31 research outputs found
Substitutions, tiling dynamical systems and minimal self-joinings
We investigate substitution subshifts and tiling dynamical systems arising
from the substitutions (1) \theta : 0 \rightarrow 001,1 \rightarrow 11001 and
(2) \eta : 0 \rightarrow 001,1 \rightarrow 11100. We show that the substitution
subshifts arising from \theta and \eta have minimal self-joinings and are
mildly mixing. We also give a criterion for 1-dimensional tiling systems
arising from \theta or \eta to have minimal self-joinings. We apply this to
obtain examples of mildly mixing 1-dimensional tiling systems
Joint ergodicity of piecewise monotone interval maps
For , let be a Borel probability measure on
which is equivalent to Lebesgue measure and let be -preserving ergodic transformations.
We say that transformations are uniformly jointly
ergodic with respect to if for any
,
We establish convenient criteria for uniform joint ergodicity and obtain
numerous applications, most of which deal with interval maps. Here is a
description of one such application. Let denote the Gauss map, , and, for , let denote the
-transformation defined by . Let
be an ergodic interval exchange transformation. Let be distinct real numbers with and assume that for all . Then for any
, \begin{equation*}
\begin{split} \lim\limits_{N -M \rightarrow \infty} \frac{1}{N -M }
\sum\limits_{n=M}^{N-1} & f_{0} (T_0^n x) \cdot f_{1} (T_{\beta_1}^n x) \cdots
f_{k} (T_{\beta_k}^n x) \cdot f_{k+1} (T_G^n x)
&= \int f_{0} \, d \lambda \cdot \prod_{i=1}^k \int f_{i} \, d \mu_{\beta_i}
\cdot \int f_{k+1} \, d \mu_G \quad \text{in } L^{2}(\lambda). \end{split}
\end{equation*}
We also study the phenomenon of joint mixing. Among other things we establish
joint mixing for skew tent maps and for restrictions of finite Blaschke
products to the unit circle.Comment: 38 page
Uniform Distribution of Prime Powers and sets of Recurrence and van der Corput sets in Z^k
We establish new results on sets of recurrence and van der Corput sets in Z^k
which refine and unify some of the previous results obtained by Sarkozy,
Furstenberg, Kamae and Mendes France, and Bergelson and Lesigne. The proofs
utilize a general equidistribution result involving prime powers which is of
independent interest
An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications
Author's accepted manuscript.This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Bergelson, V., Knutson, I. J. H. & Son, Y. (2020). An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications. International Mathematics Research Notices, 2021(19), 14965-15018 is available online at: https://academic.oup.com/imrn/article/2021/19/14965/5775499 and https://doi.org/10.1093/imrn/rnaa035.Generalized polynomials are mappings obtained from the conventional polynomials by the use of the operations of addition and multiplication and taking the integer part. Extending the classical theorem of Weyl on equidistribution of polynomials, we show that a generalized polynomial q(n) has the property that the sequence (q(n)λ)n∈Z is well-distributed mod1 for all but countably many λ∈R if and only if lim|n|→∞n∉J|q(n)|=∞ for some (possibly empty) set J having zero natural density in Z. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of Vinogradov and Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets.publishedVersio
Joint ergodicity of actions of an Abelian group
Let be a countable abelian group and let be measure preserving -actions on a probability space. We prove that joint ergodicity of implies total joint ergodicity if each is totally ergodic. We also show that if , and the actions commute, then total joint ergodicity of follows from joint ergodicity. This can be seen as a generalization of Berend’s result for commuting -actions.11Nsciescopu
Birkhoff sum fluctuations in substitution dynamical systems
We consider the deviation of Birkhoff sums along fixed orbits of substitution dynamical systems. We show distributional convergence for the Birkhoff sums of eigen functions of the substitution matrix. For non-coboundary eigen functions with eigen value of modulus 1, we obtain a central limit theorem. For other eigen functions, we show convergence to distributions supported on Cantor sets. We also give a new criterion for such an eigen function to be a coboundary, as well as a new characterization of substitution dynamical systems with bounded discrepancy.11Nsciescopu