Search

We say that $E$ is a microset of the compact set $K\subset \mathbb{R}^d$ if there exist sequences $\lambda_n\geq 1$ and $u_n\in \mathbb{R}^d$ such that $(\lambda_n K + u_n ) \cap [0,1]^d$ converges to $E$ in the Hausdorff metric, and moreover, $E \cap (0, 1)^d \neq \emptyset$. The main result of the paper is that for a non-empty set $A\subset [0,d]$ there is a compact set $K\subset \mathbb{R}^d$ such that the set of Hausdorff dimensions attained by the microsets of $K$ equals $A$ if and only if $A$ is analytic and contains its infimum and supremum. This answers a question of Fraser, Howroyd, K\"aenm\"aki, and Yu. We show that for every compact set $K\subset \mathbb{R}^d$ and non-empty analytic set $A\subset [0,\dim_H K]$ there is a set $\mathcal{C}$ of compact subsets of $K$ which is compact in the Hausdorff metric and $\{\dim_H C: C\in \mathcal{C} \}=A$. The proof relies on the technique of stochastic co-dimension applied for a suitable coupling of fractal percolations with generation dependent retention probabilities. We also examine the analogous problems for packing and box dimensions., Comment: 21 pages, the proofs of Theorems 4.7 and 4.8 were improved, also some minor modifications

The lower dimension $\dim_L$ is the dual concept of the Assouad dimension. As it fails to be monotonic, Fraser and Yu introduced the modified lower dimension $\dim_{ML}$ by making the lower dimension monotonic with the simple formula $\dim_{ML} X=\sup\{\dim_L E: E\subset X\}$. As our first result we prove that the modified lower dimension is finitely stable in any metric space, answering a question of Fraser and Yu. We prove a new, simple characterization for the modified lower dimension. For a metric space $X$ let $\mathcal{K}(X)$ denote the metric space of the non-empty compact subsets of $X$ endowed with the Hausdorff metric. As an application of our characterization, we show that the map $\dim_{ML} \colon \mathcal{K}(X)\to [0,\infty]$ is Borel measurable. More precisely, it is of Baire class $2$, but in general not of Baire class $1$. This answers another question of Fraser and Yu. Finally, we prove that the modified lower dimension is not Borel measurable defined on the closed sets of $\ell^1$ endowed with the Effros Borel structure., Comment: 10 pages. A new paragraph added to the introduction

S. Banach pointed out that the graph of the generic (in the sense of Baire category) element of $\text{Homeo}([0,1])$ has length $2$. J. Mycielski asked if the measure theoretic dual holds, i.e., if the graph of all but Haar null many (in the sense of Christensen) elements of $\text{Homeo}([0,1])$ have length $2$. We answer this question in the affirmative. We call $f \in \text{Homeo}([0,1]^d)$ singular if it takes a suitable set of full measure to a nullset, and strongly singular if it is almost everywhere differentiable with singular derivative matrix. Since the graph of $f \in \text{Homeo}([0,1])$ has length $2$ iff $f$ is singular iff $f$ is strongly singular, the following results are the higher dimensional analogues of Banach's observation and our solution to Mycielski's problem. We show that for $d \ge 2$ the graph of the generic element of $\text{Homeo}([0,1]^d)$ has infinite $d$-dimensional Hausdorff measure, contrasting the above result of Banach. The measure theoretic dual remains open, but we show that the set of elements of $\text{Homeo}([0,1]^d)$ with infinite $d$-dimensional Hausdorff measure is not Haar null. We show that for $d \ge 2$ the generic element of $\text{Homeo}([0,1]^d)$ is singular but not strongly singular. We also show that for $d \ge 2$ almost every element of $\text{Homeo}([0,1]^d)$ is singular, but the set of strongly singular elements form a so called Haar ambivalent set (neither Haar null, nor co-Haar null). Finally, in order to clarify the situation, we investigate the various possible definitions of singularity for maps of several variables, and explore the connections between them., Comment: 31 pages. Final version with minor modifications

We answer a question of Banakh, Jab\l{}o\'nska and Jab\l{}o\'nski by showing that for $d\ge 2$ there exists a compact set $K \subseteq \mathbb{R}^d$ such that the projection of $K$ onto each hyperplane is of non-empty interior, but $K+K$ is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a $K$ in the unit cube with full projections, that is, such that the projections of $K$ agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for $\ell$-fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar., Comment: 26 pages, 3 figures. Small modifications, final version

The notions of shyness and prevalence generalize the property of being zero and full Haar measure to arbitrary (not necessarily locally compact) Polish groups. The main goal of the paper is to answer the following question: What can we say about the Hausdorff and packing dimension of the fibers of prevalent continuous maps? Let $K$ be an uncountable compact metric space. We prove that the prevalent $f\in C(K,\mathbb{R}^d)$ has many fibers with almost maximal Hausdorff dimension. This generalizes a theorem of Dougherty and yields that the prevalent $f\in C(K,\mathbb{R}^d)$ has graph of maximal Hausdorff dimension, generalizing a result of Bayart and Heurteaux. We obtain similar results for the packing dimension. We show that for the prevalent $f\in C([0,1]^m,\mathbb{R}^d)$ the set of $y\in f([0,1]^m)$ for which $\dim_H f^{-1}(y)=m$ contains a dense open set having full measure with respect to the occupation measure $\lambda^m \circ f^{-1}$, where $\dim_H$ and $\lambda^m$ denote the Hausdorff dimension and the $m$-dimensional Lebesgue measure, respectively. We also prove an analogous result when $[0,1]^m$ is replaced by any self-similar set satisfying the open set condition. We cannot replace the occupation measure with Lebesgue measure in the above statement: We show that the functions $f\in C[0,1]$ for which positively many level sets are singletons form a non-shy set in $C[0,1]$. In order to do so, we generalize a theorem of Antunovi\'c, Burdzy, Peres and Ruscher. As a complementary result we prove that the functions $f\in C[0,1]$ for which $\dim_H f^{-1}(y)=1$ for all $y\in (\min f,\max f)$ form a non-shy set in $C[0,1]$. We also prove sharper results in which large Hausdorff dimension is replaced by positive measure with respect to generalized Hausdorff measures, which answers a problem of Fraser and Hyde., Comment: 42 pages

A set $\mathcal{A}\subset C[0,1]$ is \emph{shy} or \emph{Haar null } (in the sense of Christensen) if there exists a Borel set $\mathcal{B}\subset C[0,1]$ and a Borel probability measure $\mu$ on $C[0,1]$ such that $\mathcal{A}\subset \mathcal{B}$ and $\mu\left(\mathcal{B}+f\right) = 0$ for all $f \in C[0,1]$. The complement of a shy set is called a \emph{prevalent} set. We say that a set is \emph{Haar ambivalent} if it is neither shy nor prevalent. The main goal of the paper is to answer the following question: What can we say about the topological properties of the level sets of the prevalent/non-shy many $f\in C[0,1]$? The classical Bruckner--Garg Theorem characterizes the level sets of the generic (in the sense of Baire category) $f\in C[0,1]$ from the topological point of view. We prove that the functions $f\in C[0,1]$ for which the same characterization holds form a Haar ambivalent set. In an earlier paper we proved that the functions $f\in C[0,1]$ for which positively many level sets with respect to the Lebesgue measure $\lambda$ are singletons form a non-shy set in $C[0,1]$. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions $f\in C[0,1]$ for which positively many level sets with respect to the occupation measure $\lambda\circ f^{-1}$ are not perfect form a Haar ambivalent set in $C[0,1]$. We show that for the prevalent $f\in C[0,1]$ for the generic $y\in f([0,1])$ the level set $f^{-1}(y)$ is perfect. Finally, we answer a question of Darji and White by showing that the set of functions $f \in C[0,1]$ for which there exists a perfect $P_f\subset [0,1]$ such that $f'(x) = \infty$ for all $x \in P_f$ is Haar ambivalent., Comment: 12 pages

More than 80 years ago Kolmogorov asked the following question. Let $E\subseteq \mathbb{R}^{2}$ be a measurable set with $\lambda^{2}(E)<\infty$, where $\lambda^2$ denotes the two-dimensional Lebesgue measure. Does there exist for every $\varepsilon>0$ a contraction $f\colon E\to \mathbb{R}^{2}$ such that $\lambda^{2}(f(E))\geq \lambda^{2}(E)-\varepsilon$ and $f(E)$ is a polygon? We answer this question in the negative by constructing a bounded, simply connected open counterexample. Our construction can easily be modified to yield the analogous result in higher dimensions., Comment: 5 pages

A subset $X$ of a Polish group $G$ is called \emph{Haar null} if there exists a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that $\mu(gBh)=0$ for every $g,h \in G$. We prove that there exists a set $X \subset \mathbb{R}$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in \mathbb{R}$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C \subset G$ such that for every non-meagre set $X \subset G$ there exists a $t \in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This essentially generalises results of Bartoszy\'nski and Burke-Miller.

Let us say that an element of a given family $\A$ of subsets of $\R^d$ can be reconstructed using $n$ test sets if there exist $T_1,...,T_n \subset \R^d$ such that whenever $A,B\in \A$ and the Lebesgue measures of $A \cap T_i$ and $B \cap T_i$ agree for each $i=1,...,n$ then $A=B$. Our goal will be to find the least such $n$. We prove that if $\A$ consists of the translates of a fixed reasonably nice subset of $\R^d$ then this minimum is $n=d$. In order to obtain this result we reconstruct a translate of a fixed function using $d$ test sets as well, and also prove that under rather mild conditions the measure function $f_{K,\theta} (r) = \la^{d-1} (K \cap \{x \in \RR^d : = r\})$ of the sections of $K$ is absolutely continuous for almost every direction $\theta$. These proofs are based on techniques of harmonic analysis. We also show that if $\A$ consists of the magnified copies $rE+t$ $(r\ge 1, t\in\R^d)$ of a fixed reasonably nice set $E\subset \R^d$, where $d\ge 2$, then $d+1$ test sets reconstruct an element of $\A$. This fails in $\R$: we prove that an interval, and even an interval of length at least 1 cannot be reconstructed using 2 test sets. Finally, using randomly constructed test sets, we prove that an element of a reasonably nice $k$-dimensional family of geometric objects can be reconstructed using $2k+1$ test sets. A example from algebraic topology shows that $2k+1$ is sharp in general.

R.D.Mauldin asked if every translation invariant $\sigma$-finite Borel measure on $\RR^d$ is a constant multiple of Lebesgue measure. The aim of this paper is to show that the answer is "yes and no", since surprisingly the answer depends on what we mean by Borel measure and by constant. We present Mauldin's proof of what he called a folklore result, stating that if the measure is only defined for Borel sets then the answer is affirmative. Then we show that if the measure is defined on a $\sigma$-algebra \emph{containing} the Borel sets then the answer is negative. However, if we allow the multiplicative constant to be infinity, then the answer is affirmative in this case as well. Moreover, our construction also shows that an isometry invariant $\sigma$-finite Borel measure (in the wider sense) on $\RR^d$ can be non-$\sigma$-finite when we restrict it to the Borel sets.

We show that the Continuum Hypothesis implies that for every $0

We give an affirmative answer to a question of K. Ciesielski by showing that the composition $f\circ g$ of two derivatives $f,g:[0,1]\to[0,1]$ always has a fixed point. Using Maximoff's Theorem we obtain that the composition of two $[0,1]\to[0,1]$ Darboux Baire-1 functions must also have a fixed point.

We show that the existence of measurable envelopes of all subsets of $\RR^n$ with respect to the $d$-dimensional Hausdorff measure $(0

We answer a question of Darji and Keleti by proving that there exists a compact set $C_0\subset\RR$ of measure zero such that for every perfect set $P\subset\RR$ there exists $x\in\RR$ such that $(C_0+x)\cap P$ is uncountable. Using this $C_0$ we answer a question of Gruenhage by showing that it is consistent with $ZFC$ (as it follows e.g. from $\textrm{cof}(\iN)<2^\omega$) that less than $2^\omega$ many translates of a compact set of measure zero can cover $\RR$.

We show that the set of Liouville numbers is either null or non-$\sigma$-finite with respect to every translation invariant Borel measure on $\RR$, in particular, with respect to every Hausdorff measure $\iH^g$ with gauge function $g$. This answers a question of D. Mauldin. We also show that some other simply defined Borel sets like non-normal or some Besicovitch-Eggleston numbers, as well as all Borel subgroups of $\RR$ that are not $F_\sigma$ possess the above property. We prove that, apart from some trivial cases, the Borel class, Hausdorff or packing dimension of a Borel set with no such measure on it can be arbitrary.

Following Laczkovich we consider the partially ordered set $\iB_1(\RR)$ of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komj\'ath and Kunen we show (in $ZFC$) that special Aronszajn lines are embeddable into $\iB_1(\RR)$. We also show that under Martin's Axiom a linearly ordered set $\mathbb{L}$ with $|\mathbb{L}|<2^\omega$ is embeddable into $\iB_1(\RR)$ iff $\mathbb{L}$ does not contain a copy of $\omega_1$ or $\omega_1^*$. We present a $ZFC$-example of a linear order of size $2^\omega$ showing that this characterisation is not valid for orders of size continuum. These results are obtained using the notion of a compact-special tree; that is, a tree that is embeddable into the class of compact subsets of the reals partially ordered under reverse inclusion. We investigate how this notion is related to the well-known notion of an $\RR$-special tree and also to some other notions of specialness.

The set of continuous or Baire class 1 functions defined on a metric space $X$ is endowed with the natural pointwise partial order. We investigate how the possible lengths of well-ordered monotone sequences (with respect to this order) depend on the space $X$.

We consider the set of Baire 1 functions endowed with the pointwise partial ordering and investigate the structure of the linearly ordered subsets.

We show that if the gradient of $f:\RR^2\rightarrow\RR$ exists everywhere and is nowhere zero, then in a neighbourhood of each of its points the level set $\{x\in\RR^2:f(x)=c\}$ is homeomorphic either to an open interval or to the union of finitely many open segments passing through a point. The second case holds only at the points of a discrete set. We also investigate the global structure of the level sets.

A function $f:\RR^n \to \RR$ is called \emph{vertically rigid} if $graph(cf)$ is isometric to $graph (f)$ for all $c \neq 0$. We settled Jankovi\'c's conjecture in a separate paper by showing that a continuous function $f:\RR \to \RR$ is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \in \RR$). Now we prove that a continuous function $f:\RR^2 \to \RR$ is vertically rigid if and only if after a suitable rotation around the z-axis $f(x,y)$ is of the form $a + bx + dy$, $a + s(y)e^{kx}$ or $a + be^{kx} + dy$ ($a,b,d,k \in \RR$, $k \neq 0$, $s : \RR \to \RR$ continuous). The problem remains open in higher dimensions.

A function $f:\RR \to \RR$ is called \emph{vertically rigid} if $graph(cf)$ is isometric to $graph (f)$ for all $c \neq 0$. We prove Jankovi\'c's conjecture by showing that a continuous function is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \in \RR$). We answer a question of Cain, Clark and Rose by showing that there exists a Borel measurable vertically rigid function which is not of the above form. We discuss the Lebesgue and Baire measurable case, consider functions bounded on some interval and functions with at least one point of continuity. We also introduce horizontally rigid functions, and show that a certain structure theorem can be proved without assuming any regularity.

Cain, Clark and Rose defined a function $f\colon \RR^n \to \RR$ to be \emph{vertically rigid} if $\graph(cf)$ is isometric to $\graph (f)$ for every $c \neq 0$. It is \emph{horizontally rigid} if $\graph(f(c \vec{x}))$ is isometric to $\graph (f)$ for every $c \neq 0$ (see \cite{CCR}). In an earlier paper the authors of the present paper settled Jankovi\'c's conjecture by showing that a continuous function of one variable is vertically rigid if and only if it is of the form $a+bx$ or $a+be^{kx}$ ($a,b,k \in \RR$). Later they proved that a continuous function of two variables is vertically rigid if and only if after a suitable rotation around the z-axis it is of the form $a + bx + dy$, $a + s(y)e^{kx}$ or $a + be^{kx} + dy$ ($a,b,d,k \in \RR$, $k \neq 0$, $s : \RR \to \RR$ continuous). The problem remained open in higher dimensions. The characterization in the case of horizontal rigidity is surprisingly simpler. C. Richter proved that a continuous function of one variable is horizontally rigid if and only if it is of the form $a+bx$ ($a,b\in \RR$). The goal of the present paper is to prove that a continuous function of two variables is horizontally rigid if and only if it is of the form $a + bx + dy$ ($a,b,d \in \RR$). This problem also remains open in higher dimensions. The main new ingredient of the present paper is the use of functional equations.

A \emph{hull} of $A \subset [0,1]$ is a set $H$ containing $A$ such that $\lambda^*(H)=\lambda^*(A)$. We investigate all four versions of the following problem. Does there exist a monotone (wrt. inclusion) map that assigns a Borel/$G_\delta$ hull to every negligible/measurable subset of $[0,1]$? Three versions turn out to be independent of ZFC (the usual Zermelo-Fraenkel axioms with the Axiom of Choice), while in the fourth case we only prove that the nonexistence of a monotone $G_\delta$ hull operation for all measurable sets is consistent. It remains open whether existence here is also consistent. We also answer a question of Z. Gyenes and D. P\'alv\"olgyi which asks if monotone hulls can be defined for every chain (wrt. inclusion) of measurable sets. We also comment on the problem of hulls of all subsets of $[0,1]$.

Let $\R^\R$ denote the set of real valued functions defined on the real line. A map $D: \R^\R \to \R^\R$ is a {\it difference operator}, if there are real numbers $a_i, b_i \ (i=1,..., n)$ such that $(Df)(x)=\sum_{i=1}^n a_i f(x+b_i)$ for every $f\in \R^\R $ and $x\in \R$. A {\it system of difference equations} is a set of equations $S={D_i f=g_i : i\in I}$, where $I$ is an arbitrary set of indices, $D_i$ is a difference operator and $g_i$ is a given function for every $i\in I$, and $f$ is the unknown function. One can prove that a system $S$ is solvable if and only if every finite subsystem of $S$ is solvable. However, if we look for solutions belonging to a given class of functions, then the analogous statement fails. For example, there exists a system $S$ such that every finite subsystem of $S$ has a solution which is a trigonometric polynomial, but $S$ has no such solution. This phenomenon motivates the following definition. Let ${\cal F}$ be a class of functions. The {\it solvability cardinal} $\solc (\iF)$ of ${\cal F}$ is the smallest cardinal $\kappa$ such that whenever $S$ is a system of difference equations and each subsystem of $S$ of cardinality less than $\kappa$ has a solution in ${\cal F}$, then $S$ itselfhas a solution in ${\cal F}$. In this paper we determine the solvability cardinals of most function classes that occur in analysis. As it turns out, the behaviour of $\solc ({\cal F})$ is rather erratic. For example, $\solc (\text{polynomials})=3$ but $\solc (\text{trigonometric polynomials})=\omega_1$, $\solc ({f: f\ \text{is continuous}}) = \omega_1$ but $\solc ({f: f\ \text{is Darboux}}) =(2^\omega)^+$, and $\solc (\R^\R)=\omega$. We consistently determine the solvability cardinals of the classes of Borel, Lebesgue and Baire measurable functions, and give some partial answers for the Baire class 1 and Baire class $\alpha$ functions.

The main goal of this note is to prove the following theorem. If $A_n$ is a sequence of measurable sets in a $\sigma$-finite measure space $(X, \mathcal{A}, \mu)$ that covers $\mu$-a.e. $x \in X$ infinitely many times, then there exists a sequence of integers $n_i$ of density zero so that $A_{n_i}$ still covers $\mu$-a.e. $x \in X$ infinitely many times. The proof is a probabilistic construction. As an application we give a simple direct proof of the known theorem that the ideal of density zero subsets of the natural numbers is random-indestructible, that is, random forcing does not add a co-infinite set of naturals that almost contains every ground model density zero set. This answers a question of B. Farkas.

We say that a metric space $(X,d)$ possesses the \emph{Banach Fixed Point Property (BFPP)} if every contraction $f:X\to X$ has a fixed point. The Banach Fixed Point Theorem says that every complete metric space has the BFPP. However, E. Behrends pointed out \cite{Be1} that the converse implication does not hold; that is, the BFPP does not imply completeness, in particular, there is a non-closed subset of $\RR^2$ possessing the BFPP. He also asked \cite{Be2} if there is even an open example in $\RR^n$, and whether there is a 'nice' example in $\RR$. In this note we answer the first question in the negative, the second one in the affirmative, and determine the simplest such examples in the sense of descriptive set theoretic complexity. Specifically, first we prove that if $X\su\RR^n$ is open or $X\su\RR$ is simultaneously $F_\si$ and $G_\de$ and $X$ has the BFPP then $X$ is closed. Then we show that these results are optimal, as we give an $F_\si$ and also a $G_\de$ non-closed example in $\RR$ with the BFPP. We also show that a nonmeasurable set can have the BFPP. Our non-$G_\de$ examples provide metric spaces with the BFPP that cannot be remetrised by any compatible complete metric. All examples are in addition bounded.

In an earlier paper (arxiv:1108.4292) we introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. For a compact metric space $K$ let $\dim_{H}K$ and $\dim_{tH} K$ denote its Hausdorff and topological Hausdorff dimension, respectively. We proved that this new dimension describes the Hausdorff dimension of the level sets of the generic continuous function on $K$, namely $\sup{\dim_{H}f^{-1}(y) : y \in \mathbb{R}} = \dim_{tH} K - 1$ for the generic $f \in C(K)$, provided that $K$ is not totally disconnected, otherwise every non-empty level set is a singleton. We also proved that if $K$ is not totally disconnected and sufficiently homogeneous then $\dim_{H}f^{-1}(y) = \dim_{tH} K - 1$ for the generic $f \in C(K)$ and the generic $y \in f(K)$. The most important goal of this paper is to make these theorems more precise. As for the first result, we prove that the supremum is actually attained on the left hand side of the first equation above, and also show that there may only be a unique level set of maximal Hausdorff dimension. As for the second result, we characterize those compact metric spaces for which for the generic $f\in C(K)$ and the generic $y\in f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. We also generalize a result of B. Kirchheim by showing that if $K$ is self-similar then for the generic $f\in C(K)$ for every $y\in \inter f(K)$ we have $\dim_{H} f^{-1}(y)=\dim_{tH}K-1$. Finally, we prove that the graph of the generic $f\in C(K)$ has the same Hausdorff and topological Hausdorff dimension as $K$., Comment: 20 pages

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. We examine the basic properties of this new notion of dimension, compare it to other well-known notions, determine its value for some classical fractals such as the Sierpinski carpet, the von Koch snowflake curve, Kakeya sets, the trail of the Brownian motion, etc. As our first application, we generalize the celebrated result of Chayes, Chayes and Durrett about the phase transition of the connectedness of the limit set of Mandelbrot's fractal percolation process. They proved that certain curves show up in the limit set when passing a critical probability, and we prove that actually `thick' families of curves show up, where roughly speaking the word thick means that the curves can be parametrized in a natural way by a set of large Hausdorff dimension. The proof of this is basically a lower estimate of the topological Hausdorff dimension of the limit set. For the sake of completeness, we also give an upper estimate and conclude that in the non-trivial cases the topological Hausdorff dimension is almost surely strictly below the Hausdorff dimension. Finally, as our second application, we show that the topological Hausdorff dimension is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic continuous function (in the sense of Baire category) defined on a compact metric space., Comment: 39 pages, 5 figures, to appear in Advances in Mathematics