Ergodic Properties of Randomly Coloured Point Sets

Müller, Peter; Richard, Christoph

doi:10.4153/cjm-2012-009-7

Cited by 35 publications

(52 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…At the end of that proof additional arguments are provided to prove uniform convergence when m H (∂W) = 0. The same arguments yield the semi-uniform convergence for general compact windows W, see also [60,44].…”

Section: Measure Theoretic Resultsmentioning

confidence: 65%

Dynamics on the graph of the torus parametrization

Keller¹,

Richard²

2016

Ergod. Th. Dynam. Sys.

Self Cite

110

View full text Add to dashboard Cite

Model sets are projections of certain lattice subsets. It was realised by Moody that dynamical properties of such a set 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 of maximal density together with the push forward of the torus Haar measure under the torus parametrisation map, and we derive a formula for its pattern frequencies. * These notes profited enormously from talks by and/or discussions with Michael Baake, Tobias Hartnick, Johannes Kellendonk, Mariusz Lemańczyk, Daniel Lenz, Tobias Oertel-Jäger and Nicolae Strungaru, from several inspiring workshops in the framework of the DFG Scientific Network "Skew Product Dynamics and Multifractal Analysis" organised by Tobias Oertel-Jäger, and finally from very helpful and supporting comments of an anonymous referee.• we obtain as a warm-up some relatives to well known topological results on maximal equicontinuous factors and the lack of weak mixing (Theorem 1),• we can introduce a general notion of Mirsky measures, namely the push forward of Haar measure onX under the map ν W , compare [19,17,34,51],• we show that the systems equipped with this Mirsky measure have pure point dynamical spectrum (Theorem 2),• we prove strict ergodicity when m H (∂W) = 0 (Theorem 2),• we identify the Mirsky measure as the unique invariant measure with maximal density for typical configurations (Theorem 4),• we show that the configurations with maximal density are precisely the generic points for the Mirsky measure (Theorem 5),• and we deduce from this a formula for the pattern frequencies of configurations with maximal density (Remark 3.12), which is also discussed in [5, Rem. 5].While the measure theoretic assertions from this list "survive" the passage to configurations on G even if the relevant projection is not 1-1, some finer information can be transferred in this way only if that projection is 1-1 when restricted to sufficiently large subsets. This issue is discussed in Section 4 for model sets with interval windows, topologically regular windows and B-free systems.

show abstract

Section: Measure Theoretic Resultsmentioning

confidence: 65%

Dynamics on the graph of the torus parametrization

Keller¹,

Richard²

2016

Ergod. Th. Dynam. Sys.

Self Cite

110

View full text Add to dashboard Cite

show abstract

“…Also, by the results of [20], we have that weighted return time measures for mixing transformations are, in general, not weakly almost periodic. Other works where the autocorrelation of group actions are investigated include [19,24,26], and works discussing the effect of mixing conditions on the autocorrelation of tilings include [7,23,32].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Diffraction of Return Time Measures

et al. 2018

View full text Add to dashboard Cite

Letting T denote an ergodic transformation of the unit interval and letting f : [0, 1) → R denote an observable, we construct the f -weighted return time measure µ y for a reference point y ∈ [0, 1) as the weighted Dirac comb with support in Z and weights f • T z (y) at z ∈ Z, and if T is non-invertible, then we set the weights equal to zero for all z < 0. Given such a Dirac comb, we are interested in its diffraction spectrum which emerges from the Fourier transform of its autocorrelation and analyse it for the dependence on the underlying transformation. For certain rapidly mixing transformations and observables of bounded variation, we show that the diffraction of µ y consists of a trivial atom and an absolutely continuous part, almost surely with respect to y. This contrasts what occurs in the setting of regular model sets arising from cut and project schemes and deterministic incommensurate structures. As a prominent example of non-mixing transformations, we consider the family of rigid rotations T α : x → x + α mod 1 with rotation number α ∈ R + . In contrast to when T is mixing, we observe that the diffraction of µ y is pure point, almost surely with respect to y. Moreover, if α is irrational and the observable f is Riemann integrable, then the diffraction of µ y is independent of y. Finally, for a converging sequence (α i ) i∈N of rotation numbers, we provide new results concerning the limiting behaviour of the associated diffractions.Here, for z ∈ Z, we let δ z denote the Dirac point mass at z. In this note we answer the following questions.

show abstract

“…For amenable groups tempered Følner sequences will do the job, as is ensured by a general ergodic theorem of Lindenstrauss [43]. We refer to [39,49,50] for more details in the present context and sum up the main points in the following definition and the subsequent results.…”

Section: The Integrated Density Of Statesmentioning

confidence: 97%

“…This structure will not be seen by the linear clusters of percolation! The first class of models [49,50] consists of graphs G which are embedded into R d (or, more generally, into a suitable locally compact, complete metric space) with some form of aperiodic order. The celebrated Penrose tiling in R 2 constitutes a prime example.…”

Section: Outlook: Some Further Modelsmentioning

confidence: 99%

“…As such it carries at least one R d -ergodic probability measure µ, and the expectation in (2.3) will be replaced by a two-stage expectation: one with respect to µ over all graphs G in the hull of G, and inside of it, for each graph G , the expectation E (G ) p over all realisations of percolation subgraphs of G . The interested reader is referred to [37,50] for more details.…”

Section: Outlook: Some Further Modelsmentioning

confidence: 99%

See 1 more Smart Citation

Percolation Hamiltonians

Müller

Stollmann

2011

Random Walks, Boundaries and Spectra

Self Cite

View full text Add to dashboard Cite

Abstract. There has been quite some activity and progress concerning spectral asymptotics of random operators that are defined on percolation subgraphs of different types of graphs. In this short survey we record some of these results and explain the necessary background coming from different areas in mathematics: graph theory, group theory, probability theory and random operators. Mathematics Subject Classification (2000). Primary 05C25; Secondary 82B43. Keywords. Random graphs, random operators, percolation, phase transitions. PreliminariesHere we record basic notions, mostly to fix notation. Since this survey is meant to be readable by experts from different communities, this will lead to the effect that many readers might find parts of the material in this section pretty trivialnever mind. GraphsA graph is a pair G = (V, E) consisting of a countable set of vertices V together with a set E of edges. Since we consider undirected graphs without loops, edges can and will be regarded as subsets e = {x, y} ⊆ V . In this case we say that e is an edge between x and y, respectively adjacent to x and y. Sometimes we write x ∼ y to indicate that {x, y} ∈ E. The degree, the number of edges adjacent to x, is denoted byA graph with constant degree equal to k is called a k-regular graph.A path is a finite family γ := (e 1 , e 2 , ..., e n ) of consecutive edges, i.e., such that e k ∩ e k+1 = ∅; the set of points visited by γ is denoted by γ * := e 1 ∪ ... ∪ e n .

show abstract

Ergodic Properties of Randomly Coloured Point Sets

Cited by 35 publications

References 27 publications

Dynamics on the graph of the torus parametrization

Dynamics on the graph of the torus parametrization

Diffraction of Return Time Measures

Percolation Hamiltonians

Contact Info

Product

Resources

About