Hausdorff dimension of the graphs of the classical Weierstrass functions

Abstract

We show that the graph of the classical Weierstrass function \(\sum _{n=0}^\infty \lambda ^n \cos (2\pi b^n x)\) has Hausdorff dimension \(2+\log \lambda /\log b\), for every integer \(b\ge 2\) and every \(\lambda \in (1/b,1)\). Replacing \(\cos (2\pi x)\) by a general non-constant \(C^2\) periodic function, we obtain the same result under a further assumption that \(\lambda b\) is close to 1.

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Acknowledgements

I would like to thank D. Feng, W. Huang and J. Wu for drawing my attention to the recent work [2]. I would also like to thank H. Ruan and Y. Wang for reading carefully a first version of the manuscript and pointing out a number of errors.

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Correspondence to Weixiao Shen.

Appendix: A proof of Ledrappier’s theorem

Appendix: A proof of Ledrappier’s theorem

This appendix is devoted to a proof of Ledrappier’s theorem under a further assumption that \(\phi '\) is continuous (for simplicity). The proof is motivated by the original proof given in [10] and also the recent paper [9].

Let \(b\ge 2\) be an integer and \(\lambda \in (1/b,1)\) and let \(f(x)=\sum _{n=0}^\infty \lambda ^n \phi (b^n x)\). We use z to denote a point in \(\mathbb {R}^2\) and B(zr) denote the open ball in \(\mathbb {R}^2\) centered at z and of radius r. We assume that \(\dim (m_x)=1\) holds for Lebesgue a.e. \(x\in [0,1)\), which means for Lebesgue a.e. \(x\in [0,1)\) and \(\mathbb {P}\)-a.e. \(\mathbf u \in \mathcal {A}^{\mathbb {Z}^+}\),

$$\begin{aligned} \lim _{r\rightarrow 0} \frac{\log {\mathbb {P}}\left( \{\mathbf{v }: |S(x,\mathbf{u })-S(x,\mathbf{v })|\le r\}\right) }{\log r}=1. \end{aligned}$$

Let \(\mu \) be the pushforward of the Lebesgue measure on [0, 1) under the map \(x\mapsto (x, f(x))\). Let

$$\begin{aligned} \underline{d}(\mu , z)=\liminf _{r\rightarrow 0}\frac{\log \mu (B(z,r))}{\log r} \end{aligned}$$

and

$$\begin{aligned} \underline{D}=\text {essinf}\,\, \underline{d} (\mu , z). \end{aligned}$$

We shall prove that

$$\begin{aligned} \underline{D}\ge D=2+\frac{\log \lambda }{\log b}. \end{aligned}$$

This is enough to conclude that the Hausdorff dimension of the graph of f is D. Indeed, by the mass distribution principle, it implies that the Hausdorff dimension is at least D. On the other hand, it is easy to check that f is a \(C^{2-D}\) function which implies that the Hausdorff dimension is at most D (see for example Theorem 8.1 of [5]).

Telescope

Define \(\Phi : [0,1)\times \mathbb {R}\rightarrow [0,1)\times \mathbb {R}\) as

$$\begin{aligned} \Phi (x,y)=(bx \mod 1, (y-\phi (x))/\lambda ). \end{aligned}$$

Define \(G: [0,1)\times \mathbb {R}\times \mathcal {A}^{\mathbb {Z}^+}\rightarrow [0,1)\times \mathcal {A}^{\mathbb {Z}^+}\) as

$$\begin{aligned} G(x,y, \mathbf u )=(\Phi (x,y), u_0\mathbf u ), \text { if } bx\in [u_0, u_0+1). \end{aligned}$$

The graph of f is an invariant repeller of the expanding map \(\Phi \). We shall use neighborhoods bounded by unstable manifolds. For each \(z_0=(x_0,y_0)\in \mathbb {R}^2\) and \(\mathbf u \in \mathcal {A}^{\mathbb {Z}^+}\), let \(\ell _{z_0,\mathbf u }(x)\) denote the unique solution of the initial value problem:

$$\begin{aligned} y'=-\gamma S(x,\mathbf u ), y(x_0)=y_0. \end{aligned}$$

These curves are strong unstable manifolds of \(\Phi \) and they satisfy the following property: for \(z=(x,y),z_0=(x_0,y_0)\in [u_0/b, (u_0+1)/b)\times \mathbb {R}\), \(u_0\in \mathcal {A}\),

$$\begin{aligned} \Phi (x, \ell _{z_0, \mathbf u }(x))=(bx-u_0, \ell _{\Phi (z_0), u_0\mathbf u }(bx-u_0)). \end{aligned}$$

For \(z_0=(x_0,y_0)\in [0,1)\times \mathbb {R}\), \(\mathbf u \in \mathcal {A}^{\mathbb {Z}^+}\) and \(\delta _1,\delta _2>0\), let

$$\begin{aligned} Q(z_0, \mathbf u ,\delta _1,\delta _2)=\{(x,y): x\in [0,1), |x-x_0|\le \delta _1, |y-\ell _{z_0,\mathbf u }(x)|\le \delta _2\}. \end{aligned}$$

The following observation was taken from [9].

Lemma 5.17

(Telescope) Let \(\{(z_i,\mathbf u _i)\}_{i=0}^n\) be a G-orbit and let \(x_i\) denote the first coordinate of \(z_i\). For any \(\delta _1,\delta _2>0\), if \(\delta _1\le x_n<1-\delta _1\), then

$$\begin{aligned} \mu (Q(z_0,\mathbf u , \delta _1 b^{-n}, \delta _2 \lambda ^n))=b^{-n} \mu (Q(z_n,\mathbf u _n, \delta _1,\delta _2)). \end{aligned}$$

Proof

Let \(J_i=[x_i-\delta _1 b^{i-n}, x_i+\delta _1 b^{i-n}]\), \(Q_i=Q(z_i,\mathbf u _i, \delta _1 b^{i-n},\delta _2 \lambda ^{n-i})\) and let \(E_i=\{x\in J_i: (x, f(x))\in Q_i\}\). Then \(\mu (Q_i)=|E_i|\). Under the assumption \(\delta _1\le x_n<1-\delta _1\), \(Q_0\) is mapped onto \(Q_n\) diffeomorphically by \(\Phi ^n\). Thus \(J_0\) is mapped onto \(J_n\) and \(E_0\) is mapped onto \(E_n\) diffeomorphically by the linear map \(x\mapsto b^n x\). Thus \(|E_0|=b^{-n}|E_n|\).

\(\square \)

A version of Marstrand’s estimate

Fix a constant \(t\in (1/(1+\alpha ),1)\).

Proposition 5.18

For \(\mu \times \mathbb {P}\)-a.e. \((z_0,\mathbf u )\),

$$\begin{aligned} \liminf _{r\rightarrow 0}\frac{\log \mu (Q(z_0,\mathbf u , r^t, r))}{\log r} \ge 1+t(\underline{D}-1). \end{aligned}$$
(6.1)

Proof

It suffices to prove that for each \(\xi >0\) and \(\eta >0\), there is a subset \(\Sigma \) of \([0,1)\times \mathbb {R}\times \mathcal {A}^{\mathbb {Z}^+}\) with \((\mu \times \mathbb {P})(\Sigma )>1-\eta \) such that

$$\begin{aligned} \liminf _{r\rightarrow 0} \frac{\log \mu (Q(z_0,\mathbf u , r^t, r))}{\log r} \ge 1+t(\underline{D}-1)-3\xi \end{aligned}$$
(6.2)

holds for all \((z_0,\mathbf u )\in \Sigma \). By Egoroff’s theorem, we can choose \(\Sigma \) with \((\mu \times \mathbb {P})(\Sigma )>1-\eta \) for which there is \(r_0>0\) such that for each \((z_0,\mathbf u )\in \Sigma \),

  1. (S1)

    \(\mathbb {P}\left( \left\{ \mathbf v : |S(x_0,\mathbf u )-S(x_0,\mathbf v )|\le r\right\} \right) \le r^{1-\xi }\) for each \(0< r\le r_0\), where \(x_0\) is the first coordinate of \(z_0\);

  2. (S2)

    \(\mu (B(z_0, r))\le r^{\underline{D}-\xi }\) for each \(0<r\le r_0\).

In the following we shall prove that for \(r>0\) small enough,

$$\begin{aligned} \int _\mathbf{u : (z_0,\mathbf u )\in \Sigma } \mu (Q(z_0, \mathbf u , r^t, r))d\mathbb {P}\le r^{1+t(\underline{D}-1)-2\xi }, \end{aligned}$$
(6.3)

holds for every \(z_0\in [0,1)\times \mathbb {R}\). This is enough to conclude the proof. Indeed, let \(\tau \in (0,1)\) be an arbitrary constant. Then by (6.3), there is N such that for \(n>N\),

$$\begin{aligned} \mathbb {P}\left( \left\{ \mathbf u : (z_0,\mathbf u )\in \Sigma , \mu (Q(z_0,\mathbf u ,\tau ^{nt}, \tau ^n))> (\tau ^n)^{1+t(\underline{D}-1)-3\xi }\right\} \right) \le \tau ^{n\xi } \end{aligned}$$

holds for every \(z_0\in [0,1)\times \mathbb {R}\). By Fubini’s theorem, this implies that

$$\begin{aligned} \mu \times \mathbb {P}\left( \left\{ (z_0,\mathbf u )\in \Sigma : \mu (Q(z_0,\mathbf u ,\tau ^{nt}, \tau ^n))> (\tau ^n)^{1+t(\underline{D}-1)-3\xi }\right\} \right) \le \tau ^{n\xi }. \end{aligned}$$

By Borel–Cantelli, it follows that for almost every \((z_0,\mathbf u ) \in \Sigma \), \(\mu (Q(z_0,\mathbf u ,\tau ^{nt},\tau ^n))\le (\tau ^n)^{1+t(\underline{D}-1)-3\xi }\) holds for all n large enough. The inequality (6.2) follows.

Let us now prove (6.3). We first prove

Claim Provided that \(r>0\) is small enough, for every \(z_0, z\in [0,1)\times \mathbb {R}\), we have

$$\begin{aligned} \mathbb {P}(\left\{ \mathbf u : (z_0,\mathbf u )\in \Sigma , z\in Q(z_0,\mathbf u , r^t, r)\right\} )\le C_1 \left( \frac{r}{|z-z_0|}\right) ^{1-2\xi }, \end{aligned}$$
(6.4)

where \(C_1>0\) is a constant.

To prove this claim, let \(z=(x,y)\), \(z_0=(x_0,y_0)\) and \(h(x)=\ell _{z_0,\mathbf u }(x)\). Then h(x) is \(C^{1+\alpha }\) with uniformly bounded norm. So

$$\begin{aligned} |y-y_0+\gamma S(x_0,\mathbf u ) (x-x_0)|\le & {} |y-h(x)|+|h(x)-h(x_0)-h'(x_0)(x-x_0)|\\\le & {} r+|\int _{x_0}^x (h'(s)-h'(x_0))ds |\le r+ C r^{t(1+\alpha )}< 2r, \end{aligned}$$

provided that r is small enough. Thus

$$\begin{aligned} \{S(x_0,\mathbf u ):(z_0,\mathbf u )\in \Sigma , z\in Q(z_0, \mathbf u , r^t, r)\} \end{aligned}$$

is contained in an interval of length \(2r/(\gamma |x-x_0|)\). Since

$$\begin{aligned} |z-z_0|\le |x-x_0|+|y-y_0|\le (1+\gamma |S(x_0,\mathbf u )|)|x-x_0|+2r, \end{aligned}$$

and \(|S(x,\mathbf u )|\) is uniformly bounded, the inequality (6.4) follows from the property (S1). Note that if \(2r/(\gamma |x-x_0|)>r_0\), then \(r/|z-z_0|\) is bounded away from zero, so (6.4) holds for sufficiently large \(C_1\), since the left hand side of this inequality does not exceed one.

We continue the proof of (6.3). Note that there is a constant \(C_2>0\) such that for every \(r>0\) and any \(z_0\in [0,1)\times \mathbb {R}\),

$$\begin{aligned} \bigcup _\mathbf{u \in \mathcal {A}^{\mathbb {Z}^+}} Q(z_0, \mathbf u , r^t, r)\subset B(z_0, C_2r^t). \end{aligned}$$

Of course we may assume there is \(\mathbf u \) such that \((z_0,\mathbf u )\in \Sigma \). Thus for \(R>0\) small enough, we may apply (S2) and obtain

$$\begin{aligned} \int _{B(z_0,R)} \frac{d\mu (z)}{|z-z_0|^{1-2\xi }}&=\sum _{n=0}^\infty \int _{e^{-n-1}R\le |z-z_0|< e^{-n} R} \frac{d\mu (z)}{|z-z_0|^{1-2\xi }}\\&\le \sum _{n=0}^\infty \frac{\mu (B(z_0, e^{-n}R))}{(e^{-n-1}R)^{1-2\xi }}\le \sum _{n=0}^\infty \frac{(e^{-n}R)^{\underline{D}-\xi }}{(e^{-n-1}R)^{1-2\xi }}\\&=C(\xi ) R^{\underline{D}-1+\xi }, \end{aligned}$$

where \(C(\xi )\) is a constant depending on \(\xi \) and \(\underline{D}\). By Fubini’s theorem,

$$\begin{aligned}&\int _\mathbf{u : (z_0,\mathbf u )\in \Sigma } \mu (Q(z_0, \mathbf u , r^t, r) d\mathbb {P}(\mathbf u )\\&\quad = \int _\mathbf{u : (z_0,\mathbf u )\in \Sigma }\int _{\mathbb {R}^2} 1_{Q(z_0, \mathbf u , r^t, r)}(z) d\mu (z) d\mathbb {P}(\mathbf u )\\&\quad = \int _{B(z_0, C_2r^t)} \mathbb {P}\left( \left\{ \mathbf u : (z_0,\mathbf u )\in \Sigma , z\in Q(z_0,\mathbf u , r^t, r)\right\} \right) d\mu (z)\\&\quad \le C_1 r^{1-2\xi } \int _{B(z_0, C_2 r^t)} \frac{d\mu (z)}{|z-z_0|^{1-2\xi }} \\&\quad \le C' r^{1+t(\underline{D}-1)-(2-t)\xi }< r^{1+t(\underline{D}-1)-2\xi }, \end{aligned}$$

provided that r is small enough. \(\square \)

We are ready to complete the proof of Ledrappier’s theorem. For any \(\xi >0\), \(\eta >0\), by Proposition 5.18 and Egroff’s theorem, we can pick up a subset \(\Sigma \) of \(\mathbb {R}^2\times \mathcal {A}^{\mathbb {Z}^+}\) and a constant \(r_*>0\) such that \((\mu \times \mathbb {P})(\Sigma )>1-3\eta \) and such that for each \((z,\mathbf u )\in \Sigma \),

$$\begin{aligned} \mu (Q(z,\mathbf u , r^t, r))\le r^{1+t(\underline{D}-1)-\xi } \text { for each } 0<r<r_*. \end{aligned}$$

We may further assume that \(\Sigma \subset [\eta , 1-\eta ]\times \mathbb {R}\times \mathcal {A}^{\mathbb {Z}^+}\).

Note that \(\mu \times \mathbb {P}\) is an ergodic invariant measure for the map G. By Birkhorff’s Ergodic Theorem, for almost every \((z_0,\mathbf u _0)\), there is an increasing sequence \(\{n_k\}_{k=1}^\infty \) of positive integers such that \(G^{n_k}(z_0,\mathbf u _0)\in \Sigma \) and

$$\begin{aligned} \liminf _{k\rightarrow \infty } n_k/n_{k+1} > 1-3\eta . \end{aligned}$$
(5.5)

For each \(n=1,2,\ldots \), put \(\delta _n=\gamma ^{nt/(1-t)}b^{-n}\), \(r_n=\gamma ^{n/(1-t)}\), so that

$$\begin{aligned} r_n=\delta _n \lambda ^{-n},\quad \text { and }\quad r_n^t=\delta _n b^n. \end{aligned}$$

Let us prove that for k sufficiently large,

$$\begin{aligned} \frac{\log \mu (Q(z_0,\mathbf u _0, \delta _{n_k}, \delta _{n_k}))}{\log \delta _{n_k}}\ge \underline{D}+(D-\underline{D})A_1-A_2\xi , \end{aligned}$$
(5.6)

where \(A_1, A_2\) are positive constants depending only on \(\lambda \) and b.

Indeed, by Lemma 5.17, for k large enough,

$$\begin{aligned} \mu (Q(z_0,\mathbf u _0, \delta _{n_k}, \delta _{n_k}))=\frac{\mu (Q(G^{n_k}(z_0,\mathbf u _0),r_{n_k}^t, r_{n_k}))}{b^{n_k}} \le \frac{r_{n_k}^{1+t(\underline{D}-1)-\xi }}{b^{n_k}}. \end{aligned}$$

Using definition of \(r_n\) and \(\delta _n\), this gives us

$$\begin{aligned} \mu (Q(z_0,\mathbf u _0,\delta _{n_k},\delta _{n_k}))\le \delta _{n_k}^{\underline{D}}\times (b^{-n_k})^{D-\underline{D}} r_{n_k}^{-\xi }, \end{aligned}$$

Thus (5.6) holds with \(A_1=\log b/(\log b+t\log \gamma ^{-1}/(1-t))\) and \(A_2=\log \gamma /(t\log \gamma +(1-t)\log b^{-1})\).

By (5.5), for each n large enough, there is k such that \((1-3\eta )n_k<n_{k-1}<n\le n_k\). It follows that

$$\begin{aligned} \liminf _{n\rightarrow \infty }\frac{\log \mu (Q(z_0,\mathbf u _0,\delta _n, \delta _n))}{\log \delta _n}\ge (1-3\eta ) (\underline{D}+(D-\underline{D})A_1-A_2\xi ). \end{aligned}$$

Since \(\ell _{x_0,\mathbf u _0}\) is a smooth curve, there exists \(\kappa \in (0,1)\) such that \(Q(z_0,\mathbf u _0,\delta _k,\delta _k)\) contains \(B(z_0, \kappa \delta _k)\) for each k. Therefore,

$$\begin{aligned} \underline{d}(\mu , z_0)=\liminf _{n\rightarrow \infty }\frac{\log \mu (Q(z_0,\mathbf u _0,\delta _n, \delta _n))}{\log \delta _n}\ge (1-3\eta ) (\underline{D}+(D-\underline{D})A_1-A_2\xi ). \end{aligned}$$

Since this estimate holds for \(\mu \)-a.e. \(z_0\), we obtain

$$\begin{aligned} \underline{D}\ge (1-3\eta )\left( \underline{D}+ A_1(D-\underline{D})-A_2\xi \right) . \end{aligned}$$

As \(\xi , \eta \) can be chosen arbitrarily small, we conclude

$$\begin{aligned} \underline{D}\ge \underline{D}+ A_1(D-\underline{D}), \end{aligned}$$

which means \(\underline{D}\ge D\), as desired.

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Shen, W. Hausdorff dimension of the graphs of the classical Weierstrass functions. Math. Z. 289, 223–266 (2018). https://doi.org/10.1007/s00209-017-1949-1

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  • Secondary 37D20
  • 28A80
  • 37C45