Bounded remainder sets for rotations on higher-dimensional adelic tori

It has been a long-standing problem to find good symbolic codings for translations on the d -dimensional torus that enjoy the beautiful properties of Sturmian sequences like low factor complexity and good local discrepancy properties. Inspired by Rauzy’s approach we construct such codings in terms of multidimensional continued fraction algorithms that are realized by sequences of substitutions. In particular, given any exponentially convergent continued fraction algorithm, these sequences lead to renormalization schemes which produce symbolic codings of toral translations and bounded remainder sets at all scales in a natural way. The exponential convergence properties of a continued fraction algorithm can be viewed in terms of a Pisot type condition imposed on an attached symbolic dynamical system. Using this fact, our approach provides a systematic way to confirm purely discrete spectrum results for wide classes of symbolic dynamical systems. Indeed, as our examples illustrate, we are able to confirm the Pisot conjecture for many well-known families of sequences of substitutions. These examples include classical algorithms like the Jacobi–Perron, Brun, Cassaigne–Selmer, and Arnoux–Rauzy algorithms. As a consequence, we gain symbolic codings of almost all translations of the 2-dimensional torus having factor complexity 2n + 1 that are balanced for words, which leads to multiscale bounded remainder sets. Using the Brun algorithm, we also give symbolic codings of almost all 3-dimensional toral translations having multiscale bounded remainder sets.

Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary

The paper is devoted to the important problem of number theory: bounded remainder sets. We consider the point orbits on low-dimensional tori. Any starting point generates the orbit under an irrational shift of the torus. The orbit is everywhere dense and uniformly distributed on the torus if the translation vector is irrational. Denote by r(i) a function that gives the number of the orbit points which get some domain T. Then we have the formula r(i) = i vol(T) + δ(i), where δ(i) = o(i) is the remainder. If the boundaries of the remainder are limited by a constant, then T is a bounded remainder set ( BR-set). The article introduces a new BR-sets construction method, it is based on tilings parametric polyhedra. Сonsidered polyhedra are the torus development. Torus development should be to tile into figures, that can be exchanged, and we again obtain our torus development. This figures exchange equivalent shift of the torus. Author have constructed tillings with this property and two-dimensional BR-sets. The considered method gives exact estimates and the average value of the remainder. Also we obtain the optimal BR-sets which have minimal values of the remainder. These BR-sets generate the strong balanced words ( a multi-dimensional analogue of the Sturmian words). The above method is applied to the case of three-dimensional torus in this paper. Also we obtain exact estimates and the average value of the remainder for constructed sets.

We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by $α$ where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point $β$. When $α$ is badly approximable and $β$ is badly approximable with respect to $α$, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D.Dolgopyat and O.Sarig), namely we show that for any fixed initial point, the occupancy random variables, suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when $α$ is quadratic irrational, $β$ is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig (Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use renormalization, but in order to treat irrational values of $β$, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains.

The dynamical system on $$\mathbb{T}^2$$ which is a group extension over an irrational rotation on $$\mathbb{T}^1$$ is investigated. The criterion when the extension is minimal, a system of order 2 and when the maximal equicontinuous factor is the irrational rotation is given. The topological complexity of the extension is computed, and a negative answer to the latter part of an open question raised by Host et al. (2014) is obtained.

Bounded remainder sets

This is the proceedings of theworkshop on recent developments in ergodic theory and dynamical systemson March 2011and March 2012 at the University of North Carolina at Chapel Hill. Thearticles in this volume cover several aspects of vibrant research in ergodic theory and dynamical systems. It contains contributions to Teichmuller dynamics, interval exchange transformations, continued fractions, return times averages, Furstenberg Fractals, fractal geometry of non-uniformly hyperbolic horseshoes, convergence along the sequence of squares, adic and horocycle flows, and topological flows. These contributions illustrate the connections between ergodic theory and dynamical systems, number theory, harmonic analysis, probability, andalgebra. Two surveys are included which give a nice introduction for interested young or senior researcher to some active research areas. Overall this volume provides a very useful blend of techniques and methods as well as directions of research on general convergence phenomena in ergodic theory and dynamical systems.

For homeomorphisms $$\left( {z,w} \right)\mathop \to \limits^{T\varphi } \left( {z . e^{2xi\alpha } ,\varphi \left( z \right)w} \right)$$ (z, w ∈S 1,α is irrational,ϕ:S 1→S 1) of the torusS 1×S 1 it is proved thatTϕ has countable Lebesgue spectrum in the orthocomplement of the eigenfunctions wheneverϕ is absolutely continuous with nonzero topological degree and the derivative ofϕ is of bounded variation. Some other cocycles with bounded variation are studied and generalizations of the above result to certain distal homeomorphisms on finite dimensional tori are presented.

We consider random walks on $\Z$ in a stationary random medium, defined by an ergodic dynamical system, in case when possible jumps are $\{-L,\ldots,-1,0,+1\}$ for some fixed integer L. We provide a recurrence criterion expressed in terms of sign of maximal Liapounov exponent of a certain random matrix and give an algorithm of calculation of that exponent. Next, we characterize existence of absolutely continuous invariant measure for Markov chain of the environments viewed from particle and also characterize, in transient cases, existence of a nonzero drift. To study validity of central limit theorem, we consider notion of harmonic coordinates introduced by Kozlov. We characterize existence of both invariant measure and harmonic coordinates and show in recurrent case that existence of those two objects is equivalent to validity of an invariance principle. We give sufficient conditions for validity of central limit theorem in transient cases. Finally, we consider previous results in context of a random medium defined by an irrational rotation on circle and study their realization in terms of regularity and Diophantine approximation.

We establish new characterizations for (pseudo)isometric extensions of topological dynamical systems. For such extensions, we also extend results about relatively invariant measures and Fourier analysis that were previously only known in the minimal case to a significantly larger class, including all transitive systems. To bypass the reliance on minimality of the classical approaches to isometric extensions via the Ellis semigroup, we show that extensions of topological dynamical systems can be described as groupoid actions and then adapt the concept of enveloping semigroups to construct a uniform enveloping semigroupoid for groupoid actions. This approach allows to deal with the more complex orbit structures of nonminimal systems.We study uniform enveloping semigroupoids of general groupoid actions and translate the results back to the special case of extensions of dynamical systems. In particular, we show that, under appropriate assumptions, a groupoid action is (pseudo)isometric if and only if the uniform enveloping semigroupoid is actually a compact groupoid. We also provide an operator theoretic characterization based on an abstract Peter—Weyl-type theorem for representations of compact, transitive groupoids on Banach bundles which is of independent interest.

The Rudin-Shapiro sequence appears both in Fourier Analysis and Number Theory. The Ising chain is a crude model for magnetic substance and plays a fundamental rôle in Statistical Mechanics. The study of patterns of folds on a sheet of paper is linked to many domains including Number Theory and Dynamical Systems. These three concepts are different facets of one and the same object.KeywordsPartition FunctionIsing ModelCrude ModelBinary DigitBinary ExpansionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Recent findings of a genome-wide oscillation involving the transcriptome of the budding yeast Saccharomyces cerevisiae suggest that the most promising path to an understanding of the cell as a dynamic system will proceed from carefully designed time-series sampling followed by the development of signal-processing methods suited to molecular biological datasets. When everything oscillates, conventional biostatistical approaches fall short in identifying functional relationships among genes and their transcripts. Worse, based as they are on steady-state assumptions, such approaches may be misleading. In this chapter, we describe the continuous gated synchrony system and the experiments leading to the concept of genome-wide oscillations, and suggest methods of analysis better suited to dissection of oscillating systems. Using a yeast continuous-culture system, the most precise and stable biological system extant, we explore analytical tools such as wavelet multiresolution decomposition, Fourier analysis, and singular value decomposition to uncover the dynamic architecture of phenotype.

The theory of aperiodic order expanded and developed significantly since the discovery of quasicrystals, and continues to bring many mathematical disciplines together. The focus of this workshop was on harmonic analysis and spectral theory, dynamical systems and group actions, Schrödinger operators, and their roles in aperiodic order – with links into a full range of problems from number theory to operator theory.

The study of geodesic flows on homogenous spaces is an area of research that has yielded some fascinating developments. This book, first published in 2000, focuses on many of these, and one of its highlights is an elementary and complete proof (due to Margulis and Dani) of Oppenheim's conjecture. Also included here: an exposition of Ratner's work on Raghunathan's conjectures; a complete proof of the Howe-Moore vanishing theorem for general semisimple Lie groups; a new treatment of Mautner's result on the geodesic flow of a Riemannian symmetric space; Mozes' result about mixing of all orders and the asymptotic distribution of lattice points in the hyperbolic plane; Ledrappier's example of a mixing action which is not a mixing of all orders. The treatment is as self-contained and elementary as possible. It should appeal to graduate students and researchers interested in dynamical systems, harmonic analysis, differential geometry, Lie theory and number theory

Many areas of mathematics were deeply influenced or even founded by Hermann Weyl, including geometric foundations of manifolds and physics, topological groups, Lie groups and representation theory, harmonic analysis and analytic number theory as well as foundations of mathematics. In this volume, leading experts present his lasting influence on current mathematics, often connecting Weyl's theorems with cutting edge research in dynamical systems, invariant theory, and partial differential equations. In a broad and accessible presentation, survey chapters describe the historical development of each area alongside up-to-the-minute results, focussing on the mathematical roots evident within Weyl's work.