Hausdorff dimension and the dynamics of diffeomorphisms

A proof of the Hilbert-Smith conjecture for a free Lipschitz action is given. The proof is elementary in the sense that it does not rely on Yang’s theorem about the cohomology dimension of the orbit space of thep-acid action. The result turns out to be true for the class of spaces of finite Hausdorff volume, which is considerably wider than Riemannian manifolds. As a corollary to the Lipschitz version of the Hilbert-Smith conjecture, the theorem asserting that the diffeomorphism group of a finite-dimensional manifold has no small subgroups is obtained. Translated fromMatermaticheskie Zametki, Vol. 65, No. 3, pp. 457–463, March, 1999.     

Transfinite Hausdorff dimension

Making extensive use of small transfinite topological dimension trind, we ascribe to every metric space X an ordinal number (or −1 or Ω) tHD(X), and we call it the transfinite Hausdorff dimension of X. This ordinal number shares many common features with Hausdorff dimension. It is monotone with respect to subspaces, it is invariant under bi-Lipschitz maps (but in general not under homeomorphisms), in fact like Hausdorff dimension, it does not increase under Lipschitz maps, and it also satisfies the intermediate dimension property (Theorem 2.7). The primary goal of transfinite Hausdorff dimension is to classify metric spaces with infinite Hausdorff dimension. Indeed, if tHD(X)?ω₀, then HD(X)=+∞. We prove that tHD(X)?ω₁ for every separable metric space X, and, as our main theorem, we show that for every ordinal number α<ω₁ there exists a compact metric space Xα (a subspace of the Hilbert space l₂) with tHD(Xα)=α and which is a topological Cantor set, thus of topological dimension 0. In our proof we develop a metric version of Smirnov topological spaces and we establish several properties of transfinite Hausdorff dimension, including its relations with classical Hausdorff dimension.     

A new fractal dimension: The topological Hausdorff dimension

We introduce a new concept of dimension for metric spaces, the so-called topological Hausdorff dimension. It is defined by a very natural combination of the definitions of the topological dimension and the Hausdorff dimension. The value of the topological Hausdorff dimension is always between the topological dimension and the Hausdorff dimension, in particular, this new dimension is a non-trivial lower estimate for the Hausdorff dimension.     

关于测度的Hausdorff维数的局部化

主要研究测度的豪斯道夫维数的局部化.通过定义一个测度μx,ε,从而给出dim·Hμ在点x的局部化维数dim·Hμ(x).进而得到局部化维数dim·Hμ(x)与dim·Hμ之间的关系,并得到了一个等式关系.     

The Hausdorff dimension of graphs of Weierstrass functions

The Weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples of functions whose graph is a fractal. Though there is a simple formula for the Hausdorff dimension of the graph which is widely accepted, it has not been rigorously proved to hold. We prove that if arbitrary phases are included in each term of the summation for the Weierstrass function, the Hausdorff dimension of the graph of the function has the conjectured value for almost every sequence of phases. The argument extends to a much wider class of Weierstrass-like functions.

     

Hausdorff dimension and distance sets

According to a result of K. Falconer (1985), the setD(A)={|x−y|;x, y ∈A} of distances for a Souslin setA of ℝ ⁿ has positive 1-dimensional measure provided the Hausdorff dimension ofA is larger than (n+1)/2.* We give an improvement of this statement in dimensionsn=2,n=3. The method is based on the fine theory of Fourier restriction phenomena to spheres. Variants of it permit further improvements which we don’t plan to describe here. This research originated from some discussions with P. Mattila on the subject. dimA >n/2 would be the optimal result forn ≥ 2.     

Hausdorff dimension and asymptotic cycles

We carry out a multifractal analysis for the asymptotic cycles for compact Riemann surfaces of genus . This describes the set of unit tangent vectors for which the associated orbit has a given asymptotic cycle in homology.

     

Cremona transformations and diffeomorphisms of surfaces

We show that the action of Cremona transformations on the real points of quadrics exhibits the full complexity of the diffeomorphisms of the sphere, the torus, and of all non-orientable surfaces. The main result says that if X is rational, then Aut(X), the group of algebraic automorphisms, is dense in Diff(X), the group of self-diffeomorphisms of X.     

The Hausdorff dimension of a class of recurrent sets

In this paper we obtain a lower bound for the Hausdorff dimension of recurrent sets and, in a general setting, we show that a conjecture of Dekking [F.M. Dekking, Recurrent sets: A fractal formalism, Report 82-32, Technische Hogeschool, Delft, 1982] holds.     

Univoque bases and Hausdorff dimension

Given a positive integer M and a real number \(q >1\), a q -expansion of a real number x is a sequence \((c_i)=c_1c_2\ldots \) with \((c_i) \in \{0,\ldots ,M\}^\infty \) such that

$$\begin{aligned} x=\sum _{i=1}^{\infty } c_iq^{-i}. \end{aligned}$$

It is well known that if \(q \in (1,M+1]\), then each \(x \in I_q:=\left[ 0,M/(q-1)\right] \) has a q-expansion. Let \(\mathcal {U}=\mathcal {U}(M)\) be the set of univoque bases \(q>1\) for which 1 has a unique q-expansion. The main object of this paper is to provide new characterizations of \(\mathcal {U}\) and to show that the Hausdorff dimension of the set of numbers \(x \in I_q\) with a unique q-expansion changes the most if q “crosses” a univoque base. Denote by \(\mathcal {B}_2=\mathcal {B}_2(M)\) the set of \(q \in (1,M+1]\) such that there exist numbers having precisely two distinct q-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (J Number Theory 129:741–754, 2009) and prove that

$$\begin{aligned} \dim _H(\mathcal {B}_2\cap (q',q'+\delta ))>0\quad \text {for any}\quad \delta >0, \end{aligned}$$

where \(q'=q'(M)\) is the Komornik–Loreti constant.

     

Prime-representing functions and Hausdorff dimension

Acta Mathematica Hungarica - Let $$c \geq 2$$ be any fixed real number. Matomäki [4] inverstigated the set of $$A &gt; 1$$ such that the integer part of $$ A^{c^k}$$ is a prime number for...     

The Hausdorff dimension of visible sets of planar continua

For a compact set and a point , we define the visible part of from to be the set

(Here denotes the closed line segment joining to .)

In this paper, we use energies to show that if is a compact connected set of Hausdorff dimension greater than one, then for (Lebesgue) almost every point , the Hausdorff dimension of is strictly less than the Hausdorff dimension of . In fact, for almost every ,

We also give an estimate of the Hausdorff dimension of those points where the visible set has dimension greater than for some .

     

On the Hausdorff dimension of fibres

It is well known that there are planar sets of Hausdorff dimension greater than 1 which are graphs of functions, i.e., all their vertical fibres consist of 1 point. We show this phenomenon does not occur for sets constructed in a certain “regular” fashion. Specifically, we consider sets obtained by partitioning a square into 4 subsquares, discarding 1 of them and repeating this on each of the 3 remaining squares, etc.; then almost all vertical fibres of a set so obtained have Hausdorff dimension at least 1/2. Sharp bounds on the dimensions of sets of exceptional fibres are presented. Partially supported by a grant from the Landau Centre for Mathematical Analysis.     

Hausdorff dimension of homogeneous perfect sets

Summary We introduce the notion of homogeneous perfect sets as a generalization of Cantor type sets and determine their exact dimension based on the length of their fundamental intervals and the gaps between them. Some earlier results regarding the dimension of Cantor type sets are shown to be special cases of our main theorem.     

The Hausdorff dimension of multivariate operator-self-similar Gaussian random fields

Let $\{X\left(t\right):t\in {\mathbb{R}}^d\}$ be a multivariate operator-self-similar random field with values in ${\mathbb{R}}^m$. Such fields were introduced in [22] and satisfy the scaling property $\{X\left(c^{E}t\right):t\in {\mathbb{R}}^d\}\stackrel{\mathrm{d}}{=}\{c^{D}X\left(t\right):t\in {\mathbb{R}}^d\}$ for all $c>0$, where $E$ is a $d\times d$ real matrix and $D$ is an $m\times m$ real matrix. We solve an open problem in [22] by calculating the Hausdorff dimension of the range and graph of a trajectory over the unit cube $K={[0,1]}^d$ in the Gaussian case. In particular, we enlighten the property that the Hausdorff dimension is determined by the real parts of the eigenvalues of $E$ and $D$ as well as the multiplicity of the eigenvalues of $E$ and $D$.