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Dawid Huczek

Publications and source records attributed to Dawid Huczek.

12 recordsLinked to original sources
arXiv2024

Residuality of Dynamical Morphisms for Amenable Group Actions

We extend the classical Baire category approach, used in proving the finite generator theorem of Krieger, the homomorphism theorem of Sinai and the isomorphism theorem of Ornstein, applying a similar reasoning to the case of actions of countably infinite amenable groups. In principle we follow the lines of the paper by Burton, Keane and Serafin (\cite{BKS}), showing that measures defining homomorphisms or isomorphisms form residual subsets in suitably chosen spaces of joinings.

math.DS↗
arXiv2019

Zero-dimensional extensions of amenable group actions

We prove that every dynamical system $X$ with free action of a countable amenable group $G$ by homeomorphisms has a zero-dimensional extension $Y$ which is faithful and principal, i.e. every $G$-invariant measure $μ$ on $X$ has exactly one preimage $ν$ on $Y$ and the conditional entropy of $ν$ with respect to $X$ is zero. This is a version of an earlier result by T. Downarowicz and D. Huczek, which establishes the existence of zero-dimensional principal and faithful extensions for general actions of the group of integers.

math.DS↗
arXiv2019

Empirical approach to the x2, x3 conjecture

We study atomic measures on $[0,1]$ which are invariant both under multiplication by $2\mod 1$ and by $3\mod 1$, since such measures play an important role in deciding Furstenberg's $\times 2, \times 3$ conjecture. Our specific focus was finding atomic measures whose supports are far from being uniformly distributed, and we used computer software to discover a number of such measures (which we call \emph{outlier measures}). The structure of these measures indicates the possibility that a sequence of atomic measures may converge to a non-Lebesgue measure; likely one which is a combination of the Lebesgue measure and one or more atomic measures.

math.DS↗
arXiv2019

Extensions of Full Shifts with Group Actions

We give a sufficient condition for a symbolic topological dynamical system with action of a countable amenable group to be an extension of the full shift, a problem analogous to those studied by Ashley, Marcus, Johnson and others for actions of $\mathbb{Z}$ and $\mathbb{Z}^d$ .

math.DS↗
arXiv2019

A comment on ergodic theorem for amenable groups

We prove a version of ergodic theorem for an action of an amenable group, where a Fø lner sequence needs not to be tempered. Instead, it is assumed that a function satisfies certain mixing condition.

math.DS↗
arXiv2018

Doubly stochastic operators with zero entropy

We study doubly stochastic operators with zero entropy. We generalize three famous theorems: the Rokhlin's theorem on genericity of zero entropy, the Kushnirenko's theorem on equivalence of discrete spectrum and nullity and the Halmos-von Neumann's theorem on representation of maps with discrete spectrum as group rotations.

math.DS↗
arXiv2017

Dynamical quasitilings of amenable group

We prove that for any compact zero-dimensional metric space $X$ on which an infinite countable amenable group $G$ acts freely by homeomorphisms, there exists a dynamical quasitiling with good covering, continuity, Følner and dynamical properties, i.e to every $x\in X$ we can assign a quasitiling $\mathcal{T}_x$ of $G$ (with all the $\mathcal{T}_x$ using the same, finite set of shapes) such that the tiles of $\mathcal{T}_x$ are disjoint, their union has arbitrarily high lower Banach Density, all the shapes of $\mathcal{T}_x$ are large subsets of an arbitrarily large Følner set, and if we consider $\mathcal{T}_x$ to be an element of a shift space over a certain finite alphabet, then the mapping $x\mapsto \mathcal{T}_x$ is a factor map.

math.DS↗
arXiv2015

Tilings of amenable groups

We prove that for any infinite countable amenable group $G$, any $ε> 0$ and any finite subset $K\subset G$, there exists a tiling (partition of $G$ into finite "tiles" using only finitely many "shapes"), where all the tiles are $(K; ε)$-invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of $G$ (in the sense that the mappings, associated to different from unity elements of $G$, have no fixpoints), on a zero-dimensional space, and which has topological entropy zero.

math.GR↗
arXiv2014

Minimal models for actions of amenable groups

We prove that on a metrizable, compact, zero-dimensional space every free action of an amenable group is measurably isomorphic to a minimal $G$-action with the same, i.e. affinely homeomorphic, simplex of measures.

math.DS↗
arXiv2012

Rank as a function of measure

We establish certain topological properties of rank understood as a function on the set of invariant measures on a topological dynamical system. To be exact, we show that rank is of Young class LU (i.e., it is the limit of an increasing sequence of upper semicontinuous functions)

math.DS↗