33 research outputs found

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

    Get PDF
    A set AC[0,1]\mathcal{A}\subset C[0,1] is \emph{shy} or \emph{Haar null } (in the sense of Christensen) if there exists a Borel set BC[0,1]\mathcal{B}\subset C[0,1] and a Borel probability measure μ\mu on C[0,1]C[0,1] such that AB\mathcal{A}\subset \mathcal{B} and μ(B+f)=0\mu\left(\mathcal{B}+f\right) = 0 for all fC[0,1]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 fC[0,1]f\in C[0,1]? The classical Bruckner--Garg Theorem characterizes the level sets of the generic (in the sense of Baire category) fC[0,1]f\in C[0,1] from the topological point of view. We prove that the functions fC[0,1]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 fC[0,1]f\in C[0,1] for which positively many level sets with respect to the Lebesgue measure λ\lambda are singletons form a non-shy set in C[0,1]C[0,1]. The above result yields that this set is actually Haar ambivalent. Now we prove that the functions fC[0,1]f\in C[0,1] for which positively many level sets with respect to the occupation measure λf1\lambda\circ f^{-1} are not perfect form a Haar ambivalent set in C[0,1]C[0,1]. We show that for the prevalent fC[0,1]f\in C[0,1] for the generic yf([0,1])y\in f([0,1]) the level set f1(y)f^{-1}(y) is perfect. Finally, we answer a question of Darji and White by showing that the set of functions fC[0,1]f \in C[0,1] for which there exists a perfect Pf[0,1]P_f\subset [0,1] such that f(x)=f'(x) = \infty for all xPfx \in P_f is Haar ambivalent.Comment: 12 page

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

    Get PDF
    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 KK be an uncountable compact metric space. We prove that the prevalent fC(K,Rd)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 fC(K,Rd)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 fC([0,1]m,Rd)f\in C([0,1]^m,\mathbb{R}^d) the set of yf([0,1]m)y\in f([0,1]^m) for which dimHf1(y)=m\dim_H f^{-1}(y)=m contains a dense open set having full measure with respect to the occupation measure λmf1\lambda^m \circ f^{-1}, where dimH\dim_H and λm\lambda^m denote the Hausdorff dimension and the mm-dimensional Lebesgue measure, respectively. We also prove an analogous result when [0,1]m[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 fC[0,1]f\in C[0,1] for which positively many level sets are singletons form a non-shy set in C[0,1]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 fC[0,1]f\in C[0,1] for which dimHf1(y)=1\dim_H f^{-1}(y)=1 for all y(minf,maxf)y\in (\min f,\max f) form a non-shy set in C[0,1]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

    Get PDF
    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
    corecore