In this paper, we obtain the explicit value of the Hausdorff dimension of the graphs of the classical Weierstrass functions, by proving absolute continuity of the SRB measures of the associated solenoidal attractors.
In Real Analysis, the classical Weierstrass function is
Note that the Weierstrass functions have the form
where is a -periodic -function.
Weierstrass (1872) and Hardy (1916) were interested in because they are concrete examples of continuous but nowhere differentiable functions.
Remark 1 The graph of tends to be a “fractal object” because is self-similar in the sense that
We will come back to this point later.
Remark 2 is a -function for all . In fact, for all , we have
whenever , i.e., .
The study of the graphs of as fractal sets started with the work of Besicovitch-Ursell in 1937.
Remark 3 The Hausdorff dimension of the graph of a -function is
Indeed, for each , the Hölder continuity condition
leads us to the “natural cover” of by the family of rectangles given by
Nevertheless, a direct calculation with the family does not give us an appropriate bound on . In fact, since for each , we have
for . Because is arbitrary, we deduce that . Of course, this bound is certainly suboptimal for (because we know that anyway).Fortunately, we can refine the covering by taking into account that each rectangle tends to be more vertical than horizontal (i.e., its height is usually larger than its width ). More precisely, we can divide each rectangle into squares, say
such that every square has diameter . In this way, we obtain a covering of such that
for . Since is arbitrary, we conclude the desired bound
A long-standing conjecture about the fractal geometry of is:
Conjecture (Mandelbrot 1977): The Hausdorff dimension of the graph of is
Remark 4 In view of remarks 2 and 3, the whole point of Mandelbrot’s conjecture is to establish the lower bound
Remark 5 The analog of Mandelbrot conjecture for the box and packing dimensions is known to be true: see, e.g., these papers here and here).
In a recent paper (see here), Shen proved the following result:
Theorem 1 (Shen) For any integer and for all , the Mandelbrot conjecture is true, i.e.,
Remark 6 The techniques employed by Shen also allow him to show that given a -periodic, non-constant, function, and given integer, there exists such that
for all .
Remark 7 A previous important result towards Mandelbrot’s conjecture was obtained by Barańsky-Barány-Romanowska (in 2014): they proved that for all integer, there exists such that
for all .
The remainder of this post is dedicated to give some ideas of Shen’s proof of Theorem1 by discussing the particular case when and is large.
2. Ledrappier’s dynamical approach
If is an integer, then the self-similar function (cf. Remark 1) is also -periodic, i.e., for all . In particular, if is an integer, then is an invariant repeller for the endomorphism given by
This dynamical characterization of led Ledrappier to the following criterion for the validity of Mandelbrot’s conjecture when is an integer.
Denote by the alphabet . The unstable manifolds of through have slopes of the form
where , , , and
In this context, the push-forwards of the Bernoulli measure on (induced by the discrete measure assigning weight to each letter of the alphabet ) play the role of conditional measures along vertical fibers of the unique Sinai-Ruelle-Bowen (SRB) measure of the expanding endomorphism ,
where and . In plain terms, this means that
where is the unique -invariant probability measure which is absolutely continuous along unstable manifolds (see Tsujii’s paper).
As it was shown by Ledrappier in 1992, the fractal geometry of the conditional measures have important consequences for the fractal geometry of the graph :
Theorem 2 (Ledrappier) Suppose that for Lebesgue almost every the conditional measures have dimension , i.e.,
Then, the graph has Hausdorff dimension
Remark 8 Very roughly speaking, the proof of Ledrappier theorem goes as follows. By Remark 4, it suffices to prove that . By Frostman lemma, we need to construct a Borel measure supported on such that
where . Finally, the main point is that the assumptions in Ledrappier theorem allow to prove that the measure given by the lift to of the Lebesgue measure on via the map satisfies
An interesting consequence of Ledrappier theorem and the equation 1 is the following criterion for Mandelbrot’s conjecture:
Corollary 3 If is absolutely continuous with respect to the Lebesgue measure , then
Proof: By (1), the absolute continuity of implies that is absolutely continuous with respect to for Lebesgue almost every .
Since for almost every implies that for almost every , the desired corollary now follows from Ledrappier’s theorem.
3. Tsujii’s theorem
The relevance of Corollary 3 is explained by the fact that Tsujii found an explicittransversality condition implying the absolute continuity of .
More precisely, Tsujii firstly introduced the following definition:
Next, Tsujii proves the following result:
Theorem 5 (Tsujii) If there exists integer such that , then
Remark 9 Intuitively, Tsujii’s theorem says the following. The transversality condition implies that the majority of strong unstable manifolds are mutually transverse, so that they almost fill a small neighborhood of some point (see the figure below extracted from this paper of Tsujii). Since the SRB measure is absolutely continuous along strong unstable manifolds, the fact that the ‘s almost fill implies that becomes “comparable” to the restriction of the Lebesgue measure to .
Remark 10 In this setting, Barańsky-Barány-Romanowska obtained their main result by showing that, for adequate choices of the parameters and , one has . Indeed, once we know that , since , they can apply Tsujii’s theorem and Ledrappier’s theorem (or rather Corollary 3) to derive the validity of Mandelbrot’s conjecture for certain parameters and .
For the sake of exposition, we will give just a flavor of the proof of Theorem 1 by sketching the derivation of the following result:
Proposition 6 Let . If and is sufficiently large, then
In particular, by Corollary 3 and Tsujii’s theorem, if and is sufficiently large, then Mandelbrot’s conjecture is valid, i.e.,
Remark 11 The proof of Theorem 1 in full generality (i.e., for integer and ) requires the introduction of a modified version of Tsujii’s transversality condition: roughly speaking, Shen defines a function (inspired from Peter-Paul inequality) and he proves
- (a) a variant of Proposition 6: if integer and , then for some integer ;
- (b) a variant of Tsujii’s theorem: if for some integer , then .
See Sections 2, 3, 4 and 5 of Shen’s paper for more details.
We start the (sketch of) proof of Proposition 6 by recalling that the slopes of unstable manifolds are given by
for , , so that
Remark 12 Since , the series defining converges faster than the series defining .
By studying the first term of the expansion of and (while treating the remaining terms as a “small error term”), it is possible to show that if , then
(cf. Lemma 3.2 in Shen’s paper).
Using these estimates, we can find an upper bound for as follows. Take with , and let be such that distinct elements listed in such a way that
for all , where .
From (3), we see that
for all .
it follows that
Now, we observe that
for large enough. Indeed, this happens because
- if ;
- if ;
- as , and as (here we used ).
By combining (4) and (5), we deduce that
for all .
Since , the previous estimate implies that
Thus, it follows from our assumptions (, large) that
This completes the (sketch of) proof of Proposition 6 (and our discussion of Shen’s talk).