33 research outputs found

### Bruckner--Garg-type results with respect to Haar null sets in $C[0,1]$

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 page

### Hausdorff and packing dimension of fibers and graphs of prevalent continuous maps

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 page

### Chaos among self-maps of the Cantor space

AbstractGlasner and Weiss have shown that a generic homeomorphism of the Cantor space has zero topological entropy. Hochman has shown that a generic transitive homeomorphism of the Cantor space has the property that it is topologically conjugate to the universal odometer and hence far from being chaotic in any sense. We show that a generic self-map of the Cantor space has zero topological entropy. Moreover, we show that a generic self-map of the Cantor space has no periodic points and hence is not Devaney chaotic nor Devaney chaotic on any subsystem. However, we exhibit a dense subset of the space of all self-maps of the Cantor space each element of which has infinite topological entropy and is Devaney chaotic on some subsystem