arXiv subjects
Publications and source records attributed to Zoltan Buczolich.
We investigate the growth rate of the Birkhoff sums $S_{n,α} f(x)=\sum_{k=0}^{n-1} f(x+kα)$, where $f$ is a continuous function with zero mean defined on the unit circle $\mathbb T$ and $(α,x)$ is a "typical" element of $\mathbb T^2$. The answer depends on the meaning given to the word "typical". Part of the work will be done in a more general context.
Suppose that $f$ belongs to a suitably defined complete metric space $ {\cal C}^{α}$ of Hölder $ α$-functions defined on $[0,1]$. We are interested in whether one can find large (in the sense of Hausdorff, or lower/upper Minkowski dimension) sets $A {\subset} [0,1]$ such that $f|_{A}$ is monotone, or convex/concave. Some of our results are about generic functions in $ {\cal C}^{α}$ like the following one: we prove that for the generic $f\in C_{1}^{α}[0,1]$, $0\leq α<2$ for any $A {\subset} [0,1]$ such that $f|_{A}$ is convex, or concave we have ${\mathrm{dim}}_{\mathrm H} A\leq \underline{\mathrm{dim}}_M A\leq \max \{0, α-1 \}.$ On the other hand, we also have some results about all functions belonging to a certain space. For example the previous result is complemented by the following one: for $1< α\leq 2$ for any $f\in C^{α}[0,1]$ there is always a set $A {\subset}[0,1]$ such that ${\mathrm{dim}}_{\mathrm H} A= α-1$ and $f|_{A}$ is convex, or concave on $A$.
We show that there are functions $f$ in the Hölder class $C^{ { α}}[0,1]$, $1< { α}<2$ such that $f|_{A}$ is not convex, nor concave for any $A { \subset } [0,1]$ with $ { \bar { dim }_M } A> { α}-1$. Our earlier result shows that for the typical/generic $f\in { C_ { 1 } ^ { { α} } [0,1] }$, $0\leq { α}<2$ there is always a set $A { \subset } [0,1]$ such that $f|_A$ is convex and $ { \bar { dim }_M } A=1$. The analogous statement for monotone restrictions is the following: there are functions $f$ in the Hölder class $C^{ { α}}[0,1]$, $1/2 \leq { α}<1$ such that $f|_{A}$ is not monotone on $A { \subset } [0,1]$ with $ { \bar { dim }_M } A> { α}$. This statement is not true for the range of parameters $ { α}<1/2$ and our theorem for the parameter range $1\leq { α} <3/2$ cannot be obtained by integration of the result about monotone restrictions.
We show that $ { ω}(n)$ and $ { Ω}(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\sum_{n\leq K}g(n)$ then for every ergodic dynamical system $(X, { { \cal A } },μ, { τ})$ and every $f\in L^{1}(X)$ $$\lim_{K\to { \infty }} \frac{1}{S_{g,K}}\sum_{n=1}^{K} g(n)f( { τ}^{n}x)=\int_{X}fdμ\text{ for $μ$ a.e. }x\in X. $$ This answers a question raised by C. Cuny and M. Weber who showed this result for $L^{p}$, $p>1$.
We study the singularity (multifractal) spectrum of the convex hull of the typical/generic continuous functions defined on $[0,1]^{d}$. We denote by ${\mathbf E}_ { { φ} }^{h} $ the set of points at which $ φ: [0,1]^d\to {\mathbb R}$ has a pointwise Hölder exponent equal to $h$. Let $H_{f}$ be the convex hull of the graph of $f$, the concave function on the top of $H_{f}$ is denoted by $ { { φ} }_{1,f}( { { \mathbf x } })=\max \{y:( { { \mathbf x } },y)\in H_{f} \}$ and $ { { φ} }_{2,f}( { { \mathbf x } })=\min \{y:( { { \mathbf x } },y)\in H_{f} \}$ denotes the convex function on the bottom of $H_{f}$. We show that there is a dense $G_δ$ subset $ { { \cal G } } { \subset } {C[0,1]^d}$ such that for $f\in { { \cal G } }$ the following properties are satisfied. For $i=1,2$ the functions $ { { { φ} }_ {i,f}}$ and $f$ coincide only on a set of zero Hausdorff dimension, the functions $ { { { φ} }_ {i,f}}$ are continuously differentiable on $(0,1)^{d}$, ${\mathbf E}_{ { { φ} }_{i,f}}^{0} $ equals the boundary of $ {[0,1]^d}$, $\dim_{H}{\mathbf E}_{ { { φ} }_{i,f}}^{1}=d-1 $, $\dim_{H}{\mathbf E}_{ { { φ} }_{i,f}}^{+ { \infty }}=d $ and ${\mathbf E}_{ { { φ} }_{i,f}}^{h}= { \emptyset }$ if $h\in(0,+ { \infty }) { \setminus } \{1 \}$.
We consider skew tent maps $T_{α, β}(x)$ such that $(α, β)\in[0,1]^{2}$ is the turning point of $T {_ {α, β}}$, that is, $T_{α, β}=\frac{β}{α}x$ for $0\leq x \leq α$ and $T_{α, β}(x)=\frac{β}{1-α}(1-x)$ for $ α<x\leq 1$. We denote by $ {\underline{M}}=K(α, β)$ the kneading sequence of $T_ {α, β}$ and by $h(α, β)$ its topological entropy. For a given kneading squence $ {\underline{M}}$ we consider equi-kneading, (or equi-topological entropy, or isentrope) curves $(α, φ_{\underline{M}}(α))$ such that $K(α, φ_{\underline{M}}(α))= {\underline{M}}$. To study the behavior of these curves an auxiliary function $ Θ_{\underline{M}}(α, β)$ is introduced. For this function $ Θ_{\underline{M}}(α, φ_{\underline{M}}(α))=0$, but it may happen that for some kneading sequences $Θ_{\underline{M}}(α, β)=0$ for some $ β< φ_{\underline{M}}(α)$ with $(α, β)$ still in the interesting region. Using $ Θ_{\underline{M}}$ we show that the curves $(α,φ_{\underline{M}}(α))$ hit the diagonal $\{(β, β): 0.5< β<1 \}$ almost perpendicularly if $(β, β)$ is close to $(1,1)$. Answering a question asked by M. Misiurewicz at a conference we show that these curves are not necessarily exactly orthogonal to the diagonal, for example for $ {\underline{M}}=RLLRC$ the curve $(α, φ_{\underline{M}}(α))$ is not orthogonal to the diagonal. On the other hand, for $ {\underline{M}}=RLC$ it is. With different parametrization properties of equi-kneading maps for skew tent maps were considered by J.C. Marcuard, M. Misiurewicz and E. Visinescu.
Suppose that G is a compact Abelian topological group, m is the Haar measure on G and f is a measurable function. Given (n_k), a strictly monotone increasing sequence of integers we consider the nonconventional ergodic/Birkhoff averages M_N^αf(x). The f-rotation set is Gamma_f={α\in G: M_N^α f(x) converges for m a.e. x as N\to \infty .} We prove that if G is a compact locally connected Abelian group and f: G -> R is a measurable function then from m(Gamma_f)>0 it follows that f \in L^1(G). A similar result is established for ordinary Birkhoff averages if G=Z_{p}, the group of p-adic integers. However, if the dual group, \hat{G} contains "infinitely many multiple torsion" then such results do not hold if one considers non-conventional Birkhoff averages along ergodic sequences. What really matters in our results is the boundedness of the tail, f(x+n_{k} α)/k, k=1,... for a.e. x for many α, hence some of our theorems are stated by using instead of Gamma_f slightly larger sets, denoted by Gamma_{f,b}.
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 $K$.
We answer a question of J. Bourgain. We show that the sequence n^2 is L^1-universally bad.
Let $(X,\mathcal{B}, μ, T)$ be an ergodic dynamical system on a non-atomic finite measure space. Consider the maximal function $\dis R^*:(f, g) \in L^1\times L^1 \to R^*(f, g)(x) = \sup_{n} \frac{f(T^nx)g(T^{2n}x)}{n}.$ We show that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost everywhere. Two consequences are derived. The bilinear Hardy--Littlewood maximal function fails to be a.e. finite for all functions $(f, g)\in L^1\times L^1.$ The Furstenberg averages do not converge for all pairs of $(L^{1},L^{1})$ functions, while by a result of J. Bourgain these averages converge for all pairs of $(L^{p},L^{q})$ functions with $\frac{1}{p}+\frac{1}{q}\leq 1.$
A universally L^1 good sequence n_k is constructed with n_{k+1}-n_k tending to infinity. For ergodic transformations non-conventional ergodic averages of L^1 functions computed by using this sequence converge to the integral of the function.
We solve the following counting problem for measure preserving transformations. For $f\in L_+^1(μ)$, is it true that $\ds \sup_n\frac{\bN_n(f)(x)}{n} <\infty,$ where $$\ds\bN_n(f)(x)= # {k: \frac{f(T^k x)}{k}>\frac 1 n}?$$ One of the consequences is the nonvalidity of J. Bourgain's Return Time Theorem for pairs of $(L^1, L^1)$ functions.