arXiv subjects

Gerhard Keller

Publications and source records attributed to Gerhard Keller.

At least 19 recordsLinked to original sources

arXiv2025

Besicovitch covering numbers for $\mathcal B$-free and other shifts

For a finite alphabet $A$ define by $d_1(x,y):=\limsup_{n\to\infty}\frac{1}{2n+1}\#\{|i|\le n: x_i\neq y_i\}$ the Besicovitch pseudo-metric on $A^{\mathbb Z}$. It is well known that a closed subshift of $A^{\mathbb Z}$ has finite covering numbers w.r.t. $d_1$ if and only if it is mean-equicontinuous. Here we study, more generally, the scaling behavior of these covering numbers for individual orbits which are generic for an ergodic measure $\mu$ on $A^{\mathbb Z}$ with discrete spectrum, and we explore their usefulness as invariants for block code equivalence. We illustrate this by developing tools to determine these covering numbers for various classes of $\mathcal B$-free numbers (in particular also for square-free numbers), and we provide a continuous family of measures $\mu_s$, all with the same discrete spectrum generated by a single number, but such that $\mu_s$- and $\mu_{s'}$-typical $x$ resp. $x'\in A^{\mathbb Z}$ have sufficiently different growth of covering numbers such that there are no finite block codes mapping $x\to x'$ and $x'\to x$. (Indeed, both orbits have different amorphic complexities.)

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arXiv2022

Automorphisms of $\mathcal{B}$-free and other Toeplitz shifts

We present sufficient conditions for the triviality of the automorphism group of regular Toeplitz subshifts and give a broad class of examples from the class of $\mathcal{B}$-free subshifts satisfying them, extending [10]. On the other hand we provide an example of a $\mathcal{B}$-free Toeplitz subshift whose automorphism group has elements of arbitrarily large finite order, answering Question 11 in [13].

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arXiv2021

Spectrum of weak model sets with Borel windows

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article "Periods and factors of weak model sets" we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody's work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

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arXiv2021

On the Garden of Eden theorem for B-free subshifts

We prove that on B-free subshifts, with B satisfying the Erd\"os condition, all cellular automata are determined by monotone sliding block codes. In particular, this implies the validity of the Garden of Eden theorem for such systems.

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arXiv2021

Irregular $\mathcal{B}$-free Toeplitz sequences via Besicovitch's construction of sets of multiples without density

Modifying Besicovitch's construction of a set $\mathcal{B}$ of positive integers whose set of multiples $\mathcal{M}_{\mathcal{B}}$ has no asymptotic density, we provide examples of such sets $\mathcal{B}$ for which $η:=1_{\mathbb{Z}\setminus\mathcal{M}_{\mathcal{B}}}\in\{0,1\}^{\mathbb{Z}}$ is a Toeplitz sequence. Moreover our construction produces examples, for which $η$ is not only quasi-generic for the Mirsky measure (which has discrete dynamical spectrum), but also for some measure of positive entropy. On the other hand, modifying slightly an example from Kasjan, Keller, and Lemańczyk, we construct a set $\mathcal{B}$ for which $η$ is an irregular Toeplitz sequence but for which the orbit closure of $η$ in $\{0,1\}^{\mathbb{Z}}$ is uniquely ergodic.

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arXiv2021

Tautness for sets of multiples and applications to $\mathcal B$-free dynamics

For any set $\mathcal B\subseteq\mathbb N=\{1,2,\dots\}$ one can define its \emph{set of multiples} $\mathcal M_{\mathcal B}:=\bigcup_{b\in\mathcal B}b\mathbb Z$ and the set of \emph{$\mathcal B$-free numbers} $\mathcal F_{\mathcal B}:=\mathbb Z\setminus\mathcal M_{\mathcal B}$. Tautness of the set $\mathcal B$ is a basic property related to questions around the asymptotic density of $\mathcal M_{\mathcal B}\subseteq\mathbb Z$. From a dynamical systems point of view (originated by Sarnak) one studies $η$, the indicator function of $\mathcal F_{\mathcal B}\subseteq\mathbb Z$, its shift-orbit closure $X_η\subseteq\{0,1\}^{\mathbb Z}$ and the stationary probability measure $ν_η$ defined on $X_η$ by the frequencies of finite blocks in $η$. In this paper we prove that tautness implies the following two properties of $η$: (1) The measure $ν_η$ has full topological support in $X_η$. (2) If $X_η$ is proximal, i.e. if the one-point set $\{\dots000\dots\}$ is contained in $X_η$ and is the unique minimal subset of $X_η$, then $X_η$ is hereditary, i.e. if $x\in X_η$ and if $w$ is an arbitrary element of $\{0,1\}^{\mathbb Z}$, then also the coordinate-wise product $w\cdot x$ belongs to $X_η$. This strengthens two results from [Bartnicka et al. 2015] which need the stronger assumption that $\mathcal B$ has light tails for the same conclusions.

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arXiv2019

Generalized heredity in $\mathcal B$-free systems

Let $\mathcal B\subseteq\mathbb N$ be a primitive set. We complement results on heredity of the $\mathcal B$-free subshift $X_η$ from [arxiv:1509.08010] in two directions: In the proximal case we prove that a subshift $X_φ$, which micht be slightly larger than the subshift $X_η$, is always hereditary. (There is no need to assume that the set $\mathcal B$ is taut or even has light tails, but if $\mathcal B$ is taut, then $X_φ=X_η$.) We also generalize the the concept of heredity to the non-proximal (and hence non-hereditary) case by proving that $X_φ$ is always "hereditary away from its unique minimal subsystem" (which is always Toeplitz). Finally we characterize regularity of this Toeplitz subsystem equivalently by the condition $m_H(\overline{\operatorname{int}(W)})=0$, where $W$ ("the window") is a subset of a compact abelian group $H$ canonically associated with the set $\mathcal B$, and $m_H$ denotes Haar measure on $H$. Throughout, results from [arxiv:1702.02375] are heavily used.

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arXiv2019

Maximal equicontinuous generic factors and weak model sets

The orbit closures of regular model sets generated from a cut-and-project scheme given by a co-compact lattice $\mathcal{L}\subset G\times H$ and compact and aperiodic window $W\subseteq H$, have the maximal equicontinuous factor (MEF) $(G\times H)/\mathcal{L}$, if the window is toplogically regular. This picture breaks down completely, when the window has empty interior, in which case the MEF is always trivial, although $(G\times H)/\mathcal{L}$ continues to be the Kronecker factor for the Mirsky measure. As this situation occurs for many interesting examples like the square-free numbers or the visible lattice points, there is some need for a slightly weaker concept of topological factors that is still strong enough to capture basic properties of the system. Here we propose to use the concept of a generic factor \cite{HuangYe2012} for this purpose. For so called ergodic topological dynamical systems we prove the existence of a maximal equicontinuous generic factor (MEGF) and characterize it in terms of the regional proximal relation. For such systems we also show that the MEGF is trivial if and only if the system is topologically weakly mixing. This part of the paper profits strongly from previous work by McMahon \cite{McMahon1978} and Auslander \cite{Auslander1988}. In Section 3 we show that $(G\times H)/\mathcal{L}$ is indeed the MEGF of the orbit closure of each weak model set with an aperiodic Haar regulaar window, and in Section 4 we apply this fact to give an alternative proof of the finding \cite{Mentzen2017} that the centralizer of any $\mathcal{B}$-free dynamical system of Erdös type is trivial.

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arXiv2018

Synchronization versus stability of the invariant distribution for a class of globally coupled maps

We study a class of globally coupled maps in the continuum limit, where the individual maps are expanding maps of the circle. The circle maps in question are such that the uncoupled system admits a unique absolutely continuous invariant measure (acim), which is furthermore mixing. Interaction arises in the form of diffusive coupling, which involves a function that is discontinuous on the circle. We show that for sufficiently small coupling strength the coupled map system admits a unique absolutely continuous invariant distribution, which depends on the coupling strength $\varepsilon$. Furthermore, the invariant density exponentially attracts all initial distributions considered in our framework. We also show that the dependence of the invariant density on the coupling strength $\varepsilon$ is Lipschitz continuous in the BV norm. When the coupling is sufficiently strong, the limit behavior of the system is more complex. We prove that a wide class of initial measures approach a point mass with support moving chaotically on the circle. This can be interpreted as synchronization in a chaotic state.

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arXiv2018

Invariant Graphs for Chaotically Driven Maps

This paper investigates the geometrical structures of invariant graphs of skew product systems of the form $F : Θ\times I \to Θ\times I , (θ,y)\mapsto (Sθ,f_θ(y))$ driven by a hyperbolic base map $S : Θ\to Θ$ (e.g. a baker map or an Anosov surface diffeomorphism) and with monotone increasing fibre maps $(f_θ)_{θ\in Θ}$ having negative Schwarzian derivatives. We recall a classification, with respect to the number and to the Lyapunov exponents of invariant graphs, for this class of systems. Our major goal here is to describe the structure of invariant graphs and study the properties of the pinching set, the set of points where the values of all of the invariant graphs coincide. In arXiv:1610.10010, the authors studied skew product systems driven by a generalized Baker map $S$ with the restrictive assumption that $f_θ$ depends on $θ=(ξ,x)$ only through the stable coordinate $x$ of $θ$. Our aim is to relax this assumption and construct a fibre-wise conjugation between the original system and a new system for which the fibre maps depend only on the stable coordinate of the derive.

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arXiv2018

Periods and factors of weak model sets

There is a renewed interest in weak model sets due to their connection to $\mathcal B$-free systems, which emerged from Sarnak's program on the Möbius disjointness conjecture. Here we continue our recent investigation [arXiv:1511.06137] of the extended hull ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, a dynamical system naturally associated to a weak model set in an abelian group $G$ with relatively compact window $W$. For windows having a nowhere dense boundary (this includes compact windows), we identify the maximal equicontinuous factor of ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ and give a sufficient condition when ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ is an almost 1:1 extension of its maximal equicontinuous factor. If the window is measurable with positive Haar measure and is almost compact, then the system ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$ equipped with its Mirsky measure is isomorphic to its Kronecker factor. For general nontrivial ergodic probability measures on ${\mathcal M}^{\scriptscriptstyle G}_{\scriptscriptstyle W}$, we provide a kind of lower bound for the Kronecker factor. All relevant factor systems are natural $G$-actions on quotient subgroups of the torus underlying the weak model set. These are obtained by factoring out suitable window periods. Our results are specialised to the usual hull of the weak model set, and they are also interpreted for ${\mathcal B}$-free systems.

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arXiv2017

Dynamics of $\mathcal B$-free sets: a view through the window

Let $\mathcal B$ be an infinite subset of $\{1,2,\dots\}$. We characterize arithmetic and dynamical properties of the $\mathcal B$-free set $\mathcal F_{\mathcal B}$ through group theoretical, topological and measure theoretic properties of a set $W$ (called the window) associated with $\mathcal B$. This point of view stems from the interpretation of the set $\mathcal F_{\mathcal B}$ as a weak model set. Our main results are: $\mathcal B$ is taut if and only if the window is Haar regular; the dynamical system associated to $\mathcal F_{\mathcal B}$ is a Toeplitz system if and only if the window is topologically regular; the dynamical system associated to $\mathcal F_{\mathcal B}$ is proximal if and only if the window has empty interior; and the dynamical system associated to $\mathcal F_{\mathcal B}$ has the "naïvely expected" maximal equicontinuous factor if and only if the interior of the window is aperiodic.

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arXiv2017

Stability index, uncertainty exponent, and thermodynamic formalism for intermingled basins of chaotic attractors

Skew product systems with monotone one-dimensional fibre maps driven by piecewise expanding Markov interval maps may show the phenomenon of intermingled basins. To quantify the degree of intermingledness the uncertainty exponent and the stability index were suggested by various authors and characterized (partially). Here we present an approach to evaluate/estimate these two quantities rigorously using thermodynamic formalism for the driving Markov map.

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arXiv2016

Chaotically driven sigmoidal maps

We consider skew product dynamical systems $f:Θ\times\mathbb{R}\toΘ\times\mathbb{R}, f(θ,y)=(Tθ,f_θ(y))$ with a (generalized) baker transformation $T$ at the base and uniformly bounded increasing $C^3$ fibre maps $f_θ$ with negative Schwarzian derivative. Under a partial hyperbolicity assumption that ensures the existence of strong stable fibres for $f$ we prove that the presence of these fibres restricts considerably the possible structures of invariant measures - both topologically and measure theoretically, and that this finally allows to provide a "thermodynamic formula" for the Hausdorff dimension of set of those base points over which the dynamics are synchronized, i.e. over which the global attractor consists of just one point.

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arXiv2016

Dynamics on the graph of the torus parametrisation

Model sets are projections of certain lattice subsets. It was realised by Moody that dynamical properties of such sets are induced from the torus associated with the lattice. We follow and extend this approach by studying dynamics on the graph of the map which associates lattice subsets to points of the torus and then transferring the results to their projections. This not only leads to transparent proofs of known results on model sets, but we also obtain new results on so called weak model sets. In particular we prove pure point dynamical spectrum for the hull of a weak model set together with the push forward of the torus Haar measure under the torus parametrisation map, and we derive a formula for the pattern frequencies of configurations with maximal density.

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arXiv2015

An elementary proof for the dimension of the graph of the classical Weierstrass function

Let $W_{λ,b}(x)=\sum_{n=0}^\inftyλ^n g(b^n x)$ where $b\geqslant2$ is an integer and $g(u)=\cos(2πu)$ (classical Weierstrass function). Building on work by Ledrappier (1992), Baránsky, Bárány and Romanowska (2013) and Tsujii (2001), we provide an elementary proof that the Hausdorff dimension of $W_{λ,b}$ equals $2+\frac{\logλ}{\log b}$ for all $λ\in(λ_b,1)$ with a suitable $λ_b<1$. This reproduces results by Baránsky, Bárány and Romanowska without using the dimension theory for hyperbolic measures of Ledrappier and Young (1985,1988), which is replaced by a simple telescoping argument together with a recursive multi-scale estimate.

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arXiv2013

A model for the nonautonomous Hopf bifurcation

Inspired by an example of Grebogi et al [1], we study a class of model systems which exhibit the full two-step scenario for the nonautonomous Hopf bifurcation, as proposed by Arnold [2]. The specific structure of these models allows a rigorous and thorough analysis of the bifurcation pattern. In particular, we show the existence of an invariant 'generalised torus' splitting off a previously stable central manifold after the second bifurcation point. The scenario is described in two different settings. First, we consider deterministically forced models, which can be treated as continuous skew product systems on a compact product space. Secondly, we treat randomly forced systems, which lead to skew products over a measure-preserving base transformation. In the random case, a semiuniform ergodic theorem for random dynamical systems is required, to make up for the lack of compactness.

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arXiv2013

Random minimality and continuity of invariant graphs in random dynamical systems

We study dynamical systems forced by a combination of random and deterministic noise and provide criteria, in terms of Lyapunov exponents, for the existence of random attractors with continuous structure in the fibres. For this purpose, we provide suitable random versions of the semiuniform ergodic theorem and also introduce and discuss some basic concepts of random topological dynamics.

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