holds with some constant.

(ii) is immediate from (i).

(iii) the left inequality follows from Lemma 3.1, and the right is a consequence of (i) and (3.1);

(iv) easily follows from (1.7) and (1.5). ∎

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

[TABLE]

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 (3.2) and (1.13) imply

[TABLE]

Consider the symbolic space $\Lambda^{\mathbb{N}}$ with the measure $\mu=p^{\mathbb{N}}$, and the shift transformation $\sigma:{\mathbf{i}}=i_{1}i_{2}i_{3}\ldots\mapsto i_{2}i_{3}\ldots$, which is $\mu$-invariant and ergodic. The natural projection $\Pi:\Lambda^{\mathbb{N}}\to{\mathbb{RP}}^{1}$ is defined by

[TABLE]

where the limit exists and is independent of $x_{0}$ by contraction properties of the IFS. Observe that

[TABLE]

Consider the function $h:\Lambda^{\mathbb{N}}\to{\mathbb{R}}$ defined by

[TABLE]

Since $|\varphi_{A}^{\prime}|$ is bounded away from [math] and $\infty$ on $\overline{U}$, we have that $h$ is integrable with respect to $\mu$. The following lemma is standard, but we include the proof for completeness.

Lemma 3.4**.**

We have

[TABLE]

Proof.

In view of the bounded distortion (3.3), for $\mu$-a.e. ${\mathbf{i}}$,

[TABLE]

using the chain rule and (3.7) in the last step. The proof is finished by an application of the Birkhoff Ergodic Theorem. ∎

In order to apply Theorem 1.11 we need the assumption of total irreducibility. It is achieved with the help of the following lemma.

Lemma 3.5**.**

Let ${\mathcal{A}}$ be a finite set of $SL_{2}({\mathbb{R}})$ matrices having a strictly invariant multicone $U$, and let $\Phi_{\mathcal{A}}$ be the associated IFS on $U$. If the attractor of $\Phi_{\mathcal{A}}$ is not a singleton (equivalently, not all attracting fixed points of $\varphi_{A}$, $A\in{\mathcal{A}}$, coincide), then either

(a)* the group generated by ${\mathcal{A}}$ is totally irreducible, or*

(b)* ${\mathcal{A}}$ is conjugate in $SL_{2}({\mathbb{R}})$ to a finite subset of upper triangular matrices with the upper-left entry of absolute value less than one. In the latter case the IFS $\Phi_{\mathcal{A}}$ is conjugate, via a linear-fractional transformation, to a linear contracting IFS (an IFS of contracting affine linear maps) on ${\mathbb{R}}$.*

Proof.

By assumption, the attractor of $\Phi_{\mathcal{A}}$ is not a singleton, hence it contains at least two distinct fixed points of some $\varphi_{A_{1}}$ and $\varphi_{A_{2}}$, and then it is perfect. Suppose that the group generated by ${\mathcal{A}}$ is not totally irreducible. Then there is a finite invariant subset $F$ for $\Phi_{\mathcal{A}}$. If $F$ intersects $\overline{U}$, we get a contradiction, since the forward orbit of any point in $\overline{U}$ under the semigroup generated by $\Phi_{\mathcal{A}}$ is dense in the attractor. Thus $F$ is contained in the complement $\overline{U}^{c}$. However, notice that the IFS $\Phi_{{\mathcal{A}}^{-1}}$, generated by the inverses of the matrices from ${\mathcal{A}}$, is contracting on $U^{c}$ (this is because $f(\overline{U})\subset U$ implies $f^{-1}(U^{c})\subset\overline{U}^{c}$). By the previous argument, since $F$ is finite, it must be a singleton. Hence all the attracting points of $\varphi_{A^{-1}}$, that is, all the repelling points of $\varphi_{A},\ A\in{\mathcal{A}}$, coincide. However, the latter implies that we can conjugate the set of matrices in such a way that the repelling point is $0\in[0,\pi)\approx{\mathbb{RP}}^{1}$. The resulting matrices are upper-triangular, with the upper-left entry of absolute value less than one. The corresponding IFS on ${\mathbb{R}}^{*}$ has the repelling point at $\infty$, which means that we get a contracting linear IFS on ${\mathbb{R}}$. ∎

Proof of Theorem 1.7.

Recall that ${\mathcal{A}}$ is a finite set of $SL_{2}({\mathbb{R}})$ matrices satisfying the strong Diophantine condition and having a strictly invariant multicone $U$, and $\Phi=\Phi_{\mathcal{A}}$ is the associated IFS on $U$. Then $\Phi$ has a compact attractor $K$ (not a singleton by assumption), and our goal is to show that $\dim_{H}(K)=\min\{1,s\}$, where $s$ is the unique solution of Bowen’s equation (1.7). We have two cases to consider. Note that both the assumption and the conclusion of the theorem are invariant under conjugacy. If we are in the case (b) of Lemma 3.5, then applying Hochman’s Theorem [16, Cor 1.2] on the dimension of self-similar sets in ${\mathbb{R}}$, yields the result. One only needs to note that the strong Diophantine condition on the set of $SL_{2}({\mathbb{R}})$ matrices described in case (b) implies the strong exponential separation condition on a finite subset of ${\mathbb{R}}$. We postpone the proof of this fact to Lemma 4.1 in the Appendix. In the case (a) of Lemma 3.5, the group generated by ${\mathcal{A}}$ is totally irreducible, and we will assume this for the rest of the proof.

It is known that

[TABLE]

see the appendix for a short proof. Let us show the opposite inequality.

Let $d_{n}>0$ be the solution of the equation

[TABLE]

where $U_{\mathbf{i}}=\varphi_{\mathbf{i}}(U)$. It is not hard to see that

[TABLE]

For convenience of the reader, we include the proof in the appendix, following [25].

Let $p^{(n)}=(p^{(n)}_{\mathbf{i}})_{{\mathbf{i}}\in\Lambda^{n}}$ be the probability vector such that $p^{(n)}_{\mathbf{i}}=|U_{\mathbf{i}}|^{d_{n}}$. Denote

[TABLE]

Let $\eta^{(n)}$ be the invariant probability measure for the IFS $\Phi^{n}$ on $\overline{U}$, corresponding to $p^{(n)}$. Since $\eta^{(n)}$ is supported on $K$, we have $\dim\eta^{(n)}\leqslant\dim_{H}(K)$.

Note that ${\mathcal{A}}$ satisfies the assumptions of Theorem 1.11. Indeed, the existence of a strictly invariant multicone implies that all the matrices in ${\mathcal{A}}$ are hyperbolic, hence the group generated by ${\mathcal{A}}$ is unbounded, and we are now under the total irreducibility assumption. Thus the Furstenberg measure for $({\mathcal{A}}^{n},p^{(n)})$ is unique, and it coincides with $\eta^{(n)}$. Since ${\mathcal{A}}$ is strongly Diophantine, we have that ${\mathcal{A}}^{n}$ is strongly Diophantine as well. Now, by Theorem 1.11 and (3.6) we have

[TABLE]

We claim that there exists $C>0$ such that

[TABLE]

Indeed, by Lemma 3.4,

[TABLE]

where $[{\mathbf{i}}]$ is the cylinder set of sequences starting with ${\mathbf{i}}$. Here $\mu^{(n)}$ is the Bernoulli measure on $(\Lambda^{n})^{\mathbb{N}}$, with $\mu^{(n)}([{\mathbf{i}}])=|U_{\mathbf{i}}|^{d_{n}}$ for ${\mathbf{i}}\in\Lambda^{n}$. We have

[TABLE]

by the Bounded Distortion Property (3.3). Therefore,

[TABLE]

confirming (3.10). Now we can estimate

[TABLE]

Since $\lim_{n\to\infty}d_{n}=s$ and

[TABLE]

by (3.11) and Lemma 3.3(iii), we obtain $\min\{1,s\}\leqslant\dim_{H}(K)$, as desired. Finally, $s=s_{\mathcal{A}}/2$ by Lemma 3.3(iv). ∎

4. Appendix: miscellaneous proofs

4.1. Proof of (3.9) [25]

We have a projective IFS $\Phi=\{\varphi_{i}\}_{i\in\Lambda}$ on a strictly invariant multicone $U$. Observe that

[TABLE]

by the Bounded Distortion Property (3.3). Let

[TABLE]

Since $P_{\Phi}(s)=0$, we have $\lim_{n\to\infty}Q_{n}=0$. Recall (3.4), which says that

[TABLE]

for $r_{1}\in(0,1),C^{\prime}_{2}>0$, and $\lambda>1$. Then $r^{n}_{1}|U|\leqslant|U_{\mathbf{i}}|\leqslant C_{2}\lambda^{-2n}|U|$ for ${\mathbf{i}}\in\Lambda^{n}$, and hence we have

[TABLE]

where we use the convention that $[a,b]=[b,a]$ if $a>b$. In view of $\sum_{{\mathbf{i}}\in\Lambda^{n}}|U_{\mathbf{i}}|^{d_{n}}=1$, we have

[TABLE]

whence for $n$ sufficiently large,

[TABLE]

which implies $d_{n}\to s$, as desired. ∎

4.2. Proof of (3.8)

Fix ${\varepsilon}>0$. Then for sufficiently large $n$ we have $d_{n}<s+{\varepsilon}/2$, and

[TABLE]

Therefore, the $(s+{\varepsilon})$-dimensional Hausdorff measure of $K$ is zero. By the definition of the Hausdorff dimension, this proves (3.8). ∎

4.3. From the Diophantine property to exponential separation

Lemma 4.1**.**

Suppose that ${\mathcal{A}}$ is a finite strongly Diophantine set of upper-triangular $SL_{2}({\mathbb{R}})$ matrices with the upper-left entry of absolute value less than one. Then the associated IFS satisfies the strong exponential separation condition on the set $\{0,1\}$.

Instead of $\{0,1\}$ one can take any two distinct points in ${\mathbb{R}}$. We start with the following general elementary claim.

Claim. Let $G$ be a matrix group, and ${\mathcal{A}}=\{A_{1},\ldots,A_{m}\}\subset G$ is a finite strongly Diophantine set. Suppose that $\widetilde{{\mathcal{A}}}=\{\widetilde{A}_{1},\ldots,\widetilde{A}_{m}\}$ is such that $\widetilde{A}_{j}=\pm A_{j}$ for all $j\leqslant m$. Then $\widetilde{\mathcal{A}}$ is strongly Diophantine as well.

Proof of the claim.

We argue by contradiction and assume that for any ${\varepsilon}>0$ there exist $n\in{\mathbb{N}}$ and ${\mathbf{i}}\neq{\mathbf{j}}$ in $\Lambda^{n}$ such that

[TABLE]

For ${\varepsilon}>0$ sufficiently small, $\|A_{\mathbf{i}}-A_{\mathbf{j}}\|\leqslant{\varepsilon}^{n}$ is impossible, thus $\|A_{\mathbf{i}}+A_{\mathbf{j}}\|\leqslant{\varepsilon}^{n}$. But then we have

[TABLE]

where $C=\max_{A\in{\mathcal{A}}}\|A\|$. This contradicts the strong Diophantine property of ${\mathcal{A}}$ for ${\varepsilon}>0$ sufficiently small. ∎

Proof of the lemma.

Observe that $A$ and $-A$ act identically on ${\mathbb{RP}}^{1}$. In view of the Claim, we can assume that the matrices of ${\mathcal{A}}$ have the form

[TABLE]

All matrices of the semigroup generated by ${\mathcal{A}}$ have a similar upper-triangular form:

[TABLE]

The matrices of ${\mathcal{A}}$ act on ${\mathbb{RP}}^{1}\approx{\mathbb{R}}^{*}$ by $A_{j}\cdot x=\lambda_{j}x+a_{j}$. Assume that $\Phi_{\mathcal{A}}$ does not satisfy the strong exponential separation condition on $\{0,1\}$. Then for any ${\varepsilon}>0$ there exist $n\in{\mathbb{N}}$ and ${\mathbf{i}}\neq{\mathbf{j}}$ in $\Lambda^{n}$ such that $|a_{\mathbf{i}}-a_{\mathbf{j}}|\leqslant{\varepsilon}^{n}$ and

[TABLE]

It follows that

[TABLE]

In a similar elementary way one can estimate from above the expressions $|1/\sqrt{\lambda_{\mathbf{i}}}-1/\sqrt{\lambda_{\mathbf{j}}}|$ and $|a_{\mathbf{i}}\sqrt{\lambda_{\mathbf{i}}}-a_{\mathbf{j}}/\sqrt{\lambda_{\mathbf{j}}}|$ to conclude

[TABLE]

for some $C^{\prime}>0$, depending only on ${\mathcal{A}}=\{A_{j}\}_{j\in\Lambda}$. This contradicts the strong Diophantine property of ${\mathcal{A}}$. ∎

Acknowledgements

Thanks to Balazs Barańy for helpful discussions and for telling us about the papers [3, 6]. We are deeply grateful to the referees for many helpful comments and suggestions, as well as pointing out several gaps in the original proofs.