Adic Transformation Research Articles

Notes on Ergodic $$2$$-Adic Transformations

In this work we provide a new representation of isometric transformations on the group of $$2$$ -adic integers, then establish an appropriate ergodicity test.

On a Universal Borel Adic Space

We prove that the so-called uniadic graph and its adic automorphism are Borel universal, i.e., every aperiodic Borel automorphism is isomorphic to the restriction of this automorphism to a subset invariant under the adic transformation, the isomorphism being defined on a universal (with respect to the measure) set. We develop the concept of basic filtrations and combinatorial definiteness of automorphisms suggested in our previous paper.

Limiting Curves for the Pascal Adic Transformation

The paper generalizes results by E. Janvresse, T. de la Rue, and Y. Velenik and results by the second author on the fluctuations in the ergodic sums for the Pascal adic transformation in the case of an arbitrary ergodic invariant measure and arbitrary cylinder function.

A Higher Order Asymptotic Expansion of the Krawtchouk Polynomials

The paper extends a classical result on the convergence of the Krawtchouk polynomials to the Hermite polynomials. We provide a uniform asymptotic expansion in terms of Hermite polynomials and obtain explicit expressions for a few first terms of this expansion. The research is motivated by the study of ergodic sums of the Pascal adic transformation. Bibliography: 10 titles.

Random Deviations of Ergodic Sums for the Pascal Adic Transformation in the Case of the Lebesgue Measure

The paper generalizes the results by E. Janvresse, T. de la Rue, and Y. Velenik on fluctuations in ergodic sums for the Pascal adic transformation in the case of the Lebesgue measure for a wide class of functions. In particular, we answer several questions from that paper.

On the L p discrepancy of two-dimensional folded Hammersley point sets

We give an explicit construction of two-dimensional point sets whose $L_p$ discrepancy is of best possible order for all $1\le p\le \infty$. It is provided by folding Hammersley point sets in base $b$ by means of the $b$-adic baker's transformation which has been introduced by Hickernell (2002) for $b=2$ and Goda, Suzuki and Yoshiki (2013) for arbitrary $b\in \mathbb{N}$, $b\ge 2$. We prove that both the minimum Niederreiter-Rosenbloom-Tsfasman weight and the minimum Dick weight of folded Hammersley point sets are large enough to achieve the best possible order of $L_p$ discrepancy for all $1\le p\le \infty$.

Orbit equivalence of $$p$$ p -adic transformations and their iterates

Orbit equivalence is a weaker notion of equivalence than isomorphism for measurable systems. We examine $$p$$ -adic transformations in various orbit equivalence classes, including an example that preserves an infinite measure. Although $$p$$ -adic transformations have been studied with respect to Haar measure, we use other i.i.d. product measures to see examples of different orbit equivalence classes. Since translation by an integer is an iterate of translation by 1, we gain an understanding of the possible behaviors of orbit equivalence classes under iteration.

Remarks on a class of periodic points for <i>p</i>-adic transformations and stylized Chebyshev polynomials

Remarks on a class of periodic points for <i>p</i>-adic transformations and stylized Chebyshev polynomials

A Thom-Sebastiani theorem in characteristic $p$

Let $k$ be a perfect field of characteristic $p$, let $f_i:X_i\to\mathbb A_k^1$ $(i=1,2)$ be two $k$-morphism of finite type, and let $f:X_1\times_k X_2\to \mathbb A_k^1$ be the morphism defined by $f(z_1,z_2)=f_1(z_1)+f_2(z_2)$. For each $i\in\{1,2\}$, let $x_i$ be a $k$-rational point in the fiber $f_i^{-1}(0)$ such that $f_i$ is smooth on $X_i-\{x_i\}$. Using the $\ell$-adic Fourier transformation and the stationary phase principle of Laumon, we prove that the vanishing cycle of $f$ at $x=(x_1,x_2)$ is the convolution product of the vanishing cycles of $f_i$ at $x_i$ $(i=1,2)$.

The geometry of non-unit Pisot substitutions

It is known that with a non-unit Pisot substitution σ one can associate certain fractal tiles, so-called Rauzy fractals. In our setting, these fractals are subsets of a certain open subring of the adèle ring of the associated Pisot number field. We present several approaches on how to define Rauzy fractals and discuss the relations between them. In particular, we consider Rauzy fractals as the natural geometric objects of certain numeration systems, in terms of the dual of the one-dimensional realization of σ, and in the context of model sets for particular cut and project schemes. We also define stepped surfaces suited for non-unit Pisot substitutions. We provide basic topological and geometric properties of the Rauzy fractals, prove some tiling results for them, and provide relations to subshifts defined in terms of the periodic points of σ, to adic transformations, and a domain exchange.

Exponential chi-squared distributions in infinite ergodic theory

AbstractWe prove distributional limit theorems for random walk adic transformations obtaining ergodic distributional limits of exponential$\chi ^2$form.

Asymptotic behavior of the scaling entropy of the Pascal adic transformation

In this paper, we give an estimate on the growth of the scaling sequence of the Pascal adic transformation with respect to the sup-metric. We construct a special class of α-names of positive cumulative measure. The linear growth of its cardinality implies the logarithmic growth of the scaling sequence. Bibliography: 14 titles.

Generalisations of the Euler adic

AbstractWe consider generalisations of the so-called Euler adic and investigate dynamical properties like ergodicity and total ergodicity. We prove the existence of a unique fully-supported ergodic measure for these generalisations. We also investigate the structure of non-fully-supported ergodic measures and in addition show that each of these measures (fully- and non-fully-supported) is also totally ergodic. In order to determine these dynamical properties we find closed-form expressions for the generalised Eulerian numbers. Additionally we extend a result given by Frick and Petersen to a wider class of adic transformations.

Reinforced Random Walks and Adic Transformations

To a given finite graph we associate three kinds of adic, or Bratteli–Vershik, systems: stationary, symbol-count, and reinforced. We give conditions for the natural walk measure to be adic-invariant and identify the ergodic adic-invariant measures for some classes of examples. If the walk measure is adic-invariant, we relate its ergodic decomposition to the vector of limiting edge traversal frequencies. For some particular nonsimple reinforcement schemes, we calculate the density function of the edge traversal frequencies explicitly.

Minimality and unique ergodicity for adic transformations

We study the relationship between minimality and unique ergodicity for adic transformations. We show that three is the smallest alphabet size for a unimodular “adic counterexample”, an adic transformation which is minimal but not uniquely ergodic. We construct a specific family of counterexamples built from (3 × 3) nonnegative integer matrix sequences, while showing that no such (2 × 2) sequence is possible. We also consider (2 × 2) counterexamples without the unimodular restriction, describing two families of such maps.Though primitivity of the matrix sequence associated to the transformation implies minimality, the converse is false, as shown by a further example: an adic transformation with (2 × 2) stationary nonprimitive matrix, which is both minimal and uniquely ergodic.

NONSTATIONARY MIXING AND THE UNIQUE ERGODICITY OF ADIC TRANSFORMATIONS

We define topological and measure-theoretic mixing for nonstationary dynamical systems and prove that for a nonstationary subshift of finite type, topological mixing implies the minimality of any adic transformation defined on the edge space, while if the Parry measure sequence is mixing, the adic transformation is uniquely ergodic. We also show this measure theoretic mixing is equivalent to weak ergodicity of the edge matrices in the sense of inhomogeneous Markov chain theory.

Limited scope adic transformations

We introduce a family of adic transformations on diagrams that are nonstationary and nonsimple. This family includes some previously studied adic transformations such as the Pascal and Euler adic transformations. We give examples of particular adic transformations with roots of unity and we show as well that the Euler adic is totally ergodic. We show that the Euler adic and a disjoint subfamily of adic transformations are loosely Bernoulli.

Random permutations and unique fully supported ergodicity for the Euler adic transformation

There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the Eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.

Random permutations and unique fully supported ergodicity for the Euler adic transformation

Random permutations and unique fully supported ergodicity for the Euler adic transformation

Ergodicity of the adic transformation on the Euler graph

The Euler graph has vertices labelled (n,k) for n=0,1,2,... and k=0,1,...,n, with k+1 edges from (n,k) to (n+1,k) and n-k+1 edges from (n,k) to (n+1,k+1). The number of paths from (0,0) to (n,k) is the Eulerian number A(n,k), the number of permutations of 1,2,...,n+1 with exactly n-k falls and k rises. We prove that the adic (Bratteli-Vershik) transformation on the space of infinite paths in this graph is ergodic with respect to the symmetric measure.