1. Introduction
Let $\mathcal {F}=\{f_i(\underline {x})=A_i\underline {x}+\underline {t}_i\}_{i\in I}$
be a finite collection of affine self-maps of $\mathbb {R}^d$
, where $A_i$
is a $d\times d$
matrix and $\underline {t}_i\in \mathbb {R}^d$
, called iterated function system (IFS). Let us assume throughout the paper that $\|A_i\|<1$
for every $i\in I$
, where $\|\cdot \|$
denotes the usual matrix norm induced by the Euclidean-norm on $\mathbb {R}^d$
. The well-known theorem of Hutchinson [Reference Hutchinson12] states that there exists a unique non-empty compact set $\Lambda$
such that it is invariant with respect to the IFS $\mathcal {F}$
. That is,
We call the set $\Lambda$
self-affine set or the attractor of the IFS $\mathcal {F}$
. In the special case, when the affine mappings are similarity transformations, we call the set $\Lambda$
self-similar.
The dimension theory of self-affine sets has been widely studied in the past decades. Throughout the paper, we will denote the Hausdorff dimension of a set $A\subset \mathbb {R}^d$
by $\dim _H(A)$
and the box-counting and the upper box-counting dimension by $\dim _B(A)$
and $\overline {\dim }_B(A)$
, respectively. For the definition and basic properties, we direct the reader to Falconer [Reference Falconer6].
In the case, when the affine mappings are similarities and the IFS satisfies the strong separation condition, i.e.
Hutchinson [Reference Hutchinson12] showed that the Hausdorff and box dimension of $\Lambda$
equals to the similarity dimension. Determining the dimension becomes significantly harder when overlaps occur between the cylinder sets $f_i(\Lambda )$
, $i\in I$
. This problem was studied in several papers, like Bárány [Reference Bárány1], Hochman [Reference Hochman9, Reference Hochman10], Simon and Solomyak [Reference Simon and Solomyak16], Solomyak [Reference Solomyak17], etc. for typical systems in some proper sense. In general, the similarity dimension serves always as an upper bound of the box dimension.
Another challenging problem is, when there is a strict affinity between the maps of the IFS $\mathcal {F}$
, that is, there exists a matrix with at least two different eigenvalues in modulus. Falconer [Reference Falconer5] generalized the similarity dimension for that case, called the affinity dimension, and showed that if all the maps are invertible then it is an upper bound of the box dimension of the attractor. Furthermore, Falconer [Reference Falconer5] showed, which was later extended by Solomyak [Reference Solomyak17], that if $\|A_i\|<1/2$
then for Lebesgue typical translation parameters the affinity dimension equals to the Hausdorff dimension.
The dimension theory of self-affine sets has been widely studied in the recent years, see Bárány, Hochman and Rapaport [Reference Bárány, Hochman and Rapaport2], Bárány, Käenmäki and Koivusalo [Reference Bárány, Käenmäki and Koivusalo3], Falconer and Kempton [Reference Falconer and Kempton4], Hochman and Rapaport [Reference Hochman and Rapaport11], Rapaport [Reference Rapaport15]. All of these papers were considering systems, where the affine mappings are invertible. The first steps in the direction of considering non-invertible mappings has been made recently by Käenmäki and Nissinen [Reference Käenmäki and Nissinen13]. They studied the relation between the dimension of the attractor and the dimension of the self-affine set formed by the invertible mappings of the IFS for typical and for separated systems.
This paper is devoted to generalize the result of Bárány, Hochman and Rapaport [Reference Bárány, Hochman and Rapaport2] for planar self-affine sets which defining IFS contains non-invertible mappings. We will show that for typical choice of parameters the Hausdorff dimension equals to the affinity dimension, but there is a relatively large set of exceptions. Before we state our main theorem in details, we need to introduce some notations and definitions.
Let us denote by $\mathbb {S}^{d-1}$
the unit sphere on $\mathbb {R}^d$
, and let us denote by
the $d$
-dimensional torus. Every planar, contracting matrix $A$
with $\mathrm {rank}(A)=1$
can be represented as $A=\rho \underline {v}\underline {w}^T$
, where $\rho \in (0,\,1)$
and $\underline {v},\,\underline {w}\in \mathbb {S}^1$
. In particular, $\underline {v}$
is the unit vector generating the image space and $\underline {w}$
is the unit vector generating the kernel of the matrix $A$
. Note that this representation is not unique.
For a $2\times 2$
matrix $A$
, denote by $\alpha _1(A)\geq \alpha _2(A)\geq 0$
the singular values. For every $t\geq 0$
, let $\varphi ^t(A)$
be the singular value function defined as
Note that if $A$
has rank one then $\varphi ^t(A)=0$
for every $t>1$
.
Let $I$
and $J$
be finite sets of indices and for every $i\in I\cup J$
let $f_i(\underline {x})=A_i\underline {x}+t_i$
be an affine map such that $\|A_i\|<1$
for every $i\in I\cup J$
, $\mathrm {rank}(A_i)=2$
for every $i\in I$
and $\mathrm {rank}(A_i)=1$
for every $i\in J$
. Let us consider the following parametrized family of affine IFS
where $\mathsf {w}=(\underline {w}_j)_{j\in J}\in (\mathbb {S}^1)^{\#J}=\mathbb {T}^{\#J}$
is considered as the parameters while all the other quantities are fixed. We assume throughout the paper that $\#J\geq 1$
and $\#(I\cup J) \geq 2$
to avoid trivial cases. Let us denote the attractor of $\mathcal {F}_{\mathsf {w}}$
by $\Lambda _{\mathsf {w}}$
. We define the affinity dimension $s(\mathcal {F}_{\mathsf {w}})$
of the self-affine IFS as
The affinity dimension serves as a natural upper bound for the upper box-counting dimension, and in particular for the Hausdorff dimension, of the attractor in the non-invertible case too, see Käenmäki and Nissinen [Reference Käenmäki and Nissinen13].
Let us define the affinity dimension of the sub-system formed by the invertible mappings $\mathcal {F}_{\mathrm {reg}}=\{f_i\}_{i\in I}$
as
Let us observe that if $s(\mathcal {F}_{\mathsf {w}})>1$
then $s(\mathcal {F}_{\mathsf {w}})=s_{\mathrm {reg}}$
. Furthermore, if $s_{\mathrm {reg}}\leq 1$
then $s(\mathcal {F}_{\mathsf {w}})\leq 1$
as well. The following was shown by Käenmäki and Nissinen [Reference Käenmäki and Nissinen13, Theorem 1.1(1)]: suppose that $s_{\mathrm {reg}}\geq 1$
and suppose that the IFS $\mathcal {F}_{\mathrm {reg}}=\{f_i\}_{i\in I}$
satisfies the strong separation condition and the matrices $\{A_i\}_{i\in I}$
do not preserve any finite collection of the proper subspaces of $\mathbb {R}^2$
then $\dim _H(\Lambda _{\mathsf {w}})=\dim _B(\Lambda _{\mathsf {w}})=s(\mathcal {F}_{\mathsf {w}})=s_{\mathrm {reg}}$
for every $\mathsf {w}\in \mathbb {T}^{\#J}$
. Hence, in the remaining part of the paper we assume that $s_{\mathrm {reg}}<1$
.
We say that satisfies the convex separation condition uniformly, if there exists a convex compact set $U\subset \mathbb {R}^2$
such that
In the second part of the assumption, the image $f_i(U)$
depends on $\mathsf {w}\in \mathbb {T}^{\#J}$
if and only if $i\in J$
, and in this case only on the corresponding coordinate of $\mathsf {w}$
. In particular, if $i\in J$
then $f_i(U)$
is a line-segment which is parallel to $\mathrm {Im}(A_i)$
, and $\bigcup _{\mathsf {w}\in \mathbb {T}^{\#J}}f_i(U)$
is the smallest line segment parallel to $\mathrm {Im}(A_i)$
containing $f_i(U)$
for every $\underline {w}_i\in \mathbb {S}^1$
. Note that because of non-invertibility, the convex separation condition does not imply that the second and higher iterates do not contain overlaps. For an example of such system, see figure 1, which IFS consists of 3 invertible and 2 non-invertible mappings.
Figure 1. The first and second level cylinder sets of an IFS $\mathcal {F} = \{f_i(\underline {x})=A_i\underline {x}+\underline {t}_i\}_{i\in \{1,2,3\}^*}\bigcup \{f_j(\underline {x})=\rho _j\underline {v}_j\underline {w}_j^T\underline {x}+\underline {t}_j\}_{j\in \{4,5\}^*}$
satisfying the convex separation condition.
Furthermore, we say that the IFS $\mathcal {F}_{\mathrm {reg}}=\{f_i(\underline {x})=A_i\underline {x}+\underline {t}_i\}_{i\in I}$
is irreducible if there is no proper subspace $V$
of $\mathbb {R}^2$
such that $V$
is preserved by the all matrices $A_i$
for $i\in I$
.
Theorem 1.1 Let $\mathcal {F}_{\mathsf {w}}$
be a family of affine IFSs as in (1.1) with attractor $\Lambda _{\mathsf {w}}$
such that $\mathcal {F}_{\mathsf {w}}$
contains at least two maps and contains at least one non-invertible affine map. Suppose that $s_{\mathrm {reg}}<1$
, $\mathcal {F}_{\mathsf {w}}$
satisfies the convex separation condition uniformly for every $\mathsf {w}\in \mathbb {T}^{\#J}$
and $\mathcal {F}_{\mathrm {reg}}$
is irreducible. Then
(1) there exists a set $E_1\subseteq \mathbb {T}^{\#J}$
such that $\dim _HE_1\leq \#J-1$ and for every $\mathsf {w}\in \mathbb {T}^{\#J}\setminus E_1$,
\[ \dim_H(\Lambda_{\mathsf{w}})=\dim_B(\Lambda_{\mathsf{w}})=s(\mathcal{F}_{\mathsf{w}}). \]
(2) If $\sup _{\mathsf {w}}s(\mathcal {F}_{\mathsf {w}})<1$
then there exists a set $E_2\subseteq \mathbb {T}^{\#J}$ such that $\dim _HE_2=\#J-1$ and for every $\mathsf {w}\in E_2$
\[ \dim_H(\Lambda_{\mathsf{w}})\leq\overline{\dim}_B(\Lambda_{\mathsf{w}})< s(\mathcal{F}_{\mathsf{w}}). \]
Let us note that it is not known whether the box-counting dimension of self-affine sets exists and equals to the Hausdorff dimension in general. Furthermore, if the matrices $\{A_i\}_{i\in I}$
do not preserve any finite collection of the proper subspaces of $\mathbb {R}^2$
then by Bárány, Hochman and Rapaport [Reference Bárány, Hochman and Rapaport2] the affinity dimension $s_{\mathrm {reg}}$
of the subsystem formed by the invertible mappings serves as a lower bound for $\dim _H(\Lambda _{\mathsf {w}})$
for every $\mathsf {w}\in \mathbb {T}^{\#J}$
, however, it is not necessarily the case if $\{A_i\}_{i\in I}$
is only irreducible.
2. Study of the affinity dimension
Let us first introduce some notations used throughout the paper. For an index set $K$
, let $K^*=\bigcup _{n=0}^\infty K^n$
be the set of every finite words formed by the symbols in $K$
. For a finite word $\bar {\imath } \in K^*$
, denote $ {|{\bar {\imath }}|}$
the length of $\bar {\imath }$
. For $\bar {\imath } = i_1 \, \dots \, i_n\in K^*$
, we denote by $A_{\bar {\imath }}$
the finite product $A_{i_1} \cdot \, \dots \, \cdot A_{i_n}$
, and by $f_{\bar {\imath }}=f_{i_1}\circ \cdots \circ f_{i_n}$
the finite composition. We also use the convention that $\emptyset \in K^*$
with $|\emptyset |=0$
, moreover, $A_{\emptyset }$
and $f_{\emptyset }$
are the identity matrix and identity map of $\mathbb {R}^2$
, respectively.
Let $A$
be a $2\times 2$
matrix and let $V$
be a proper subspace of $\mathbb {R}^2$
. Then let us define the conditional norm of $A$
on $V$
by
For a $2\times 2$
matrix $A$
with $\mathrm {rank}(A)=1$
, denote by $\mathrm {Im}(A)$
and by $\mathrm {Ker}(A)$
the image space and the kernel of $A$
, respectively. Clearly, if $A=\rho \underline {v}\underline {w}^T$
then $\mathrm {Im}(A)=\langle \underline {v}\rangle$
and $\mathrm {Ker}(A)=\langle \underline {z}\rangle$
, where $\underline {w}$
is perpendicular to $\underline {z}$
and $\langle \underline {v}\rangle$
denotes the subspace generated by $\underline {v}$
. Let us also denote the usual Euclidean scalar product on $\mathbb {R}^2$
by $\langle \underline {v},\,\underline {w}\rangle =\underline {v}^T\underline {w}$
.
Clearly, for every $2\times 2$
matrices $A,\,B$
with $\mathrm {rank}(B)=1$
, and for every subspace $V$
of $\mathbb {R}^2$
, we get
Note that if $\mathrm {Ker}(B)=V$
then $\|B|V\|=0$
.
Lemma 2.1 Let $\mathcal {F}_{\mathsf {w}}$
be a family of affine IFSs as in (1.1). Suppose that $\mathcal {F}_{\mathrm {reg}}=\{f_i(\underline {x})=A_i\underline {x}+\underline {t}_i\}_{i\in I}$
is irreducible. Then there exist constants $C>0,\,K>0$
such that for every $\bar {\imath }\in I^*$
and for every $i,\,j\in J$
, there exists $\bar {\jmath }\in I^*$
with $|\bar {\jmath }|\leq K$
such that
In particular,
The proof is a slight modification of the proof of Feng [Reference Feng7, Proposition 2.8].
Proof. For every $i\in J$
, $A_i=\rho _i\underline {v}_i\underline {w}_i^T$
. Let us argue by contradiction. Suppose that for every $C,\,K>0$
there exist $\bar {\imath }\in I^*$
and $i,\,j\in J$
such that for every $\bar {\jmath }\in I^*$
with $|\bar {\jmath }|\leq K$
Letting $C\to 0$
and $K\to \infty$
, and taking an accumulation point $A'$
of $A_{\bar {\imath }}^T/\|A_{\bar {\imath }}\|$
we get that there exists $i',\,j'\in J$
such that
for every $\bar {\jmath }\in I^*$
. Hence, $V:=\left \langle \bigcup _{\bar {\jmath }\in I^*}A_{\bar {\jmath }}\underline {v}_{j'}\right \rangle =\langle A'\underline {w}_{i'}\rangle ^{\perp }$
is a proper subspace of $\mathbb {R}^2$
invariant with respect to the matrices $\{A_i\}_{i\in I}$
, which is a contradiction.
For every $j\in J$
, let us define
We note that a simple corollary of Feng and Käenmäki [Reference Feng and Käenmäki8, Proposition 1.2] is that
Lemma 2.2 Let $\mathcal {F}_{\mathsf {w}}$
be a family of affine IFSs as in (1.1). Suppose that $s_{\mathrm {reg}}<1$
and $\mathcal {F}_{\mathrm {reg}}=\{f_i(\underline {x})=A_i\underline {x}+\underline {t}_i\}_{i\in I}$
is irreducible. Then
where $s(\mathcal {F}_{\mathsf {w}})$
is defined in (1.2).
Let us note that the claim $s(\mathcal {F}_{\mathsf {w}})>s_{\mathrm {reg}}$
follows also by [Reference Käenmäki and Nissinen13, Lemma 2.9], however, for the sake of completeness, we give an alternative proof here.
Proof. First, we show that $s_j(\mathsf {w})>s_{\mathrm {reg}}$
for every $j\in J$
. Observe that by (2.1) and lemma 2.1
By (2.2), there exists $s>s_{\mathrm {reg}}$
such that $\sum _{\bar {\imath }\in I^*}\|A_{\bar {\imath }}\|^s>C^{-1}$
and so, $s_j(\mathsf {w})\geq s>s_{\mathrm {reg}}$
.
Now, let us show that $s_i(\mathsf {w})=s_j(\mathsf {w})$
for every $i,\,j\in J$
. Again by (2.1) and lemma 2.1
Hence, the claim follows by symmetrical reasons.
Finally, let us show that $s_j(\mathsf {w})=s(\mathcal {F}_{\mathsf {w}})$
. Clearly, by (2.1)
for every $j\in J$
, and so $s_j(\mathsf {w})\leq s(\mathcal {F}_{\mathsf {w}})$
.
On the other hand, let us enumerate the elements of $J$
by $j_1,\,\ldots,\,j_{m}$
. Then similarly to previous calculations we have,
By induction, we get that for every $n\in [1,\,m-1]\cap \mathbb {N}$
Thus, we get that $s(\mathcal {F}_{\mathsf {w}})\leq \max \{s_{\mathrm {reg}},\,\max _{j\in J}s_j(\mathsf {w})\}$
, which implies the claim.
Let us observe that for every $j\in J$
, by lemma 2.1
Hence,
In other words,
Lemma 2.3 Let $\mathcal {F}_{\mathsf {w}}$
be a family of affine IFSs as in (1.1). Suppose that $s_{\mathrm {reg}}<1$
and $\mathcal {F}_{\mathrm {reg}}=\{f_i(\underline {x})=A_i\underline {x}+\underline {t}_i\}_{i\in I}$
is irreducible. Then for every $j\in J$
and $s_j(\mathsf {w})$
is the unique solution of the equation above.
Proof. For any subset $J'\subsetneq J$
and for any $j',\,j''\in J\setminus J'$
, let us define the following maps
By lemma 2.1, the map $H_{J',j',j''}$
is strictly monotone decreasing on its support. Furthermore, since $H_{J',j',j''}$
is the limit of an increasing sequence of continuous maps, we get that $H_{J',j',j''}$
is lower semi-continuous, i.e. $\liminf _{s\to s_0}H_{J',j',j''}(s)\geq H_{J',j',j''}(s_0)$
for every $s_0\in [0,\,\infty )$
. In particular, for every $s_0\in \mathbb {R}$
such that $H_{J',j',j''}(s_0)<\infty$
,
For any $j',\,j''\in J$
,
and so by (2.2), $H_{\emptyset,j',j''}(s)<\infty$
for every $s>s_{\mathrm {reg}}$
. On the other hand, by lemma 2.1
Hence, by (2.2), $\lim _{s\searrow s_{\mathrm {reg}}}H_{\emptyset,j',j''}(s)=\infty$
.
Let us recall the basic facts that for every $x\in \mathbb {R}$
, $e^x\geq 1+x$
and for every $\varepsilon >0$
there exists a $c=c(\varepsilon )>0$
such that $x^\varepsilon \log x\leq c(\varepsilon )$
for every $x\in [0,\,1]$
. Thus, for every $s_1>s_2>s_{\mathrm {reg}}$
where $s_2-s_{\mathrm {reg}}>\varepsilon$
. Thus the map $H_{\emptyset,j',j''}\colon (s_{\mathrm {reg}},\,\infty )\mapsto \mathbb {R}_+$
is strictly monotone decreasing and continuous, and so, there exists a unique $d_{\emptyset,j',j''}\in (s_{\mathrm {reg}},\,\infty )$
such that $H_{\emptyset,j',j''}(d_{\emptyset,j',j''})=1$
.
Let $j\in J$
be arbitrary but fixed. Let us enumerate the elements of $J$
by $j_1,\,\ldots,\,j_m$
such that $j_m=j$
. Let $J_{k}:=\{j_1,\,\ldots,\,j_k\}$
. Let us argue by induction. Namely, suppose that $d_{J_{k},\,j',\,j''}$
is well defined for every $j',\,j''\notin J_k$
and $H_{J_k,j',j''}(s)\colon (d_{J_{k-1},\,j_{k},\,j_{k}},\,\infty )\mapsto \mathbb {R}_+$
is strictly monotone decreasing and continuous, $\lim _{s\searrow d_{J_{k-1},\,j_{k},\,j_{k}}}H_{J_{k},\,j',\,j''}(s)=\infty$
and $H_{J_k,j',j''}(d_{J_k,j',j''})=1$
for a unique $d_{J_k,j',j''}>d_{J_{k-1},\,j_k,\,j_k}$
.
Let $j',\,j''\notin J_{k+1}$
be arbitrary but fixed. By (2.1)
Since $H_{J_k,j',j''}(s),\,H_{J_k,j',j_{k+1}}(s),\,H_{J_k,j_{k+1},\,j''}(s)<\infty$
for every $s>d_{J_{k-1},\,j_k,\,j_k}$
we get that
and
It is enough then to show that $H_{J_{k+1},\,j',\,j''}(s)$
is continuous, since the strict monotonicity follows by lemma 2.1. But similarly to the case $H_{\emptyset,j',j''}$
, we get that for any $s_1>s_2>d_{J_k,j_{k+1},\,j_{k+1}}$
where $s_2-d_{J_k,j_{k+1},\,j_{k+1}}>\varepsilon$
. Thus, the map $H_{J_{k+1},\,j',\,j''}\colon (d_{J_k,j_{k+1},\,j_{k+1}},\,\infty )\mapsto \mathbb {R}_+$
is strictly monotone decreasing and continuous. In particular, there exists a unique $d_{J_{k+1},\,j_{k+2},\,j_{k+2}}>d_{J_k,j_{k+1},\,j_{k+1}}$
such that $H_{J_{k+1},\,j_{k+2},\,j_{k+2}}(d_{J_{k+1},\,j_{k+2},\,j_{k+2}})=1$
.
3. Lower bound of the dimension
Let us introduce the natural mapping $\underline {w}\colon \mathbb {R}\mapsto \mathbb {S}^1$
as
This section is devoted to show the following proposition.
Proposition 3.1 Let $I$
and $J$
be finite collections of indices such that $J$
is non-empty. Let $\mathcal {F}_{\mathrm {reg}}=\{f_i(\underline {x})=A_i\underline {x}+\underline {t}_i\}_{i\in I}$
be an irreducible IFS of invertible affine mappings such that $s_{\mathrm {reg}}<1$
. Furthermore, let $\rho _j\in (0,\,1)$
, $\underline {v}_j\in \mathbb {S}^1$
, $c_j\in \mathbb {R}$
, $\underline {t}_j\in \mathbb {R}^2$
and $\beta _j\in \mathbb {R}_+$
be arbitrary but fixed for every $j\in J$
. Finally, for $\alpha \in \mathbb {R}$
let
be an IFS of affine mappings. Suppose that $\mathcal {F}_{\alpha }$
satisfies the convex separation condition uniformly for $\alpha \in \mathbb {R}$
. Then there exists a set $E \subset \mathbb {R}$
such that $\dim _H(E) = 0$
and for every $\alpha \in \mathbb {R} \setminus E$
where $\Lambda _{\alpha }$
is the attractor of the IFS $\mathcal {F}_{\alpha }$
.
First, let us show why theorem 1.1(1) follows from proposition 3.1.
Proof of theorem 1.1(1) Let $\mathcal {F}_{\mathsf {w}}$
be an IFS satisfying the assumptions of theorem 1.1. By Käenmäki and Nissinen [Reference Käenmäki and Nissinen13, Lemma 3.2], we have that
Thus, it is enough to verify the lower bound.
Let us argue by contradiction. That is, suppose that there exists a set $E\subset \mathbb {T}^{\#J}$
with $\dim _HE>\#J-1$
such that
By using the map $\mathsf {w}\colon [0,\,2\pi ]^{\#J}\mapsto \mathbb {T}^{\#J}$
defined as
we get that there exists a set $E'\subset [0,\,2\pi ]^{\#J}$
with $\dim _HE'>\#J-1$
such that
For a $\underline {z}\in \mathbb {S}^{\#J-1}$
, let us denote the orthogonal projection from $\mathbb {R}^{\#J}$
to the $\#J-1$
-dimensional subspace $\langle \underline {z}\rangle ^\perp$
by $\mathrm {proj}_{\underline {z}}$
. Let us denote the spherical measure on $\mathbb {S}^{\#J-1}$
by $\sigma _{\#J-1}$
. By Mattila's slicing theorem, see [Reference Mattila14, Theorem 10.10]
Let $\underline {z}\in \mathbb {S}^{\#J-1}$
and $\underline {y}\in \mathbb {R}^{\#J}$
be such that $\dim _H(E'\cap \operatorname {proj}_{\underline {z}}^{-1}(\underline {y}))>0$
. Hence, there exists a set $E''\subset \mathbb {R}$
with $\dim _H(E'')>0$
such that for every $\alpha \in E''$
which contradicts to proposition 3.1.
3.1 Hochman's theorem
Let us recall a theorem of Hochman [Reference Hochman9, Corollary 1.2], which will be used to prove proposition 3.1. Let $\mathcal {I} \subset \mathbb {R}$
be a compact interval, and let $K$
be a finite set of indices. For every $k\in K$
, let $\lambda _k \colon \mathcal {I} \to (-1,\,1) \setminus \{0\}$
and $a_k \colon \mathcal {I} \to \mathbb {R}$
be real analytic mappings, and let $\Phi _\alpha =\{g_k^{(\alpha )}(x) = \lambda _k(\alpha )x + a_k(\alpha )\}_{k\in K}$
be a parametrized family of IFS of contracting similarities on the real line with parameters $\alpha \in \mathcal {I}$
. For every $\bar {\imath }=(i_1,\,i_2,\,\ldots ),\,\bar {\jmath }=(j_1,\,j_2,\,\ldots )\in K^\mathbb {N}$
let
Theorem 3.2 Hochman
Let $\mathcal {I} \subset \mathbb {R}$
be a compact interval, and let $K$
be a finite set of indices. For every $k\in K$
, let $\lambda _k \colon \mathcal {I} \to (-1,\,1) \setminus \{0\}$
and $a_k \colon \mathcal {I} \to \mathbb {R}$
be real analytic mappings, and let $\Phi _\alpha =\{g_k^{(\alpha )}(x) = \lambda _k(\alpha )x + a_k(\alpha )\}_{k\in K}$
be a parametrized family of IFS of contracting similarities on the real line with parameters $\alpha \in \mathcal {I}$
. Denote the attractor of $\Phi _\alpha$
by $\Gamma _\alpha$
. Suppose that
Then there exists a set $E$
with $\dim _PE=0$
such that for every $\alpha \in \mathcal {I}\setminus E$
3.2 Verifying Hochman's condition
Recall that for a $\underline {z}\in \mathbb {S}^{1}$
, we denote the orthogonal projection from $\mathbb {R}^{2}$
to the $1$
-dimensional subspace $\langle \underline {z}\rangle ^\perp$
by $\mathrm {proj}_{\underline {z}}$
. The following is our main geometric lemma.
Lemma 3.3 Let $A$
and $B$
convex compact sets, such that $A \cap B = \emptyset$
. If the Convex Separation Condition holds, then there exists an open set $O \subset \mathbb {S}^1$
, such that
Proof. Let $\vec {AB} = \{ \underline {a} - \underline {b} : \underline {a} \in A \text { and } \underline {b} \in B \}$
be the set of every vectors directing from $A$
to $B$
. Then let us define cone
Then $\mathcal {C}$
and $-\mathcal {C}$
are closed and compact. It is enough to show that the open set $\mathbb {S}^1\setminus (\mathcal {C}\cup -\mathcal {C})$
is non-empty. So $\mathrm {proj}_{\underline {w}}(A) \cap \mathrm {proj}_{\underline {w}}(B) = \emptyset$
for every $\underline {w}\in O$
.
If there is no such $\underline {w}$
, i.e. $\mathcal {C}\cup -\mathcal {C}=\mathbb {S}^1$
, then $\mathcal {C}\cap -\mathcal {C}\neq \emptyset$
. Let $\underline {w}\in \mathcal {C}\cap -\mathcal {C}$
. Let $\underline {a}_1,\,\underline {a}_2\in A$
and $\underline {b}_1,\,\underline {b}_2\in B$
be point such that,
By connecting the endpoints we might define a (possibly degenerate) trapeze.
Let $P_b =\{t\underline {b}_1+(1-t)\underline {b}_2:t\in [0,\,1]\}$
be the line between $\underline {b}_1$
and $\underline {b}_2$
and let $P_a = \{t\underline {a}_1+(1-t)\underline {a}_2:t\in [0,\,1]\}$
be the line between $\underline {a}_1$
and $\underline {a}_2$
. The diagonals of a (possibly degenerate) trapeze always intersect each other and so $P_a\cap P_b\neq \emptyset$
. However, since $A$
and $B$
are convex closed sets then, $P_b(x,\,y) \subset B$
and $P_a(x,\,y) \subset A$
which implies that, these lines cannot intersect each other, which is a contradiction.
Let $\mathcal {F}_\alpha$
be the IFS defined in (3.1) satisfying the conditions of proposition 3.1. For simplicity, let us denote by $A_j^{(\alpha )}$
the matrices $\rho _j\underline {v}_j\underline {w}(c_j+\beta _j\alpha )^T$
for $j\in J$
, and the products by $A_{\bar {\imath }}^{(\alpha )}$
for $\bar {\imath }\in (I\cup J)^*$
to emphasize its possible dependence on $\alpha$
.
Let us define the natural projection $\Pi _\alpha$
from the symbolic space $\Sigma =(I\cup J)^\mathbb {N}$
to the attractor $\Lambda _\alpha$
of $\mathcal {F}_\alpha$
by
for $\bar {\imath }=(i_1,\,i_2,\,\ldots )\in \Sigma$
. For any $\bar {\imath },\,\bar {\jmath }\in \Sigma$
, let $|\bar {\imath }\wedge \bar {\jmath }|=\min \{k\geq 1:i_k\neq j_k\}-1$
, and let $\bar {\imath }\wedge \bar {\jmath }=(i_1,\,\ldots,\,i_{|\bar {\imath }\wedge \bar {\jmath }|})$
. Denote the left-shift operator on $\Sigma$
by $\sigma$
. Then clearly
Lemma 3.4 Let $\mathcal {F}_\alpha$
be the IFS defined in (3.1) satisfying the conditions of proposition 3.1. Then
Proof. Let $U\subset \mathbb {R}^2$
be the compact convex set with respect to the uniform convex separation condition holds. Let $\bar {\imath },\,\bar {\jmath }\in \Sigma$
be such that $\bar {\imath }\neq \bar {\jmath }$
. Then by using the linearity of the maps of $\mathcal {F}_\alpha$
,
By the uniform convex separation condition we get that $\Pi _{\alpha }(\sigma ^{|\bar {\imath }\wedge \bar {\jmath }|}\bar {\imath })-\Pi _{\alpha }(\sigma ^{|\bar {\imath }\wedge \bar {\jmath }|}\bar {\jmath })\neq \underline {0}$
for every $\alpha \in \mathbb {R}$
. Hence,
For a fixed $\alpha \in \mathbb {R}$
, there are three possibilities:
(I.) $\mathrm {rank}(A_{\bar {\imath }\wedge \bar {\jmath }}^{(\alpha )})=2$
(i.e. $\bar {\imath }\wedge \bar {\jmath }\in I^*$) but then $\mathrm {Ker}(A_{\bar {\imath }\wedge \bar {\jmath }}^{(\alpha )})=\{\underline {0}\}$, which cannot happen;(II.) $\mathrm {rank}(A_{\bar {\imath }\wedge \bar {\jmath }}^{(\alpha )})=1$
then $\mathrm {Ker}(A_{\bar {\imath }\wedge \bar {\jmath }}^{(\alpha )})=\mathrm {Ker}(A_{i_k}^{(\alpha )}A_{\bar {\imath }'})$, where $k=\max \{1\leq \ell \leq |\bar {\imath }\wedge \bar {\jmath }|:i_{\ell }\in J\}$ and $\bar {\imath }'\in I^{|\bar {\imath }\wedge \bar {\jmath }|-k-1}$ is the suffix of $\bar {\imath }\wedge \bar {\jmath }$;(III.) $\mathrm {Ker}(A_{\bar {\imath }\wedge \bar {\jmath }}^{(\alpha )})=\mathbb {R}$
, namely, there exists $1\leq k<\ell \leq |\bar {\imath }\wedge \bar {\jmath }|$ and $\bar {\imath }'\in I^*$ such that $\mathrm {Im}(A_{\bar {\imath }'}^{(\alpha )}A_{i_{\ell }})=\mathrm {Ker}(A_{i_k}^{(\alpha )})$ and $i_\ell,\,i_k\in J$.
By the definition of $A_j^{(\alpha )}=\rho _j\underline {v}_j\underline {w}(c_j+\beta _j\alpha )^T$
for $j\in J$
, $\mathrm {Im}(A_{\bar {\imath }'}A_{j}^{(\alpha )})$
is independent of $\alpha$
for any $\bar {\imath }'\in I^*$
and $j\in J$
, it is clear that the set $\{\beta \in \mathbb {R}:\mathrm {Ker}(A_{\bar {\imath }\wedge \bar {\jmath }}^{(\alpha )})=\mathbb {R}\}$
is a discrete and countable set for every $\bar {\imath }\neq \bar {\jmath }\in \Sigma$
. Hence, it is enough to check that for every $\bar {\imath }\in I^*$
, $j\in J$
and $k_1\neq k_2\in I\cup J$
there exists $\alpha \in \mathbb {R}$
such that
But by lemma 3.3, there exists an open set of $\alpha \in \mathbb {R}$
such that (3.2) holds, which completes the proof.
Proof of proposition 3.1 Let $j\in J$
be arbitrary but fixed. For every $\bar {\imath }\in \left(I\cup J\setminus \{j\}\right)^*$
, let us define a map $g_{\bar {\imath }}^{(\alpha )}\colon \mathbb {R}\mapsto \mathbb {R}$
such that
In particular,
Furthermore, $\|A_jA_{\bar {\imath }}|\mathrm {Im}(A_j)\|=\left |\rho _j\underline {w}(c_j+\alpha \beta _j)^TA_{\bar {\imath }}\underline {v}_j\right |$
for every $\bar {\imath }\in (I\cup J\setminus \{j\})^*$
.
For every $n\in \mathbb {N}$
, let us define an IFS on the real line as $\mathcal {G}^{n}_{\alpha }=\{g_{\bar {\imath }}^{(\alpha )}\}_{\bar {\imath }\in \bigcup _{k=0}^n(I\cup J\setminus \{j\})^k}$
. Let us denote the attractor of $\mathcal {G}^{n}_{\alpha }$
by $\Gamma _{n,\alpha }$
. Let $\pi _\alpha \colon ((\bigcup _{k=0}^nI\cup J\setminus \{j\})^k)^\mathbb {N}\mapsto \Gamma _{n,\alpha }$
be the natural projection associated to the IFS $\mathcal {G}^{n}_{\alpha }$
. Then by (3.4),
By defining the bi-Lipschitz mapping $h\colon \mathbb {R}\mapsto \mathbb {R}^2$
as $h(x)=x\underline {v}_j+\underline {t}_j$
, we see that $h(\Gamma _{n,\alpha })\subset \Lambda _\alpha$
for every $n\in \mathbb {N}$
and $\alpha \in \mathbb {R}$
. Moreover, combining lemma 3.4 with (3.5), we see that
Clearly, the contraction ratios and the translation parameters of the maps in $\mathcal {G}^{n}_{\alpha }$
are analytic maps of $\alpha \in \mathbb {R}$
. Let $R_n$
be the set of roots of the contraction ratios of the maps in $\mathcal {G}^{n}_{\alpha }$
. Then for every $a\in \mathbb {N}$
, the set $(-a,\,a)\cap R_N$
is finite. Let $I_1^a,\,\ldots,\,I_N^a$
be disjoint open subintervals of $(-a,\,a)$
such that $\|A_jA_{\bar {\imath }}|\mathrm {Im}(A_j)\|\neq 0$
for every $\alpha \in I_k$
and every $k=1,\,\ldots,\,L$
. For every $k=1,\,\ldots,\,L$
and $\ell \in \mathbb {N}$
, let $J_{k,\ell }^a\subseteq I_k^a$
be compact intervals such that $\bigcup _{\ell =1}^\infty J_{k,\ell }^a=I_k^a$
.
Applying theorem 3.2, there exists $E_{k,\ell }^a\subset J_{k,\ell }^a$
such that $\dim _H(E_{k,\ell }^a)=0$
and
where $s^{(n)}(\alpha )$
is the similarity dimension of $\mathcal {G}^{n}_{\alpha }$
, that is,
By lemma 2.2 and (2.3), $\lim _{n\to \infty }s^{(n)}(\alpha )=s(\mathcal {F}_{\alpha })$
. Hence, by choosing $E:=\bigcup _{n=1}^\infty R_n\cup \bigcup _{a,k,\ell }E_{k,\ell }^a$
, the claim of the proposition follows.
4. Exceptional parameters
The remaining part of the paper is devoted to prove theorem 1.1(2). Let $\mathcal {F}_{\mathsf {w}}$
be a family of affine IFSs as in (1.1) with attractor $\Lambda _{\mathsf {w}}$
. Suppose that $\sup _{\mathsf {w}}s(\mathcal {F}_{\mathsf {w}})<1$
and $\mathcal {F}_{\mathsf {w}}$
satisfies the convex separation condition uniformly.
Lemma 4.1 Let us fix $j\in J$
and $i\in I\cup J$
such that $i\neq j$
. Then there exists $\underline {w}_j\in \mathbb {S}^1$
such that $f_{j}$
and $f_j\circ f_i$
share the same fixed point. In particular, $f_{j}\circ f_j\circ f_i\equiv f_j\circ f_i\circ f_{j}$
.
Proof. Let $U\subset \mathbb {R}^2$
be the convex set with respect to the convex separation condition holds uniformly. Let the map $g_{\emptyset }^{(\alpha )}\colon \mathbb {R}\mapsto \mathbb {R}$
and $g_{i}^{(\alpha )}\colon \mathbb {R}\mapsto \mathbb {R}$
be as in (3.3). Namely,
So, $f_j(x\underline {v}_j+\underline {t}_j)=g_{\emptyset }^{(\alpha )}(x)\underline {v}_j+\underline {t}_j$
and $f_j\circ f_i(x\underline {v}_j+\underline {t}_j)=g_{i}^{(\alpha )}(x)\underline {v}_j+\underline {t}_j$
. By lemma 3.3, there exists $\alpha '\in [0,\,\pi ]$
such that $\mathrm {proj}_{\underline {w}(\alpha ')}(f_j(U))\cap \mathrm {proj}_{\underline {w}(\alpha ')}(f_i(U))=\emptyset$
, and similarly for $\underline {w}(\alpha '+\pi )$
, where $\underline {w}(\alpha )=(\cos (\alpha ),\,\sin (\alpha ))^T$
. This implies that
Since the fixed point of the maps $g_{\emptyset }^{(\alpha )}$
and $g_{i}^{(\alpha ')}$
are continuous functions of $\alpha$
, by Bolzano-Darboux theorem, there exist $\alpha \in [0,\,2\pi ]$
such that $g_{\emptyset }^{(\alpha )}$
and $g_{i}^{(\alpha ')}$
share the same fixed points, and in particular $f_j$
and $f_j\circ f_i$
do.
Proof of theorem 1.1(??)2) Let $\mathcal {F}_{\mathsf {w}}$
be a family of affine IFSs as in (1.1) with attractor $\Lambda _{\mathsf {w}}$
such that $\#J\geq 1$
and $\#(I\cup J)\geq 2$
. Suppose that $\sup _{\mathsf {w}}s(\mathcal {F}_{\mathsf {w}})<1$
, $\mathcal {F}_{\mathsf {w}}$
satisfies the convex separation condition uniformly for every $\mathsf {w}\in \mathbb {T}^{\#J}$
.
Let $j\in J$
and $i\in I\cup J$
be arbitrary but fixed such that $i\neq j$
. By lemma 4.1, there exists $\underline {w}_j\in \mathbb {S}^1$
such that $f_{j}\circ f_j\circ f_i\equiv f_j\circ f_i\circ f_j$
. Let us fix this $\underline {w}_j\in \mathbb {S}^1$
and choose every other $\underline {w}_{j'}$
for $j'\in J\setminus \{j\}$
arbitrarily. Let us define a new IFS
Hence, $\Lambda _\mathsf {w}'=\Lambda _\mathsf {w}$
, where $\Lambda _\mathsf {w}'$
is the attractor of $\mathcal {F}_{\mathsf {w}}'$
.
However, by lemmas 2.2 and 2.3, one can see that $s(\mathcal {F}_{\mathsf {w}}')< s(\mathcal {F}_{\mathsf {w}})$
, and by Käenmäki and Nissinen [Reference Käenmäki and Nissinen13, Lemma 3.2], we have that
which implies the desired claim.
Acknowledgements
The research was supported by the grants the grants NKFI FK134251, K142169, and the grant NKFI KKP144059 ‘Fractal geometry and applications.’
