Some remarks on the general finite type condition

Abstract

In this paper, the author discusses the iterated function system of generalized finite type conditions. First, the author constructs an iterated function system of generalized finite type condition, which satisfies the case where an invariant set is a basic set if and only if it is a subset of (0, 1). Second, the author proves that, with respect to the nested index sequence \{{\mathrm{\Lambda }}_k{\}}_k\ge \text{0}, any iterated function system in {\mathbb{R}}^dcannot satisfy the generalized finite type condition when the exponents of contractive ratios are not commensurable. Finally, the author constructs a family of self-similar sets which satisfy the generalized finite type condition and computes the Hausdorff dimensions of them.

Graphical abstract

Keywords

Self-similar set / iterated function system / generalized finite type condition / Hausdorff dimension

1 Introduction

Let {\left\{S_i\right\}}_{i=1}^N be an iterated function system on {\mathbb{R}}^d, where the similarity contraction functions S_i are given by the following expression:

(1)S_i\left(x\right)={\rho }_i{R}_{i}x+d_i,i=1,2,\dots ,N,

where the contraction ratio {\rho }_i\in \left(0,1\right), R_i is an orthogonal matrix, and d_i{\in \mathbb{R}}^d. The unique non-empty compact set F satisfying the following equation is called a self-similar set (see [3, 6]):

(2)F=\bigcup _{i=1}^N{S}_i\left(F\right).

If there exists a non-empty bounded open set \mathrm{\Omega }\subset {\mathbb{R}}^d such that \bigcup _{i=1}^N S_i\left(\mathrm{\Omega }\right)\subset \mathrm{\Omega }, then \mathrm{\Omega } is called an invariant set of the iterated function system \{S_i{\}}_{i=1}^N. If there exists an invariant set \mathrm{\Omega } such that S_i\left(\mathrm{\Omega }\right)\cap S_j\left(\mathrm{\Omega }\right)=\varnothing for i\ne j, then this iterated function system is said to satisfy the open set condition.

The computation of the Hausdorff dimension of a self-similar set is a popular topic. When the open set condition holds, the Hausdorff dimension s of the self-similar set is the unique solution to the following equation [2‒4, 6, 13, 17]:

\sum _{i=1}^N {\rho }_i^s=1.

When the iterated function system does not satisfy the open set condition, there is no unified method for calculating the Hausdorff dimension. However, through the efforts of many scholars, several important results have been obtained [1, 5, 9, 11, 12, 14]. In 1996, Liu and Ni [9] introduced a condition weaker than the open set condition—the weak separation property; in the same year, Zenner [18] provided an estimate for the Hausdorff dimension of the corresponding self-similar set but did not give an exact formula. To address this, Rao and Wen [16] in 1998, and Ni and Wang [14] in 2013, independently proposed the finite-type condition (which implied the weak separation property [15]) and provided explicit formulas for calculating the Hausdorff dimension of such self-similar sets, extending the method of Lalley [8]. However, the finite-type condition requires that the exponents of the contraction ratios be commensurable, and it is not a complete extension of the open set condition. Liu and Ni [10], and Jin and Qiu [7], independently proposed the generalized finite-type condition for cases where the exponents of the contraction ratios were not commensurable, and calculated the corresponding Hausdorff dimension. The generalized finite-type condition can derive the weak separation property [10], and it is an extension of both the open set condition and the finite-type condition: all iterated function systems that satisfy either the open set condition or the finite-type condition also satisfy the generalized finite-type condition. When the exponents of the contraction ratios are commensurable, the generalized finite-type condition is equivalent to the finite-type condition.

It is known that both finite-type and generalized finite-type depend on the choice of the invariant set from their definitions. When the invariant set changes, the structure of the iterated function system may also change. Then, there is a question: if an iterated function system has a basic set (see Definition 2.2), are all invariant sets basic sets? For iterated function systems that satisfy the finite-type condition, it is known that if such a system has a basic set that is a finite interval, then all invariant sets are basic sets. However, for the generalized finite-type condition, this property no longer holds. Based on the incomparability of the exponents of the contraction ratios, there are the following theorems:

Theorem 1.1　 There exists a function iteration system that satisfies the generalized finite-type condition, and its invariant set is a basic set if and only if this invariant set is a subset of the unit open interval.

Both finite-type and generalized finite-type conditions depend not only on the invariant set but also the choice of the index family (the definitions of the index families \mathrm{\Lambda } and \mathrm{\Sigma } are given in Section 2). For example, under the index family \mathrm{\Lambda }, the iterated function system

\left\{S_1\left(x\right)=\frac{x}{4},S_2\left(x\right)=\frac{x}{2},S_3\left(x\right)=\frac{x+1}{2}\right\}

is generalized finite-type (and also finite-type, because the exponents of the contraction ratios are commensurable) on {\mathbb{R}}^1. However, under the index family Σ, it is infinite-type [10]. This example shows that when considering iterated function systems that satisfy the finite-type condition, the index family \mathrm{\Lambda }, is superior to Σ. However, when the exponents of the contraction ratios are not commensurable, under the index family \mathrm{\Lambda }, no iterated function system is a generalized finite-type. The following theorem illustrates that when determining whether an iterated function system satisfies the generalized finite-type condition, the index family \mathrm{\Lambda }, will not provide useful information. This behaves in a completely different manner from finite-type systems.

Theorem 1.2　 Let the iterated function system {\left\{S_i\right\}}_{i=1}^N on {\mathbb{R}}^d be defined by Equation (1). If \mathrm{l}\mathrm{n}{\rho }_1,\mathrm{l}\mathrm{n}{\rho }_2,\dots ,\mathrm{l}\mathrm{n}{\rho }_N are incommensurable, then with respect to the index family \mathrm{\Lambda }, this iterated function system is not a generalized finite-type.

According to the reviewed literature, iterated function systems that satisfy the generalized finite-type condition but not the finite-type condition are rare on the real line. The main examples are of the following form:

(3)\left\{\begin{array}{c}{S}_1\left(x\right)=\rho x,S_2\left(x\right)=rx+\rho (1-r),S_3\left(x\right)=rx+(1-r)\end{array}\right\},

where \rho ,r\in \left(0,1\right) satisfy \rho +2r-\rho r\le 1. In a certain sense, the lack of concrete examples of generalized finite-type systems limits the application of the generalized finite-type condition. Therefore, another goal of this paper is to construct iterated function systems that satisfy the generalized finite-type condition. The index family \mathrm{\Sigma } is used to construct a class of iterated function systems that satisfy the generalized finite-type condition.

The structure of this paper is as follows: Section 2 introduces the generalized finite-type condition; Section 3 provides proofs for Theorems 1.1 and 1.2; and Section 4 presents a class of iterated function systems that satisfy the generalized finite-type condition.

2 Generalized finite-type

The notation in this section is based on the work in [10]. Let \{S_i{\}}_{i=1}^N be an iterated function system on {\mathbb{R}}^d as defined by Equation (1). The index sets are defined as follows:

{\mathrm{\Sigma }}^k=\{1,2,\dots ,N{\}}^k,k\ge 1,\mathrm{\Sigma }=\bigcup _{k\ge 0}{\mathrm{\Sigma }}^k,

with {\mathrm{\Sigma }}_0=\left\{\varnothing \right\}, where \varnothing represents the empty word. An element \mathbf{i}\in {\mathbf{\Sigma }}^{\mathbf{k}} is called a word of length k, and its length is denoted by \left|\mathbf{i}\right|. Clearly, \left|\varnothing \right|=0. Let \mathbf{i}\mathbf{=}{\mathbf{i}}_{\mathbf{1}}{\mathbf{i}}_{\mathbf{2}}\cdots {\mathbf{i}}_{\mathbf{n}}\in \mathbf{\Sigma }, and . Define \mathbf{i}\mathbf{|}\mathbf{k}\mathbf{=}{\mathbf{i}}_{\mathbf{1}}{\mathbf{i}}_{\mathbf{2}}\cdots {\mathbf{i}}_{\mathbf{k}}, and in particular, {\mathbf{i}}^{-}=\mathbf{i}\left|\right(\mathbf{n}\mathbf{-}\mathbf{1}\mathbf{)}. For \mathbf{i}\mathbf{=}{\mathbf{i}}_{\mathbf{1}}{\mathbf{i}}_{\mathbf{2}}\cdots {\mathbf{i}}_{\mathbf{k}}\mathbf{,}\mathbf{j}\mathbf{=}{\mathbf{j}}_{\mathbf{1}}{\mathbf{j}}_{\mathbf{2}}\cdots {\mathbf{j}}_{\mathbf{m}}\in \mathbf{\Sigma }, the concatenation of words is defined as \mathbf{i}\mathbf{j}\mathbf{=}{\mathbf{i}}_{\mathbf{1}}\cdots {\mathbf{i}}_{\mathbf{k}}{\mathbf{j}}_{\mathbf{1}}\cdots {\mathbf{j}}_{\mathbf{m}}. Let \mathbf{i}\mathbf{,}\mathbf{j}\in \mathbf{\Sigma }. If there exists \mathbf{k}\in \mathbf{\Sigma } (which can be the empty word) such that \mathbf{j}\mathbf{=}\mathbf{i}\mathbf{k}, i is said to be a prefix of j, denoted as \mathbf{i}\le \mathbf{j}. Otherwise, we denote \mathbf{i}⪇\mathbf{j}.

For \mathbf{i}\mathbf{=}{\mathbf{i}}_{\mathbf{1}}{\mathbf{i}}_{\mathbf{2}}\cdots {\mathbf{i}}_{\mathbf{k}}\in {\mathbf{\Sigma }}^{\mathbf{k}}, we use the standard notation:

S_{\mathbf{i}}=S_{i_1}\circ S_{i_2}\circ \cdots \circ S_{i_k},{\rho }_{\mathbf{i}}={\rho }_{i_1}{\rho }_{i_2}\cdots {\rho }_{i_k},R_{\mathbf{i}}=R_{i_1}{R}_{i_2}\cdots R_{i_k},

with {\rho }_{\varnothing }=1, S_{\varnothing } being the identity map on {\mathbb{R}}^d, and R_{\varnothing } being the identity matrix.

Let \mathcal{M}={\left\{{\mathcal{M}}_k\right\}}_{k\ge 0} be a sequence of index sets, where {\mathcal{M}}_k\subset \mathrm{\Sigma },k\ge 0. Define:

{\underset{\_}{m}}_k={\underset{\_}{m}}_k\left({\mathcal{M}}_k\right)=\mathrm{m}\mathrm{i}\mathrm{n}\left\{\left|\mathbf{i}\right|:\mathbf{i}\in {\mathcal{M}}_{\mathbf{k}}\right\},{\overline{m}}_k={\overline{m}}_k\left({\mathcal{M}}_k\right)=\mathrm{m}\mathrm{a}\mathrm{x}\left\{\left|\mathbf{i}\right|:\mathbf{i}\in {\mathcal{M}}_{\mathbf{k}}\right\}.

\mathcal{M}={\left\{{\mathcal{M}}_k\right\}}_{k=0}^{\mathrm{\infty }} is an index family if it satisfies the following conditions:

(1) \left\{{\underset{\_}{m}}_k\right\} and \left\{{\overline{m}}_k\right\} are non-decreasing, and {\mathrm{l}\mathrm{i}\mathrm{m}}_{k\to \mathrm{\infty }}{\underset{\_}{m}}_k={\mathrm{l}\mathrm{i}\mathrm{m}}_{k\to \mathrm{\infty }}{\overline{m}}_k=\mathrm{\infty };

(2) For every k\ge 0 and all \mathbf{i}\mathbf{,}\mathbf{j}\in {\mathcal{M}}_{\mathbf{k}}, if \mathbf{i}\ne \mathbf{j}, then \mathbf{i}⪇\mathbf{j} and \mathbf{j}⪇\mathbf{i};

(3) For every \mathbf{j}\in \mathbf{\Sigma } such that \left|\mathbf{j}\right|>{\overline{m}}_k, there exists \mathbf{i}\in {\mathcal{M}}_{\mathbf{k}} such that \mathbf{i}\le \mathbf{j}\mathbf{;}

(4) For every \mathbf{j}\in \mathbf{\Sigma } such that \left|\mathbf{j}\right|>{\overline{m}}_k, there exists \mathbf{i}\in {\mathcal{M}}_{\mathbf{k}} such that \mathbf{j}\le \mathbf{i}\mathbf{;}

(5) There exists a positive integer L (independent of k) such that for all words \mathbf{i}\in {\mathcal{M}}_{\mathbf{k}} and \mathbf{j}\in {\mathcal{M}}_{\mathbf{k}\mathbf{+}\mathbf{1}} with \mathbf{i}\le \mathbf{j}, there is \left|\mathbf{j}\right|-\left|\mathbf{i}\right|\le L.

In general, \bigcup _{\mathrm{k}\ge 0}{\mathcal{M}}_{\mathrm{k}} is a proper subset of \mathrm{\Sigma }. However, when {\mathcal{M}}_k={\mathrm{\Sigma }}^k, there are \bigcup _{k\ge 0}{\mathcal{M}}_k=\mathrm{\Sigma }, and \mathcal{M} is regarded as the index family Σ, which is the commonly used index family. Another commonly used index family is \mathrm{\Lambda }={\left\{{\mathrm{\Lambda }}_k\right\}}_{k=0}^{\mathrm{\infty }}, where {\mathrm{\Lambda }}_k=\{\mathbf{j}\in

Now, consider a chosen index family {\left\{{\mathcal{M}}_k\right\}}_{k=0}^{\mathrm{\infty }}. For each positive integer k\ge 1, we define

{\mathcal{V}}_k=\{\left(S_i,k\right):\mathbf{i}\in {\mathcal{M}}_{\mathbf{k}}\}\mathbf{.}

Let {\mathcal{V}}_0=\{{\mathbf{v}}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}=(I,0\left)\right\}. {\mathcal{V}}_k is said to be the vertex set with respect to the index family {\left\{{\mathcal{M}}_k\right\}}_{k=0}^{\mathrm{\infty }}, and {\mathbf{v}}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}} is called the root vertex. Let \mathcal{V}=\bigcup _{k\ge 0}{\mathcal{V}}_k. For \mathbf{v}\mathbf{=}\mathbf{(}{\mathbf{S}}_{\mathbf{i}}\mathbf{,}\mathbf{k}\mathbf{)}\in {\mathcal{V}}_{\mathbf{k}}, it is convenient to denote S_{\mathbf{v}}=S_{\mathbf{i}} and {\rho }_{\mathbf{v}}={\rho }_{\mathbf{i}}. Note that when \mathbf{i}\ne \mathbf{j}, it is possible for \left(S_i,k\right)=\left(S_j,k\right). Let Ω be the invariant set of {\left\{S_i\right\}}_{i=1}^N. If \mathbf{v}\mathbf{,}{\mathbf{v}}^{\prime }\in {\mathcal{V}}_{\mathbf{k}} satisfy S_{\mathbf{v}}\left(\mathrm{\Omega }\right) and S_{{\mathbf{v}}^{\prime }}\left(\mathrm{\Omega }\right) intersect, it is said that \mathbf{v}, and {\mathbf{v}}^{\prime } are adjacent (with respect to Ω). The neighborhood of v (with respect to Ω) is the vertex set

\mathrm{\Omega }\left(\mathrm{v}\right)=\{{\mathbf{v}}^{\prime }:{\mathbf{v}}^{\prime }\text{is adjacent to}\mathbf{v}\}.

Definition 2.1　Let \mathbf{v}\in {\mathcal{V}}_{\mathbf{k}} and {\mathbf{v}}^{\prime }\in {\mathcal{V}}_{k^{\prime }}. If the following conditions hold for \tau =S_{{\mathbf{v}}^{\prime }}\circ S_{\mathbf{v}}^{-1}:

(1) \{S_{{\mathbf{u}}^{\prime }}:{\mathbf{u}}^{\prime }\in \mathrm{\Omega }({\mathbf{v}}^{\prime }\left)\right\}=\{\tau \circ S_{\mathbf{u}}:\mathbf{u}\in \mathbf{\Omega }\mathbf{(}\mathbf{v}\left)\right\};

(2) Let \mathbf{u}\in \mathbf{\Omega }\left(\mathrm{v}\right) and {\mathbf{u}}^{\prime }\in \mathbf{\Omega }\left({\mathrm{v}}^{\prime }\right) such that S_{{\mathbf{u}}^{\prime }}=\tau \circ S_{\mathbf{u}}. Then for any integer l\ge 1, an index \mathbf{i}\in \mathbf{\Sigma } satisfies (S_{\mathbf{u}\mathbf{i}},k+l)\in V_{k+l} if and only if (S_{{\mathbf{u}}^{\prime }\mathbf{i}},k^{\prime }+l)\in V_{k^{\prime }+l}. Then it is said that \mathbf{v} and {\mathbf{v}}^{\prime } are equivalent, denoted as \mathbf{v}\sim {\mathbf{v}}^{\prime }. The set of all vertices equivalent to v is denoted as: {\left[\mathbf{v}\right]}_{\mathrm{\Omega }}=\{\mathbf{u}\in \mathcal{V}\mathbf{:}\mathbf{u}\sim \mathbf{v}\}.

Definition 2.2　Let the iterated function system be defined as in Equation (1). If there exists an invariant set Ω and an index family \mathcal{M} such that the quotient \mathcal{V}/ is a finite set, then it is said that this iterated function system (with respect to the invariant set Ω and the index family \mathcal{M}) satisfies the generalized finite-type condition. In this case, Ω is called a basic set for the iterated function system.

To obtain the Hausdorff dimension of the self-similar set corresponding to an iterated function system that satisfies the generalized finite-type condition, two infinite directed graphs \mathcal{G} and {\mathcal{G}}_Rare introduced in this paper. The vertex set of \mathcal{G} is \mathcal{V}, and the directed edges \mathcal{E} are defined as follows. Let \mathrm{v}\in {\mathcal{V}}_k and u\in {\mathcal{V}}_{k+1}. Assume that there exists \mathbf{i}\in {\mathcal{M}}_{\mathbf{k}}\mathbf{,}\mathbf{j}\in {\mathcal{M}}_{\mathbf{k}\mathbf{+}\mathbf{1}}, and \mathbf{k}\in \mathbf{\Sigma } such that

(4)\mathbf{v}\mathbf{=}\left({\mathbf{S}}_{\mathbf{i}}\mathbf{,}\mathbf{k}\right)\mathbf{,}\mathbf{u}\mathbf{=}\left({\mathbf{S}}_{\mathbf{j}}\mathbf{,}\mathbf{k}\mathbf{+}\mathbf{1}\right)\mathbf{,}\mathbf{j}\mathbf{=}\mathbf{i}\mathbf{k}\mathbf{,}

then a directed edge k from v to u is drawn. Let \mathcal{E} be the set of all directed edges as defined above, and let \text{G}=(\mathcal{V},\mathcal{E}). To simplify the definition of {\mathcal{G}}_R, the partial order of Σ is first fixed as lexicographical order. In the graph \mathcal{G}, all directed edges are removed except the smallest ones between any two vertices, and the resulting graph is denoted as {\mathcal{G}{}^{\prime }}_R. Then, all vertices that have no descendants are removed, as well as edges and vertices that only point to these vertices. The final graph is the simplified graph {\mathcal{G}}_R=({\mathcal{V}}_R,{\mathcal{E}}_R), where {\mathcal{V}}_R is the vertex set and {\mathcal{E}}_R is the set of directed edges.

When \mathcal{V}/ is a finite set, a finite graph is defined recursively as an iterated function system (V,E,M) and its associated matrix A_s. The vertex set is V=\{\left[\mathbf{v}\right]:\mathbf{v}\in {\mathcal{V}}_{\mathbf{R}}\}. For \left[\mathbf{v}\right],\left[\mathbf{u}\right]\in V, if there exists {\mathbf{u}}^{\prime }\in \left[\mathbf{u}\right],{\mathbf{v}}^{\prime }\in \left[\mathbf{v}\right] and a directed edge \mathbf{k}\in {\mathcal{E}}_{\mathbf{R}}connecting {\mathbf{u}}^{\prime } and {\mathbf{v}}^{\prime }, then a directed edge ([v], [u]) is drawn between [v] and [u]. As shown in [10], the map S_{\mathbf{k}} depends only on the equivalence classes [v] and [u], independent of the choices of v′ and u′, so we define the self-similar compression map for the edge ([v], [u]) as S_{(\left[\mathbf{v}\right],[\mathbf{u}\left]\right)}=S_{\mathbf{k}}, with the compression ratio denoted as {\rho }_{\left(\right[\mathbf{v}\mathbf{]}\mathbf{,}\mathbf{[}\mathbf{u}\left]\right)}. Thus, we obtain the graph recursion system (V,E,M), where V is the vertex set, E is the edge set, and M is the family of compression functions \{S_e:e\in E\}. The associated matrix is defined as

A_s={\left({\rho }_{\left(\right[\mathbf{v}\mathbf{]}\mathbf{,}\mathbf{[}\mathbf{u}\left]\right)}^s\right)}_{\left[\mathbf{v}\right],\left[\mathbf{u}\right]\in V},

with the convention that {\rho }_{\left(\right[\mathbf{v}\mathbf{]}\mathbf{,}\mathbf{[}\mathbf{u}\left]\right)}=0when \left(\left[\mathbf{v}\right],\left[\mathbf{u}\right]\right)\notin E. The following theorem is from [7, 10].

Theorem 2.1 [7, 10]　 Let the iterated function system {\left\{S_i\right\}}_{i=1}^N on {\mathbb{R}}^d satisfy the generalized finite-type condition, and let the corresponding self-similar set be F. Let {\lambda }_{\left(s\right)} be the spectral radius of the associated matrix A_s. Then

{\mathrm{d}\mathrm{i}\mathrm{m}}_{B}F={\mathrm{d}\mathrm{i}\mathrm{m}}_{H}F=s,

where s is the unique solution to \lambda \left(s\right)=1. Moreover, .

3 Proofs of Theorem 1.1 and Theorem 1.2

In this section, proofs for Theorem 1.1 and Theorem 1.2 are provided in detail. Let a_1,a_2 be real numbers. If the ratio \frac{a_1}{a_2} is rational, then it is said that a_1 and a_2 are commensurable; otherwise, they are incommensurable. If there are two incommensurable real numbers among a_1,a_2,\dots ,a_n, it is said that a_1,a_2,\dots ,a_n are incommensurable.

Proof of Theorem 1.1　Let {\left\{S_i\right\}}_{i=1}^3 be the iterated function system defined by Equation (3), and let \rho +2r-\rho r=1.In this case, the self−similar set F=\left[0,1\right], and S_2\left(1\right)=S_3\left(0\right). Let \mathrm{\Omega }\subset \left(0,1\right) be the invariant set for this iterated function system. Let Σ be the index family. Following the proof in [10], it is known that Ω is a basic set, so the iterated function system satisfies the generalized finite-type condition. Next, it is required to prove that when \text{ln}\rho and \mathrm{l}\mathrm{n}r are incommensurable (e.g., take \rho =\frac{1}{3}, r=\frac{2}{5}), if \mathrm{\Omega }\not\subset \left(0,1\right), then for any index family \mathcal{M}, Ω is not a basic set.

Assume that Ω is an invariant set and \mathrm{\Omega }\not\subset \left(0,1\right). Let x_0 be an arbitrary point in the open set \mathrm{\Omega }\setminus \left[0,1\right]. We will first consider the case where , with the proof for x_0>1 being analogous. Let x_k=S_{1^k}\left(x_0\right). From the fact that Ω is an invariant set, it is known that \left\{x_k\right\}\subset \mathrm{\Omega }, and from the expression of S_1, it is known that \left\{x_k\right\} is strictly increasing and tends to 0. Next, let \mathcal{M} be an arbitrary index family, and consider the subset of the corresponding vertex set \mathcal{V}:

H=\left\{\right(S_{{31}^n},k)\in \mathcal{V}:k\ge 1\}.

From the definition of {\left\{S_i\right\}}_{i=1}^3, it can be seen that S_i\left(1\right)=1-r holds if and only if there exists an m\ge 0 such that \mathbf{i}\mathbf{=}{\mathbf{23}}^{\mathbf{m}} (see Fig.1). Moreover, for any n\ge 1, there is S_{{31}^n}\left(0\right)=1-r. Let m and n be arbitrary non-negative integers. Since \left[0,1\right]\subset \overline{\mathrm{\Omega }}, there exists \epsilon >0 such that \left[1-r-\epsilon ,1-r\right]\subset S_{{23}^m}\left(\overline{\mathrm{\Omega }}\right)=\overline{S_{{23}^m}\left(\mathrm{\Omega }\right)}. On the other hand, there is S_{{31}^n}\left(x_k\right)=S_3\left(x_{k+n}\right)={rx}_{k+n}+(1-r), so the sequence \left\{S_{{31}^n}\right(x_k\left)\right\} is strictly increasing and tends to S_3\left(0\right)=1-r. Therefore, there exists k>0 such that

S_{{31}^n}\left(x_k\right)\in (1-r-\epsilon ,1-r)\subset \overline{S_{{23}^m}\left(\mathrm{\Omega }\right)}.

Thus, there is S_{{31}^n}\left(\mathrm{\Omega }\right)\cap \overline{S_{{23}^m}\left(\mathrm{\Omega }\right)}\ne \varnothing. Since both S_{{31}^n}\left(\mathrm{\Omega }\right) and S_{{23}^n}\left(\mathrm{\Omega }\right) are open sets, there are

S_{{31}^n}\left(\mathrm{\Omega }\right)\cap S_{{23}^m}\left(\mathrm{\Omega }\right)\ne \varnothing .

From the definition of the index family in (3) and (4), it follows that for any \mathbf{v}\in \mathbf{H}\cap {\mathcal{V}}_{\mathbf{k}}, there exists an m (depending on v) such that \mathbf{u}\mathbf{=}\mathbf{(}{\mathbf{S}}_{{\mathbf{23}}^{\mathbf{m}}}\mathbf{,}\mathbf{k}\mathbf{)}\in \mathbf{\Omega }\left(\mathbf{v}\right).

Assume that \mathbf{v}\mathbf{=}\mathbf{(}{\mathbf{S}}_{{\mathbf{31}}^{\mathbf{n}}}\mathbf{,}\mathbf{k}\mathbf{)} and {\mathbf{v}}^{\prime }\mathbf{=}\mathbf{(}{\mathbf{S}}_{{\mathbf{31}}^{{\mathbf{n}}^{\prime }}}\mathbf{,}{\mathbf{k}}^{\prime }\mathbf{)}\in \mathbf{H} satisfy \mathbf{v}\sim {\mathbf{v}}^{\prime }, and let \mathbf{u}\mathbf{=}\mathbf{(}{\mathbf{S}}_{{\mathbf{23}}^{\mathbf{m}}}\mathbf{,}\mathbf{k}\mathbf{)}\in \mathbf{\Omega }\left(\mathbf{v}\right). Let \tau =S_{{\mathbf{v}}^{\prime }}\circ S_{\mathbf{v}}^{-1}. Then, by the condition of generalized finite-type (Definition 2.1(1)), there exists {\mathbf{u}}^{\prime }\in \mathbf{\Omega }\left({\mathbf{v}}^{\prime }\right) such that S_{{\mathbf{u}}^{\prime }}=\tau \circ S_{\mathbf{u}}. Note that

S_{{\mathbf{u}}^{\prime }}\left(1\right)=S_{{\mathbf{v}}^{\prime }}\circ S_{\mathbf{v}}^{-1}\circ S_{\mathbf{u}}\left(1\right)=S_{{\mathbf{v}}^{\prime }}\circ S_{\mathbf{v}}^{-1}(1-r)=S_{{\mathbf{v}}^{\prime }}\left(0\right)=1-r,

so there exists a positive integer m^{\prime } such that {\mathbf{u}}^{\prime }={23}^{m^{\prime }}. Now consider the compression ratio on both sides of the equation S_{\mathbf{u}}^{\prime }=\tau \circ S_{\mathbf{u}}, there is {\rho }_{{\mathbf{u}}^{\prime }}={\rho }_{{\mathbf{v}}^{\prime }}{\rho }_{\mathbf{v}}^{-1}{\rho }_{\mathbf{u}}, which leads to

r^{m^{\prime }+1}=r^{m+1}{\rho }^{n^{\prime }-n}.

Taking the logarithm of both sides and simplifying gives: \left(m^{\prime }-m\right)\mathrm{l}\mathrm{n}r=\left(n^{\prime }-n\right)\mathrm{l}\mathrm{n}\rho. The incommensurability of \mathrm{l}\mathrm{n}\rho and \mathrm{l}\mathrm{n}r implies that n=n^{\prime }, and \mathbf{v}\mathbf{=}{\mathbf{v}}^{\prime }. Therefore, any two distinct elements of H are not equivalent. Since H is an infinite set, it follows that \mathcal{V}/ is an infinite set. Therefore, for the invariant set ∈ and the index family \mathcal{M}, the generalized finite-type condition is not satisfied. This proves that Ω is not a basic set. □

The proof of Theorem 1.2 is shown as follows. Let [x] denote the greatest integer less than or equal to x, and \left\{x\right\}=x-\left[x\right] be the fractional part of x.

Proof of Theorem 1.2　Without loss of generality, let {\rho }_1=\mathrm{m}\mathrm{i}\mathrm{n}\{{\rho }_i:1\le i\le N\}. From the incommensurability of \mathrm{l}\mathrm{n}{\rho }_1,\mathrm{l}\mathrm{n}{\rho }_2,\dots ,\mathrm{l}\mathrm{n}{\rho }_N, it is known that there exists 2\le i\le N such that \frac{\mathrm{l}\mathrm{n}{\rho }_1}{\mathrm{l}\mathrm{n}{\rho }_i} is irrational. Let \mathrm{\Lambda } be the index set, and let H be the infinite set

H=\left\{\right(S_i,k):\text{There exists}{m}_k\text{such that}\mathbf{i}=i^{mk}\in {\mathrm{\Lambda }}_k,k\ge 1\}.

Assume there exist \mathbf{v}\mathbf{=}\left({\mathbf{S}}_{\mathbf{i}}\mathbf{,}\mathbf{k}\right)\mathbf{,}{\mathbf{v}}^{\prime }\mathbf{=}\mathbf{(}{\mathbf{S}}_{{\mathbf{i}}^{\prime }}\mathbf{,}{\mathbf{k}}^{\prime }\mathbf{)}\in \mathbf{H}, and \mathbf{v}\ne {\mathbf{v}}^{\prime } such that \mathbf{v}\sim {\mathbf{v}}^{\prime }. From the definition of H, it is known that k\ne k^{\prime }. According to the equivalence relation \sim, it is known that for any i^n\in \Sigma and positive integer l, the following equivalence holds:

(S_{\mathbf{v}}\circ S_{i^n},k+l)\in {\mathcal{V}}_{k+l}\text{when and only when}(S_{{\mathbf{v}}^{\prime }}\circ S_{i^n},k^{\prime }+l)\in {\mathcal{V}}_{k^{\prime }+l}.

From the definition of the index set \mathrm{\Lambda }, it is known that: (S_{\mathbf{v}}\circ S_{i^n},k+l)\in {\mathcal{V}}_{k+l} is equivalent to . Taking logarithms with base {\rho }_i, there is , where t=\frac{\mathrm{l}\mathrm{n}{\rho }_1}{\mathrm{l}\mathrm{n}{\rho }_i} is irrational. Therefore, there is \left[\left(k+l\right)t\right]=m_k+n-1. Similarly, (S_{{\mathbf{v}}^{\prime }}\circ S_{i^n},k^{\prime }+l)\in {\mathcal{V}}_{k^{\prime }+l} is equivalent to \left[\left(k^{\prime }+l\right)t\right]=m_{k^{\prime }}+n-1. From this, we obtain the following equation:

(5)\left[\left(k+l\right)t\right]-\left[\left(k^{\prime }+l\right)t\right]=m_k-m_{k^{\prime }},

which is a constant independent of l. Assume that \left\{kt\right\}>\left\{k^{\prime }t\right\}. Since t is irrational, {\left\{lt\right\}}_{l\ge 1} is uniformly distributed modulo 1. Hence, there exist l_1 and l_2 such that (see Fig.2):

Thus, there is

\left[\left(k+l\right)t\right]-\left[\left(k^{\prime }+l\right)t\right]=\{\begin{array}{ll}\left[kt\right]-\left[k^{\prime }t\right],& l=l_1;\\ \left[kt\right]-\left[k^{\prime }t\right]+1,& l=l_2.\end{array}

This contradicts Equation (5), which is a constant. Therefore, the assumption that \mathbf{v}\sim {\mathbf{v}}^{\prime } is false. Hence, no two elements of H are equivalent. Since H is an infinite set, it follows that the iteration function system does not satisfy the generalized finite-type condition with respect to the index set \mathrm{\Lambda }. □

4 A class of iterated function systems satisfying the generalized finite-type condition

In this section, the index set Σ is used to present a class of iterated function systems that satisfy the generalized finite-type condition and compute the Hausdorff dimension of the corresponding self-similar sets.

Example 4.1　Let N\ge 3, and let the iterated function system \left\{S_i\right(x)={\rho }_{i}x+d_i{\}}_{i=1}^N on \mathbb{R} be defined as shown in the first iteration of Fig.3, where the constants {\rho }_i and d_i satisfy the following relations:

(6){\rho }_1,{\rho }_N\in \left(0,1\right),{\rho }_{i+1}={\rho }_1^{1-i}{\rho }_N^i\in \left(0,1\right),1\le i\le N-2;

(7)d_1=0,d_N=1-{\rho }_N,d_{i+1}=\left(1-{\rho }_N\right)\sum _{j=1}^i {\rho }_j,1\le i\le N-2.

If

(8)\sum _{i=1}^N {\rho }_i-{\rho }_N\sum _{i=1}^{N-2} {\rho }_i\le 1,

then this iterated function system {\left\{S_i\right\}}_{i=1}^N satisfies the generalized finite-type condition.

Proof　First, direct calculation shows that S_1\left(0\right)=0, S_N\left(1\right)=1, and the fixed points of the other functions are between 0 and 1. Therefore, from Equation (2), it is known that the attractor of this iterated function system includes 0 and 1 and is a subset of the interval [0,1]. Given conditions (6) and (7), it is known that for all indices S_i\left(\right(0,1\left)\right)\subset \left(0,1\right), so \mathrm{\Omega }=\left(0,1\right) is an invariant set for this iterated function system. In light of Theorem 1.2, Σ is chosen as the index set for this iterated function system.

Since S_1\left(x\right)-S_N\left(x\right) is a linear function, its extreme values occur at the endpoints, with the maximum value being This implies that . Similarly, it is proved that for 1\le i\le N-1. From condition (6), it is known that {\rho }_{i+1}{\rho }_1={\rho }_i{\rho }_N. For 1\le i\le N-2, there is

(9)S_{iN}\left(x\right)={\rho }_i{\rho }_{N}x+\left(1-{\rho }_N\right){\rho }_i+d_i={\rho }_{i+1}{\rho }_{1}x+d_{i+1}=S_{\left(i+1\right)1}\left(x\right).

According to S_1\left(0\right)=0, S_N\left(1\right)=1, and Equation (8), there is

(10)

Direct calculation shows

(11)S_N\left(0\right)-S_{N-1}\left(1\right)=1-\sum _{i=1}^N {\rho }_i+{\rho }_N\sum _{i=1}^{N-2} {\rho }_i\ge 0.

The inequality holds due to condition (7). Therefore, the monotonicity of the functions S_i implies that:

(12)S_N\left(\mathrm{\Omega }\right)\cap S_i\left(\mathrm{\Omega }\right)=\varnothing \text{,}1\le i\le N-1.

Inequality (9) and inequality (10) suggest that , which implies that Next, consider the N-3 differences S_{i+2}\left(0\right)-S_i\left(1\right), for 1\le i\le N-3. In this case, since {\rho }_i{\rho }_N={\rho }_{i+1}{\rho }_1 and , there is

S_{i+2}\left(0\right)-S_i\left(1\right)=(1-{\rho }_N)({\rho }_i+{\rho }_{i+1})-{\rho }_i={\rho }_{i+1}-{\rho }_{i+1}{\rho }_N-{\rho }_{i+1}{\rho }_1>0.

That is (see the first iteration in Fig.3)

(13)S_{i+2}\left(\mathrm{\Omega }\right)\cap S_i\left(\mathrm{\Omega }\right)=\varnothing \text{,}1\le i\le N-3.

Now, with the help of Equations (11) and (12), the neighborhood types are calculated for this iterated function system. Let {\mathbf{v}}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}=(I,0) be the root vertex, and let its corresponding neighborhood type be {\mathcal{T}}_1. After one iteration, the root vertex generates N vertices

{\mathbf{v}}_1=(S_1,1),{\mathbf{v}}_2=(S_2,1),\dots ,{\mathbf{v}}_N=(S_N,1).

Equation (11) implies that \mathrm{\Omega }\left({\mathbf{v}}_N\right)=\left\{{\mathbf{v}}_N\right\}, so {\mathbf{v}}_N\sim {\mathbf{v}}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}. Equation (12) implies that:

\mathrm{\Omega }\left({\mathbf{v}}_1\right)=\left\{{\mathbf{v}}_1,{\mathbf{v}}_2\right\},\mathrm{\Omega }\left({\mathbf{v}}_{N-1}\right)=\{{\mathbf{v}}_{N-1},{\mathbf{v}}_{N-2}\},

\mathrm{\Omega }\left({\mathbf{v}}_i\right)=\left\{{\mathbf{v}}_i,{\mathbf{v}}_{i-1,}{\mathbf{v}}_{i+1,}\right\},3\le i\le N-2.

Let {\mathcal{T}}_2,\dots ,{\mathcal{T}}_N represent the neighborhood types corresponding to \left[{\mathbf{v}}_1\right],\dots ,\left[{\mathbf{v}}_{N-1}\right]. Therefore, there are directed edges from vertex {\mathcal{T}}_1 to vertex {\mathcal{T}}_i, with the compression ratio on the edge being {\rho }_{i-1} for 1\le i\le N, where {\rho }_0={\rho }_N. The vertex relations mentioned above can be simplified as

{\mathcal{T}}_1\to {\mathcal{T}}_2\left({\rho }_1\right)+{\mathcal{T}}_3\left({\rho }_2\right)+\cdots +{\mathcal{T}}_N\left({\rho }_{N-1}\right)+{\mathcal{T}}_1\left({\rho }_N\right).

Since the functions S_i satisfy Equation (8), that is, S_{iN}=S_{(i+1)1}, and by lexicographical order, N>1, we delete the descendants of each {\mathbf{v}}_i(1\le i\le N-2) from (S_{iN},2) (see the second iteration in Fig.3). Thus, for 1\le i\le N-2, the descendants of {\mathbf{v}}_i are

{\mathbf{v}}_{ij}=\left(S_{ij},2\right),1\le j\le N-1.

The descendants of {\mathbf{v}}_{N-1} are {\mathbf{v}}_{(N-1)j}=({\mathrm{S}}_{\left(N-1\right)j},2), 1\le j\le N. From Equation (11), it is known that \mathrm{\Omega }\left({\mathbf{v}}_{\left(N-1\right)N}\right)=\left\{{\mathbf{v}}_{(N-1)N}\right\}, so {\mathbf{v}}_{(N-1)N}{\mathbf{v}}_{\mathrm{r}\mathrm{o}\mathrm{o}\mathrm{t}}. From Equation (12), it is known that for 1\le i\le N-1,

\mathrm{\Omega }\left({\mathbf{v}}_{i,j}\right)=\{\begin{array}{ll}\{{\mathbf{v}}_{i,j},{\mathbf{v}}_{i,j+1}\},& j=1;\\ \{{\mathbf{v}}_{i,j},{\mathbf{v}}_{i,j-1},{\mathbf{v}}_{i,j+1}\},& 2\le j\le N-2;\\ \{{\mathbf{v}}_{i,j},{\mathbf{v}}_{i,j-1}\},& j=N-1.\end{array}

The neighborhood expressions mentioned above indicate that for 1\le i\le N-1, \left[{\mathbf{v}}_{ij}\right]={\mathcal{T}}_{j+1}, for 1\le j\le N-1. Thus, we obtain the following neighborhood-type iteration relation:

{\mathcal{T}}_j\to \{\begin{array}{ll}{\mathcal{T}}_2\left({\rho }_1\right)+{\mathcal{T}}_3\left({\rho }_2\right)+\cdots +{\mathcal{T}}_N\left({\rho }_{N-1}\right),& 2\le j\le N-1;\\ {\mathcal{T}}_1\left({\rho }_N\right)+{\mathcal{T}}_2\left({\rho }_1\right)+\cdots +{\mathcal{T}}_N\left({\rho }_{N-1}\right),& j=N.\end{array}

No new neighborhood types are generated, so \mathcal{V}/\sim =\{{\mathcal{T}}_i{\}}_{i=1}^N is a finite set. Therefore, this iterated function system satisfies the generalized finite-type condition. The corresponding association matrix is

A_s=\left(\begin{array}{ccccc}{\rho }_N^s& {\rho }_1^s& {\rho }_2^s& \cdots & {\rho }_{N-1}^s\\ 0& {\rho }_1^s& {\rho }_2^s& \cdots & {\rho }_{N-1}^s\\ 0& {\rho }_1^s& {\rho }_2^s& \cdots & {\rho }_{N-1}^s\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& {\rho }_1^s& {\rho }_2^s& \cdots & {\rho }_{N-1}^s\\ {\rho }_N^s& {\rho }_1^s& {\rho }_2^s& \cdots & {\rho }_{N-1}^s\end{array}\right).

By Theorem 2.1, {\mathrm{d}\mathrm{i}\mathrm{m}}_{H}F=s corresponds to the solution s for which the spectral radius of the matrix A_s equals 1, i.e.,

\sum _{i=1}^N\rho {}_i{}^s{-\rho }_N^S\sum _{i=1}^{N-2}\rho {}_i{}^s=1.

Compared to condition (7), the self-similar set F has a non-empty interior if and only if s=1, which means that (7) is an equality. □

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