On multifractal formalism for self-similar measures with overlaps

Abstract

Let $\mu$ be a self-similar measure generated by an IFS $\varPhi =\{\phi _i\}_{i=1}^\ell$ of similarities on ${{\mathbb {R}}}^d$ ($d\ge 1$). When $\varPhi$ is dimensional regular (see Definition 1.1), we give an explicit formula for the $L^q$-spectrum $\tau _\mu (q)$ of $\mu$ over [0, 1], and show that $\tau _\mu$ is differentiable over (0, 1] and the multifractal formalism holds for $\mu$ at any $\alpha \in [\tau _\mu '(1),\tau _\mu '(0+)]$. We also verify the validity of the multifractal formalism of $\mu$ over $[\tau _\mu '(\infty ),\tau _\mu '(0+)]$ for two new classes of overlapping algebraic IFSs by showing that the asymptotically weak separation condition holds. For one of them, the proof appeals to the recent result of Shmerkin (Ann. Math. (2) 189(2):319–391, 2019) on the $L^q$-spectrum of self-similar measures.

Appendix: the proof of Theorem 1.2 and some extensions

Throughout this section, let $\varPhi =\{\phi _i\}_{i=1}^\ell$ be a dimensional regular IFS of similarities on ${{\mathbb {R}}}^d$ with ratios $r_1,\ldots , r_\ell$, and let $\mathbf{p}=(p_1, \ldots , p_\ell )$ be a probability vector with strictly positive entries. Let $\mu$ be the self-similar measure generated by $\varPhi$ and $\mathbf{p}$.

We begin with a simple lemma.

Lemma 7.1

Let $\eta$ be the self similar measure generated by $\varPhi$ and a probability vector $\tilde{\mathbf{p}}=({\tilde{p}}_1,\ldots , {\tilde{p}}_\ell )$. Then for $\eta$-a.e. $z\in {{\mathbb {R}}}^d$,

$$\begin{aligned} {\overline{d}}(\mu ,z)\le \frac{\sum _{i=1}^\ell {\tilde{p}}_i\log p_i}{\sum _{i=1}^\ell {\tilde{p}}_i\log r_i}. \end{aligned}$$

Proof

Let $(\varSigma , \sigma )$ denote the one-sided full shift over the alphabet $\{1,\ldots , \ell \}$, and $\pi :\varSigma \rightarrow {{\mathbb {R}}}^d$ the canonical coding map associated with $\varPhi$, i.e.

$$\begin{aligned} \pi x=\lim _{n\rightarrow \infty } \phi _{x_1}\circ \cdots \circ \phi _{x_n}(0),\quad x=(x_i)_{i=1}^\infty \in \varSigma . \end{aligned}$$

Let m be the infinite Bernoulli product measure on $\varSigma$ generated by the weight $({\tilde{p}}_1,\ldots , {\tilde{p}}_\ell )$, i.e. $m([x_1\ldots x_n])={\tilde{p}}_{x_1}\cdots {\tilde{p}}_{x_n}$ for each cylinder $[x_1\ldots x_n]$. Then $\eta =m\circ \pi ^{-1}$ [21].

Take a large $R>0$ such that the attractor of $\varPhi$ is contained in the ball B(0, R). For any $x=(x_i)_{i=1}^\infty \in \varSigma$ and $n\in {{\mathbb {N}}}$, since $\phi _{x_1}\circ \cdots \circ \phi _{x_n} (B(0, R))$ is a ball of radius $r_1\cdots r_n R$ which contains the point $\pi x$, we have

$$\begin{aligned} \phi _{x_1}\circ \cdots \circ \phi _{x_n} (B(0, R))\subset B(\pi x, 2r_1\cdots r_n R), \end{aligned}$$

so by the self-similarity of $\mu$,

$$\begin{aligned} \mu ( B(\pi x, 2r_1\cdots r_n R))\ge & {} p_{x_1}\cdots p_{x_n} \mu \left( (\phi _{x_1}\circ \cdots \circ \phi _{x_n})^{-1}B(\pi x, 2r_1\cdots r_n R) \right) ,\\\ge & {} p_{x_1}\cdots p_{x_n} \mu \left( B(0, R) \right) \\= & {} p_{x_1}\cdots p_{x_n}. \end{aligned}$$

It follows that

$$\begin{aligned} {\overline{d}}(\mu , \pi x)\le \limsup _{n\rightarrow \infty }\frac{\log (p_{x_1}\cdots p_{x_n})}{\log (r_{x_1}\cdots r_{x_n})},\quad x\in \varSigma . \end{aligned}$$

Applying Birkhoff’s ergodic theorem to the righthand side of the above inequality yields that

$$\begin{aligned} {\overline{d}}(\mu , \pi x)\le \frac{\sum _{i=1}^\ell {\tilde{p}}_i\log p_i}{\sum _{i=1}^\ell {\tilde{p}}_i\log r_i}\quad \text{ for } m\text{-a.e. } x\in \varSigma . \end{aligned}$$

This concludes the desired result since $\eta =m\circ \pi ^{-1}$. $\square$

To prove Theorem 1.2, we also need the following well-known result in multifractal analysis. For a proof, see e.g. [23, Proposition 2.5(iv)].

Lemma 7.2

Let $\nu$ be a compactly supported Borel probability measure on ${{\mathbb {R}}}^d$. Then for any $\beta \in {{\mathbb {R}}}$ and $q\ge 0$,

$$\begin{aligned} \dim _H\{z\in {{\mathbb {R}}}^d:\; {\underline{d}}(\nu , z)\le \beta \}\le \beta q-\tau _\nu (q). \end{aligned}$$

Now we are ready to prove Theorem 1.2.

Proof of Theorem 1.2

Recall that T satisfies the equation $\sum _{i=1}^\ell p_i^q r_i^{-T(q)}=1$. According to a general result of Falconer (see [5, Theorem 6.2]),

$$\begin{aligned} \tau _\mu (q)\ge T(q),\quad q\in [0,1]. \end{aligned}$$

(7.1)

This inequality will be used later.

Taking the derivative of T at q gives

$$\begin{aligned} T'(q)=\frac{\sum _{i=1}^\ell p_i^q r_i^{-T(q)}\log p_i}{\sum _{i=1}^\ell p_i^q r_i^{-T(q)}\log r_i} \end{aligned}$$

(7.2)

and in particular,

$$\begin{aligned} T'(1)=\frac{\sum _{i=1}^\ell p_i\log p_i}{\sum _{i=1}^\ell p_i\log r_i}=\dim _S\mu . \end{aligned}$$

(7.3)

Since $\varPhi$ is assumed to be dimensional regular, we have

$$\begin{aligned} \dim _H\mu =\min \{d, T'(1)\}. \end{aligned}$$

(7.4)

Meanwhile, as a general result on self-similar measures, one always has

$$\begin{aligned} \tau _\mu '(1+)=\dim _H\mu . \end{aligned}$$

(7.5)

Indeed, it is known that for every self-similar measure $\mu$, $\tau _\mu '(1+)$ equals the entropy dimension of $\mu$ (see [35, Theorem 5.1 and Remark 5.2]); but since $\mu$ is exact dimensional [10], its entropy dimension and Hausdorff dimension coincide [40].

In what follows we prove the theorem by considering 3 different cases: (i) $T'(1)\ge d$; (ii) $T'(1)<d$ and $T(0)\ge -d$; (iii) $T'(1)<d$ and $T(0)< -d$.

Case (i): $T'(1)\ge d$. Let K denote the attractor of $\varPhi$. By (7.4) we have $\dim _H\mu =d$, which implies that $\dim _B K=d$. By [25, Theorem 1.1],

$$\begin{aligned} \tau _\mu '(1-)\ge \dim _H\mu =d. \end{aligned}$$

However since $\tau _\mu (0)=-\dim _B K=-d$ and $\tau _\mu (1)=0$, the above inequality and the concavity of $\tau _\mu$ force that

$$\begin{aligned} \tau _\mu (q)=d(q-1) \quad \text{ for } \text{ every } q\in [0,1]. \end{aligned}$$

This proves part (1a) of the theorem. To show part (2) of the theorem, first notice that $\tau _\mu$ is differentiable on (0, 1) and moreover, $\tau _\mu '(1-)=d=\dim _H\mu$, so by (7.5) $\tau _\mu$ is differentiable at 1 as well. Next we analyze the multifractal structure of $\mu$. Since $\mu$ is exact dimensional with $\dim _H\mu =d$, we have

$$\begin{aligned} \dim _H\{z\in {{\mathbb {R}}}^d:\; d(\mu , z)=d\}\ge \dim _H\mu =d; \end{aligned}$$

on the other hand by Lemma 7.2,

$$\begin{aligned}\dim _H\{z:\; d(\mu , z)=d\}\le d\cdot 1-\tau _\mu (1)=d,\end{aligned}$$

so we have

$$\begin{aligned}\dim _H\{z\in {{\mathbb {R}}}^d:\; d(\mu , z)=d\}=d=dq-\tau _\mu (q)\end{aligned}$$

for $q\in [0,1]$. This verifies part (2) of the theorem.

Case (ii): $T'(1)<d$ and $T(0)\ge -d$. To prove the conclusions of the theorem, we need to show that for each $q\in [0,1]$,

$$\begin{aligned} \tau _\mu (q)= & {} T(q)\quad \text{ and } \end{aligned}$$

(7.6)

$$\begin{aligned} \dim _H\{z\in {{\mathbb {R}}}^d:\; d(\mu , z)=T'(q)\}= & {} T'(q)q-T(q). \end{aligned}$$

(7.7)

To this end, fix $q\in [0,1]$. By the concavity of T, we have

$$\begin{aligned} T'(q)q-T(q)\le T'(0)\cdot 0-T(0)\le d. \end{aligned}$$

Set $\tilde{\mathbf{p}}=({\tilde{p}}_1,\ldots , {\tilde{p}}_\ell )$ with ${\tilde{p}}_i=p_i^qr_i^{-T(q)}$, and let $\eta$ be the self-similar measure generated by $\varPhi$ and $\tilde{\mathbf{p}}$. Applying (7.3) to $\eta$ yields

$$\begin{aligned} \dim _S\eta =\frac{\sum _{i=1}^\ell {\tilde{p}}_i \log {\tilde{p}}_i}{\sum _{i=1}^\ell {\tilde{p}}_i\log r_i}=\frac{\sum _{i=1}^\ell p_i^qr_i^{-T(q)}(q\log p_i-T(q)\log r_i)}{\sum _{i=1}^\ell p_i^qr_i^{-T(q)}\log r_i}=T'(q)q-T(q)\le d, \end{aligned}$$

so by the dimensional regularity of $\varPhi$, $\dim _H\eta =\dim _S\eta =T'(q)q-T(q)$. Applying Lemma 7.1 and (7.2), we have

$$\begin{aligned} {\overline{d}}(\mu , z) \le \frac{\sum _{i=1}^\ell {\tilde{p}}_i\log p_i}{\sum _{i=1}^\ell {\tilde{p}}_i\log r_i}= \frac{\sum _{i=1}^\ell p_i^qr_i^{-T(q)}\log p_i}{\sum _{i=1}^\ell p_i^qr_i^{-T(q)}\log r_i}=T'(q) \quad \text{ for } \eta \text{-a.e. } z. \end{aligned}$$

(7.8)

Take a strictly increasing sequence $(\alpha _n)$ of real numbers so that $\lim _n\alpha _n=T'(q)$. If $q>0$, then by Lemma 7.2,

$$\begin{aligned} \dim _H\{z\in {{\mathbb {R}}}^d:\; {\underline{d}}(\mu ,z)\le \alpha _n\}\le & {} \alpha _n q-\tau _\mu (q)\\< & {} T'(q) q-T(q), \end{aligned}$$

where we have used (7.1) in the last inequality. Otherwise if $q=0$, then by Lemma 7.2, for every n and sufficiently small $\epsilon _n>0$ so that $T(\epsilon _n)-T(0)>\alpha _n\epsilon _n$, we have

$$\begin{aligned} \dim _H\{z\in {{\mathbb {R}}}^d:\; {\underline{d}}(\mu ,z)\le \alpha _n\}\le & {} \alpha _n \epsilon _n-\tau _\mu (\epsilon _n)\\\le & {} \alpha _n \epsilon _n-T(\epsilon _n)\\< & {} -T(0)\\= & {} T'(q)q-T(q), \end{aligned}$$

producing the same inequality. Since $\eta$ is exact dimensional with dimension $T'(q) q-T(q)$, the above inequality implies that

$$\begin{aligned} \eta \{z\in {{\mathbb {R}}}^d:\; {\underline{d}}(\mu ,z)\le \alpha _n\}=0 \quad \text{ for } \text{ each } n\in {{\mathbb {N}}}, \end{aligned}$$

and so

$$\begin{aligned} \eta \{z\in {{\mathbb {R}}}^d:\; {\underline{d}}(\mu ,z)<T'(q)\}=0. \end{aligned}$$

This together with (7.8) yields $d(\mu ,z)=T'(q)$ for $\eta$-a.e. z. Hence

$$\begin{aligned} \dim _H\{z\in {{\mathbb {R}}}^d:\; d(\mu ,z)=T'(q)\}\ge \dim _H\eta =T'(q) q-T(q). \end{aligned}$$

Meanwhile by Lemma 7.2,

$$\begin{aligned}\dim _H\{z\in {{\mathbb {R}}}^d:\; d(\mu ,z)=T'(q)\}\le T'(q) q-\tau _\mu (q)\le T'(q) q-T(q).\end{aligned}$$

These two equations imply immediately (7.7) and (7.6). From (7.6) we see that $\tau _\mu$ is differentiable on (0, 1) and $\tau _\mu '(1-)=T'(1)$. By (7.4)–(7.5), $\tau _\mu '(1+)=\min \{d, T'(1)\}=T'(1)$, so $\tau _\mu$ is also differentiable at 1.

Case (iii): $T'(1)<d$ and $T(0)< -d$. Set

$$\begin{aligned}{\tilde{q}}=\inf \{q\in (0,1):\; T'(q)q-T(q)\le d\}.\end{aligned}$$

By the concavity of T, the function $g(q):=T'(q)q-T(q)$ is decreasing in q with $g(0)=-T(0)>d$ and $g(1)=T'(1)<d$. Hence ${\tilde{q}}\in (0, 1)$.

If $q\in [{\tilde{q}},1]$, then $T'(q) q-T(q)\le d$; the same argument as that in Case (ii) shows that (7.7) and (7.6) hold for $q\in [{\tilde{q}},1]$ and that $\tau _\mu$ is differentiable on $({\tilde{q}},1]$.

Finally assume that $q\in [0, {\tilde{q}})$. By the definition of ${\tilde{q}}$ we have

$$\begin{aligned}T'({\tilde{q}}) {\tilde{q}}-T({\tilde{q}})=d=\tau _\mu '({\tilde{q}}+) {\tilde{q}}-\tau _\mu ({\tilde{q}}).\end{aligned}$$

It follows that the tangent line of the graph of $\tau _\mu$ at the point $({\tilde{q}}, \tau _\mu ({\tilde{q}})$) crosses at the y-axis at $(0, -d)$. Since $\tau _\mu$ is concave and $\tau _\mu (0)\ge -d$, $\tau _\mu$ must take the linear expression of the statement over $[0, {\tilde{q}})$. Notice that the slope of this line segment equals $(d+T({\tilde{q}}))/{\tilde{q}}=T'({\tilde{q}})=\tau _\mu '({\tilde{q}}+)$, it follows that $\tau _\mu$ is differentiable on $(0,{\tilde{q}}]$ and

$$\begin{aligned} {[}\tau '_\mu (1), \tau '_\mu (0+)]=\{T'(q):\; q\in [{\tilde{q}}, 1]\}. \end{aligned}$$

Since (7.7) holds for $q\in [{\tilde{q}}, 1]$, we obtain the desired result on the multifractal structure of $\mu$. This completes the proof of the theorem. $\square$

Remark 7.3

 

• (i) For a dimensional regular IFS $\varPhi$ on ${{\mathbb {R}}}^d$ and a given $q>1$ with $T'(q)q-T(q)\le d$, if we have known that $\tau _\mu (q)=T(q)$ in advance, then following the same argument as in Case (ii) of the proof of Theorem 1.2, we obtain that

  $$\begin{aligned} \dim _HE_\mu (T'(q))=T'(q)q-T(q). \end{aligned}$$

  (7.9)

• (ii) Applying the above argument to the case when $\varPhi$ is an IFS on ${{\mathbb {R}}}$ satisfying the ESC, we conclude that (7.9) holds for every $q\in (1,\infty )$ so that $T(q)\le q-1$. Indeed for such q, by Theorem 3.1(i), $\tau _\mu (q)=\min \{q-1, T(q)\}=T(q)$; meanwhile by the concavity of T,

  $$\begin{aligned}T'(q)q-T(q)\le \frac{T(q)-T(1)}{q-1}q-T(q)=\frac{T(q)}{q-1}\le 1.\end{aligned}$$

In the end of this section, we present several extensions of Theorem 1.2.

First rather than self-similar measures, there is an analogue of Theorem 1.2 for the projections of quasi-Bernoulli measures under a stronger assumption on $\varPhi$. To be more precise, assume that $\varPhi =\{\phi _i=r_iU_i+a_i\}_{i=1}^\ell$ is an IFS of similarities on ${{\mathbb {R}}}^d$ such that for any $\sigma$-invariant ergodic measure $\eta$ on $\varSigma =\{1,\ldots , \ell \}^{{\mathbb {N}}}$, the Hausdorff dimension of the projection of $\eta$ under the coding map $\pi$ satisfies

$$\begin{aligned} \dim _H\eta \circ \pi ^{-1}=\min \left\{ d,\; \frac{h(\eta )}{\lambda (\eta )}\right\} , \end{aligned}$$

where $h(\eta )$ stands for the measure theoretic entropy of $\eta$ and $\lambda (\eta ):=\sum _{i=1}^\ell \log (1/r_i)\eta ([i])$. For instance, this assumption holds for all IFS on ${{\mathbb {R}}}$ satisfying the ESC [37]. Suppose that m is a quasi Bernoulli measure on $\varSigma$, in the sense that there exists a constant $C>0$ such that

$$\begin{aligned} C^{-1}m([I])m([J])\le m([IJ])\le C m([I])m([J]) \text{ for } \text{ all } I, J\in \bigcup _{n=1}^\infty \{1,\ldots , \ell \}^n. \end{aligned}$$

Define $f, \phi _n\in C(\varSigma )$, $n\in {{\mathbb {N}}}$, by

$$\begin{aligned} f(x)=-\log r_{x_1} \text{ and } \phi _n(x)=\log m([x_1\ldots x_n]) \; \text{ for } x=(x_k)_{k=1}^\infty . \end{aligned}$$

For $q\in {{\mathbb {R}}}$, let D(q) be the unique value so that

$$\begin{aligned} P\left( (D(q) S_nf+q \phi _n)_{n=1}^\infty \right) =0, \end{aligned}$$

where $S_nf=f+f\circ \sigma +\ldots +f\circ \sigma ^{n-1}$ and $P(\cdot )$ stands for the sub-additive pressure (see e.g. [1, Section 2.1]). Then by adapting the proof of Theorem 1.2 and using some ideas of the proof of [1, Theorem 1.3 (i)], we can show that the conclusions of Theorem 1.2 (in which T is replaced by D) still holds for $\mu =m\circ \pi ^{-1}$.

Secondly, Theorem 1.2 and Remark 7.3 also extend to the convolutions of certain self-similar measures on ${{\mathbb {R}}}$. To see it, let $\mu _i$, $i=1,2$, be the self-similar measures on ${{\mathbb {R}}}$ generated by $\varPhi _i=\{\rho _ix+a_{i,j}\}_{j=1}^{\ell _i}$ and $\mathbf{p}_i=(p_{i,j})_{j=1}^{\ell _i}$. Assume that both $\varPhi _1$ and $\varPhi _2$ satisfy the OSC and $\log \rho _1/\log \rho _2\not \in {{\mathbb {Q}}}$. Then according to [34, Theorem 2.2], for every $q>1$, the $L^q$-spectrum of $\mu _1*\mu _2$ is given by

$$\begin{aligned}\tau _{\mu _1*\mu _2}(q)=\min \{q-1, T_1(q)+T_2(q)\},\end{aligned}$$

where $T_i(q):=\log (\sum _{j=1}^{\ell _i} p_{i,j}^q) /\log \rho _i$. Moreover, by [18, Theorem 1.4], $\mu _1*\mu _2$ is exact dimensional with dimension equal to $\min \{1, T_1'(1)+T_2'(1)\}$. Using the these results and modifying the proof of Theorem 1.2 correspondingly, one can show that the conclusions of Theorem 1.2 still holds (in which we replace T by $T_1+T_2$), furthermore, the multifractal formalism holds for $\mu _1*\mu _2$ for those $q>1$ so that $(T_1'(q)+T_2'(q))q-(T_1(q)+T_2(q))\le 1$.

Finally we remark that Theorem 1.2 and Remark 7.3 extend to a class of dynamically driven self-similar measures on ${{\mathbb {R}}}$ considered in [34]. To see it, let $(\mu _x)_{x\in X}$ be the dynamically driven self-similar measures generated by a pleasant model $(X, \mathbf{T}, \varDelta , \lambda )$ satisfying the ESC (see [34, Section 1.5] for the involved definitions). Under mild assumptions (see [34, Theorem 1.11]), Shmerkin showed that the $L^q$-spectrum of $\mu _x$, for each $x\in X$ and $q>1$, is given by $\tau _{\mu _x}(q)=\min \{q-1, T(q)\}$, where

$$\begin{aligned} T(q):=\frac{\int \log \Vert \varDelta (x)\Vert _q^q\; d{{\mathbb {P}}}(x)}{\log \lambda }. \end{aligned}$$

Using similar arguments, we can show that under the same assumptions as in [34, Theorem 1.11], the conclusions of Theorem 1.2 (in which $\mu$ is replaced by $\mu _x$) hold for each $x\in X$; moreover for each $q>1$ with $T'(q)q-T(q)\le 1$, we have $\dim _HE_{\mu _x}(T'(q))=T'(q)q-T(q)$ for every x.