Renewal Theorems and Their Application in Fractal Geometry

Abstract

A selection of probabilistic renewal theorems and renewal theorems in symbolic dynamics are presented. The selected renewal theorems have been widely applied. Here, we will show how they can be utilised to solve problems in fractal geometry with particular focus on counting problems and the question of Minkowski measurability. The fractal sets we consider include self-similar and self-conformal sets as well as limit sets of graph-directed systems consisting of similarities and conformal mappings.

Appendices

A Appendix: Symbolic Dynamics

Here, we provide some background from symbolic dynamics which we use in Sect. 4. Good references for the exposition below are [9, 38].

1.1 A.1 Sub-Shifts of Finite Type—Admissible Paths of a Random Walk Through Σ

Recall the following setting from Sect. 4. Σ = {1, …, M}, M ≥ 2 denotes the state space of the stochastic process $(X_n)_{n\in \mathbb N_0}$ and A denotes an irreducible (M × M)- incidence matrix of zeros and ones. The set of one-sided infinite admissible paths of $(X_n)_{n\in \mathbb N_0}$ through Σ consistent with A = (A _i,j)_{i,j ∈ Σ} is defined by

$$\displaystyle \begin{aligned} \Sigma_A:=\{x\in\Sigma^{\mathbb N}\mid A_{x_k,x_{k+1}}=1\ \forall\, k\in\mathbb N\}. \end{aligned}$$

Elements of Σ_A are interpreted as paths which describe the history of the process, supposing that the process has been going on forever.

The path of the process prior to the current time is described by σ(x), where σ: Σ_A → Σ_A denotes the (left) shift-map on Σ_A given by σ(ω ₁ ω ₂…):= ω ₂ ω ₃…. The set of admissible words of length$n\in \mathbb N$ is defined by

$$\displaystyle \begin{aligned} \Sigma_A^n:=\{\omega\in\Sigma^{n}\mid A_{\omega_k,\omega_{k+1}}=1\ \text{for}\ k\leq n-1\}. \end{aligned}$$

If ω has infinite length or length m ≥ n we define ω|_n:= ω ₁⋯ω _n to be the sub-path of length n. Further, [ω]:= {u ₁ u ₂⋯ ∈ Σ_A∣u _i = ω _i for i ≤ n} is the ω-cylinder set for $\omega \in \Sigma _A^n$.

1.2 A.2 (Hölder-)Continuous and (Non-)lattice Functions

Equip $\Sigma ^{\mathbb N}$ with the product topology of the discrete topologies on Σ and equip $\Sigma _A\subset \Sigma ^{\mathbb N}$ with the subspace topology, i. e. the weakest topology with respect to which the canonical projections onto the coordinates are continuous. Denote by $\mathcal C(\Sigma _A)$ the space of continuous real-valued functions on Σ_A. Elements of $\mathcal C(\Sigma _A)$ are called potential functions.

Definition A.1

For $\xi \in \mathcal C(\Sigma _A)$, α ∈ (0, 1) and $n\in \mathbb N_0$ define

$$\displaystyle \begin{aligned} \text{var}_n(\xi)&:=\sup\{\lvert \xi(\omega)-\xi(u)\rvert\mid \omega, u\in\Sigma_A\ \text{and}\ \omega_i=u_i\ \text{for all}\ i\in\{1,\ldots,n\}\},\\ \lvert \xi\rvert_{\alpha}&:=\sup_{n\geq 0}\frac{\text{var}_n(\xi)}{\alpha^{n}}\ \text{and}\\ \mathcal F_{\alpha}(\Sigma_A)&:=\{\xi\in\mathcal C(\Sigma_A)\mid \lvert \xi\rvert_{\alpha}<\infty\}. \end{aligned} $$

Elements of $\mathcal F_{\alpha }(\Sigma _A)$ are called α-Hölder continuous functions on Σ_A.

Definition A.2

Functions $\xi _1,\xi _2\in \mathcal {C}(\Sigma _A)$ are called co-homologous, if there exists $\psi \in \mathcal {C}(\Sigma _A)$ such that ξ ₁ − ξ ₂ = ψ − ψ ∘ σ. A function $\xi \in \mathcal {C}(\Sigma _A)$ is said to be lattice, if it is co-homologous to a function whose range is contained in a discrete subgroup of $\mathbb R$. Otherwise, we say that ξ is non-lattice.

1.3 A.3 Topological Pressure Function and Gibbs Measures

The topological pressure function$P\colon \mathcal C(\Sigma _A)\to \mathbb R$ is given by the well-defined limit

$$\displaystyle \begin{aligned} P(\xi):=\lim_{n\to\infty}n^{-1}\log\sum_{\omega\in\Sigma_A^n}\exp\sup_{u\in[\omega]}S_n\xi(u). \end{aligned} $$

(A.1)

Here, $S_n\xi : =\sum _{k=0}^{n-1}\xi \circ \sigma ^k$ denotes the n-th Birkhoff sum of ξ with $n\in \mathbb N$ and S ₀ ξ:= 0.

Proposition A.3

Let$\xi ,\eta \in \mathcal C(\Sigma _A)$be so that S _n ξ is strictly positive on Σ _A , for some$n\in \mathbb N$ . Then s↦P(η + sξ) is continuous, strictly monotonically increasing and convex with lim_s→−∞ P(η + sξ) = −∞ and lim_s→∞ P(η + sξ) = ∞. Hence, there is a unique$\delta \in \mathbb R$for which P(η − δξ) = 0.

A finite Borel measure μ on Σ_A is said to be a Gibbs measure for $\xi \in \mathcal C(\Sigma _A)$ if there exists a constant c > 0 such that

$$\displaystyle \begin{aligned} c^{-1} \leq \frac{\mu([\omega\vert_n])}{\exp(S_n \xi(\omega)-n\cdot P(\xi))} \leq c \end{aligned} $$

(A.2)

for every ω ∈ Σ_A and $n\in \mathbb N$.

1.4 A.4 Ruelle’s Perron-Frobenius Theorem

The Ruelle-Perron-Frobenius operator to a potential function $\xi \in \mathcal C(\Sigma _A)$ is defined by $\mathcal L_{\xi }\colon \mathcal C(\Sigma _A)\to \mathcal C(\Sigma _A)$,

$$\displaystyle \begin{aligned} \mathcal L_{\xi} \chi(x):= \sum_{y\in\Sigma_A:\sigma y=x} \chi(y) \mathrm{e}^{\xi(y)}. \end{aligned} $$

(A.3)

The dual operator acting on the set of Borel probability measures supported on Σ_A, is denoted by $\mathcal L_{\xi }^*$.

By [39, Thm. 2.16, Cor. 2.17] and [9, Theorem 1.7], for each $\xi \in \mathcal F_{\alpha }(\Sigma _A)$, some α ∈ (0, 1), there exists a unique Borel probability measure ν _ξ on Σ_A satisfying $\mathcal L_{\xi }^*\nu _{\xi }=\gamma _{\xi }\nu _{\xi }$ for some γ _ξ > 0. This equation uniquely determines γ _ξ, which satisfies $\gamma _{\xi }=\exp (P(\xi ))$ and which coincides with the spectral radius of $\mathcal L_{\xi }$. Further, there exists a unique strictly positive eigenfunction $h_{\xi }\in \mathcal {C}(\Sigma _A)$ satisfying $\mathcal L_{\xi } h_{\xi }=\gamma _{\xi } h_{\xi }$ and $\int h_{\xi }\mbox{d}\nu _{\xi }=1$. Define μ _ξ by dμ _ξ∕dν _ξ = h _ξ. This is the unique σ-invariant Gibbs measure for the potential function ξ.

Proposition A.3 and the relation $\gamma _{\xi }=\exp (P(\xi ))$ imply the following.

Proposition A.4

Let$\xi ,\eta \in \mathcal C(\Sigma _A)$be such that for some$n\in \mathbb N$the n-th Birkhoff sum S _n ξ of ξ is strictly positive on Σ _A . Then s↦γ _η+sξis continuous, strictly monotonically increasing, log-convex in$s\in \mathbb R$with lim_s→−∞ γ _η+sξ = 0 and satisfies lim_s→∞ γ _η+sξ = ∞. The unique$\delta \in \mathbb R$from PropositionA.3is the unique$\delta \in \mathbb R$for which γ _η−δξ = 1.

1.5 A.5 The Constants in Theorem 4.2

Using the notation from Sect. A.3 we can explicitly state the form of G(x) and G _x(t) occurring in the Renewal Theorem 4.2. For this, write ⌊t⌋ for the largest integer $k\in \mathbb Z$ satisfying k ≤ t, where $t\in \mathbb R$. Moreover, set {t}:= t −⌊t⌋∈ [0, 1). Notice, for $t\in \mathbb R$ positive, ⌊t⌋ is the integer part and {t} is the fractional part of t.

$$\displaystyle \begin{aligned} G(x) &= \frac{h_{\eta-\delta \xi}(x)}{\int \xi\mbox{d}\mu_{\eta-\delta \xi}}\int_{\Sigma_A} \chi(y)\int_{-\infty}^{\infty}\mathrm{e}^{-T\delta}g_y(T)\,\mbox{d}T\mbox{d}\nu_{\eta-\delta \xi}(y)\quad \text{and}\\ \widetilde{G}_x(t) &= \int_{\Sigma_A}\chi(y) \sum_{l=-\infty}^{\infty}\mathrm{e}^{-a l\delta} g_y\left(a l+a\left\{\tfrac{t+\psi(x)}{a}\right\}-\psi(y)\right) \mbox{d}\nu_{\eta-\delta\zeta}(y)\\ &\qquad \times \mathrm{e}^{-a\big{\{}\frac{t+\psi(x)}{a}\big{\}}\delta} \frac{a\mathrm{e}^{\delta\psi(x)}}{\int\zeta\mbox{d}\mu_{\eta-\delta\zeta}}\cdot h_{\eta-\delta\zeta}(x). \end{aligned} $$

B Appendix: Relation to the Probabilistic Renewal Theorems

The setting of Sect. 4 extends and unifies the setting of established renewal theorems. In brief: in the context of classical renewal theory for finitely supported measures (in particular of the key renewal theorem), η and ξ only depend on the first coordinate. When η and ξ only depend on the first two coordinates, we are in the setting of Markov renewal theory. If η is the constant zero-function and , where $\chi \in \mathcal F_{\alpha }(\Sigma _A)$ is non-negative, we are precisely in the setting of [28], where renewal theorems for counting measures in symbolic dynamics were developed, see Remark 4.3. The results of the infinite alphabet case obtained in [21] even yield the respective cases for general discrete measures.

In the following we expand upon the above and let $N\colon \Sigma _A\times \mathbb R\to \mathbb R$ denote the renewal function given in (4.2).

1.1 B.1 The Key Renewal Theorem for Finitely Supported Measures

The special case of Theorem 4.2 that N is independent of Σ_A gives the classical key renewal theorem for measures on [0, ∞) that are finitely supported:

N being independent of Σ_A can be achieved by the following assumptions. First, $\Sigma _A=\Sigma ^{\mathbb N}$ (i. e. full shift). Second, g _x = f is independent of $x\in \Sigma ^{\mathbb N}$ implying that equi d. R. i. of $\{t\mapsto \mathrm {e}^{-t\delta } \lvert g_x(t)\rvert \mid x\in \Sigma _A\}$ is equivalent to $z\colon \mathbb R\to \mathbb R$ with z(t):= e^−δt f(t) being absolutely d. R. i. Third, . Fourth and most importantly, ξ and η are constant on cylinder sets of length one. To emphasise local constancy, write s _u:= S _n ξ(u ₁⋯u _n ω) and $p_{u}: =\exp \left [S_n(\eta -\delta \xi )(u_1\cdots u_n\omega )\right ]$ for u = u ₁⋯u _n ∈ Σⁿ and $\omega =\omega _1\omega _2\cdots \in \Sigma ^{\mathbb N}$. Setting Z(t):= e^−δt N(t) we obtain that

$$\displaystyle \begin{aligned} Z(t)=\sum_{n=0}^{\infty}\sum_{\omega\in\Sigma^n}z(t-s_{\omega})p_{\omega} \quad \text{and}\quad Z(t) =\sum_{i=1}^M Z(t - s_i)p_i+ z(t), \end{aligned} $$

(B.1)

for $t\in \mathbb R$. Notice, the latter equation of (B.1) is the classical renewal equation (3.4). The assumption S _n ξ > 0 for some $n\in \mathbb N$ implies s _i > 0 for all i ∈ Σ. Thus, the distribution F which assigns mass p _i to s _i is concentrated on (0, ∞). On the other hand, any vector (s ₁, …, s _M) with s ₁, …, s _M > 0 determines a strictly positive function $\xi \in \mathcal F_{\alpha }(\Sigma ^{\mathbb N})$ via $\xi (\omega _1\omega _2\cdots ): = s_{\omega _1}$. Furthermore, in the setting of Theorem 4.2, (p ₁, …, p _M) is a probability vector with p _i ∈ (0, 1) since

$$\displaystyle \begin{aligned} 0 =P(\eta-\delta\xi) =\lim_{n\to\infty}n^{-1}\log\bigg{(}\sum_{i\in\Sigma}p_i\bigg{)}^n =\log\sum_{i\in\Sigma}p_i \end{aligned}$$

by Proposition A.3. Thus, F is a probability distribution. On the other hand, any probability vector (p ₁, …, p _M) with p ₁, …, p _M ∈ (0, 1) determines $\eta \in \mathcal F_{\alpha }(\Sigma ^{\mathbb N})$ via $\eta (\omega _1\omega _2\cdots ): =\log (p_{\omega _1}\mathrm {e}^{\delta s_{\omega _1}})$.

Consequently, Theorem 4.2 provides the asymptotic behaviour of Z under the assumptions that (p ₁, …, p _M) is a probability vector and that s ₁, …, s _M > 0. In order to present the asymptotic term in a common form, observe that $\mathcal L_{\eta -\delta \xi }\mathbf {1}=\mathbf {1}(x)$ for any $x\in \Sigma ^{\mathbb N}$, where . Thus,

$$\displaystyle \begin{aligned} h_{\eta-\delta\xi}=\mathbf{1} \qquad \text{and} \qquad \mu_{\eta-\delta\xi}([i])=\nu_{\eta-\delta\xi}([i])= p_i, \end{aligned}$$

where the last equality follows by considering the dual operator of $\mathcal L_{\eta -\delta \xi }$. If ξ is lattice then the range of ξ itself lies in a discrete subgroup of $\mathbb R$: If there exist $\zeta ,\psi \in \mathcal C(\Sigma ^{\mathbb N})$ with ξ − ζ = ψ − ψ ∘ σ and $\zeta (\Sigma ^{\mathbb N})\subset a\mathbb Z$ for some a > 0, then ξ and ζ need to coincide on $\{\omega \in \Sigma ^{\mathbb N}\mid \omega =\sigma \omega \}$. As every cylinder set of length one contains a periodic word of period one the claim follows. Hence, we can choose ζ = ξ and ψ to be the constant zero-function. We deduced the key renewal theorem, Theorem 3.4 for finitely supported measures on [0, ∞) and f ≥ 0. In exactly the same way [21, Thm. 3.1] yields the key renewal theorem for discrete measures.

1.2 B.2 Relation to Markov Renewal Theorems

Suppose that we are in the setting of Sect. 4.

If we assume that η and ξ are constant on cylinder sets of length two, then the point process with inter-arrival times W ₀, W ₁, … becomes a Markov random walk: To see this, define $\widetilde {\eta },\widetilde {\xi }\colon \Sigma _A^2\to \mathbb R$ by $\widetilde {\eta }(ij): =\eta (ij\omega )$ and $\widetilde {\xi }(ij): =\xi (ij\omega )$ for any ω ∈ Σ_A for which ijω ∈ Σ_A. Then

$$\displaystyle \begin{aligned} \mathbb P(X_1=i\mid X_0X_{-1}\cdots=x) = \mathrm{e}^{\eta(ix)} &= \mathrm{e}^{\widetilde{\eta}(ix_1)} =\mathbb P(X_1=i\mid X_0=x_1). \end{aligned} $$

Thus, $(X_n)_{n\in \mathbb Z}$ is a Markov chain. Further, $W_n=\xi (X_{n+1}X_nX_{n-1}\cdots ) =\widetilde {\xi }(X_{n+1}X_n)$ implies that the inter-arrival times W ₀, W ₁, … are Markov dependent on $(X_n)_{n\in \mathbb Z}$. Applying Theorem 4.2 to such Markov random walks gives the Markov renewal theorem presented in Theorem 3.9. In order to state its conclusions in the form of Theorem 3.9 we present several simplifications and conversions in the following. Set

and define $F: =(\widetilde {F}_{ij})_{i,j\in \Sigma }$ to be the matrix with entries . Then, F is irreducible if and only if A is irreducible. Moreover, $\widetilde {F}_{ij}$ is a distribution function of a discrete measure. Thus, ξ is lattice if and only if $\widetilde {F}_{ij}$ is lattice for all i, j. For $s\in \mathbb R$ and i, j ∈ Σ we have

$$\displaystyle \begin{aligned} B_{i,j}(s) &:=\int\mathrm{e}^{-s T}\widetilde{F}_{i,j}(\mbox{d}T) =\begin{cases} \exp(\widetilde{\eta}(ji)-s\widetilde{\xi}(ji))&:\ ji\in\Sigma_A^2\\ 0&:\ \text{otherwise}. \end{cases} \end{aligned} $$

Setting B(s):= (B _ij(s))_{i,j ∈ Σ} we see that the action of B(−s) on vectors coincides with the action of the Ruelle-Perron-Frobenius operator $\mathcal L_{\eta +s\xi }$ on functions $g\colon \Sigma _A\to \mathbb R$ which are constant on cylinder sets of length one. That is, setting $\widetilde {g}_i: = g(ix)$, for x ∈ Σ_A with ix ∈ Σ_A, gives

$$\displaystyle \begin{aligned} \mathcal L_{\eta+s\xi}g(ix) =\sum_{j\in\Sigma,\,ji\in\Sigma_A^2}\mathrm{e}^{\widetilde{\eta}(ji)+s\widetilde{\xi}(ji)}\widetilde{g}_j =\sum_{j\in\Sigma} B_{ij}(-s)\widetilde{g}_j =(B(-s)\widetilde{g})_i. \end{aligned}$$

By the Perron-Frobenius theorem for matrices there is a unique s for which B(s) has spectral radius one. By the above this value coincides with the unique s for which $\mathcal L_{\eta -s\xi }$ has spectral radius one, which we denoted by δ in Proposition A.4. Similarly, h _η−δξ is constant on cylinder sets of length one. Thus, setting h _i:= h _η−δξ(ix) for x ∈ Σ_A with ix ∈ Σ_A we obtain a vector (h _i)_{i ∈ Σ} with strictly positive entries which satisfies B(δ)h = h, since

$$\displaystyle \begin{aligned} (B(\delta)h)_i =\mathcal L_{\eta-\delta\xi}h_{\eta-\delta\xi}(ix) =h_{\eta-\delta\xi}(ix) =h_i. \end{aligned}$$

Moreover, the vector ν given by ν _i:= ν _η−δξ([i]) satisfies ν _i > 0 for all i ∈ Σ and νB(δ) = ν, since $\mathcal L^{*}_{\eta -\delta \xi }\nu _{\eta -\delta \xi }=\nu _{\eta -\delta \xi }$. By the Perron-Frobenius theorem h and ν are unique with these properties. Additionally assuming and that f _x only depends on the first letter of x ∈ Σ_A it follows that N(t, x) only depends on the first letter of x. Thus, for i ∈ Σ write N(t, i):= N(t, ix) with x ∈ Σ_A for which ix ∈ Σ_A. Now, the renewal equation becomes

$$\displaystyle \begin{aligned} N(t,i) &=\sum_{j\in\Sigma,\ ji\in\Sigma_A^2} N(t-\widetilde{\xi}(ji),j)\mathrm{e}^{\widetilde{\eta}(ji)} + f_i(t)\\ &=\sum_{j\in\Sigma} \int_{-\infty}^{\infty}N(t- u,j)\widetilde{F}_{i,j}(\mbox{d}u)+ f_i(t), \end{aligned} $$

(B.2)

for i ∈ Σ, where f _i(t):= f _ix(t) for x ∈ Σ_A with ix ∈ Σ_A, compare (3.8). Using the above in conjunction with the constants provided in Sect. A.5 thus yields Theorem 3.9.