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## Convergence of spherical averages for actions of Fuchsian groups

Commentarii Mathematici Helvetici. 2023. Vol. 98. No. 1. P. 41-134.

We prove pointwise convergence of spherical averages for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen–Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by the first author for the case of free groups may be applied.

Klimenko A. V., Буфетов А. И., Сириес К., Успехи математических наук 2020

В заметке представлен результат о сходимости почти всюду сферических средних для сохраняющщих вероятностную меру действий фуксовой группы, если группа и набор образующих удовлетворяют условию "ровных углов" (even corners). Именно, группа обладает фундаментальной областью, для которой граница замощения диска её образами состоит из целых геодезических, а в качестве набора образующих взяты элементы группы, переводящие эту фундаментальную ...

Added: October 31, 2019

Klimenko A. V., Bufetov A., Series C., / Cornell University. Series math "arxiv.org". 2018. No. 1805.11743.

Pointwise convergence of spherical averages is proved for a measure-preserving action of a Fuchsian group. The proof is based on a new variant of the Bowen-Series symbolic coding for Fuchsian groups that, developing a method introduced by Wroten, simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving ...

Added: September 18, 2018

Minabutdinov A., Записки научных семинаров ПОМИ РАН 2016 Т. 448 С. 177-200

We prove existence of limiting curves (describing deviations in ergodic theorem) for cylindrical functions for Polynomial adic systems. For a general ergodic measure-preserving transformation and a summable function we give necessary condition of a limiting curve to exist. Our work generalizes results by E'. Janvresse, T. de la Rue and Y. Velenik. ...

Added: October 14, 2016

Vyacheslav V. Chistyakov, Tretyachenko Y. V., Journal of Mathematical Analysis and Applications 2013 Vol. 402 No. 2 P. 648-659

Given a rectangle in the real Euclidean n-dimensional space and two maps f and g defined on it and taking values in a metric semigroup, we introduce the notion of the total joint variation TV(f , g) of these maps. This extends similar notions considered by Hildebrandt (1963) [17], Leonov (1998) [18], Chistyakov (2003, 2005) ...

Added: August 29, 2013

Vyacheslav V. Chistyakov, Svetlana A. Chistyakova, Studia Mathematica 2017 Vol. 238 No. 1 P. 37-57

Given a subset $T$ of the reals $R$ and a metric space $M$, we introduce a nondecreasing sequence $\{\nu_n\}$ of pseudometrics on $M^T$ (the set of all functions from $T$ into $M$), called the joint modulus of variation. We prove that if two sequences $\{f_j\}$ and $\{g_j\}$ of functions from $M^T$ are such that $\{f_j\}$ ...

Added: May 11, 2017

V. V. Chistyakov, S. A. Chistyakova, Lobachevskii Journal of Mathematics 2022 Vol. 43 No. 3 P. 550-563

Given a Hausdorff uniform space X with the countable gage of pseudometrics of the
uniformity of X, we introduce a concept of the approximate variation of a function f mapping a
subset T of the reals intoX: this is the infimum of the family of Jordan-type variations of all functions
g : T → X which differ from ...

Added: April 30, 2022

Vyacheslav V. Chistyakov, Svetlana A. Chistyakova, / Cornell University. Series math "arxiv.org". 2016. No. 1601.07298.

Given a subset T of real numbers and a metric space M, we introduce a nondecreasing sequence {v_n} of pseudometrics on the set M^T of all functions from T into M, called the joint modulus of variation. We prove that if two sequences of functions {f_j} and {g_j} from M^T are such that {f_j} is ...

Added: February 12, 2016

Vyacheslav V. Chistyakov, Svetlana A. Chistyakova, / Cornell University, NY, USA. Series arXiv [math.FA] "Functional Analysis". 2020. No. 2010.11410.

Let $T\subset\mathbb{R}$ and $(X,\mathcal{U})$ be a uniform space with an at most countable gage of pseudometrics $\{d_p:p\in\mathcal{P}\}$ of the uniformity $\mathcal{U}$. Given $f\in X^T$ (=\,the family of all functions from $T$ into $X$), the {\em approximate variation\/} of $f$ is the two-parameter family $\{V_{\varepsilon,p}(f):\varepsilon>0,p\in\mathcal{P}\}$, where $V_{\varepsilon,p}(f)$ is the greatest lower bound of Jordan's variations $V_p(g)$ ...

Added: October 23, 2020

Vyacheslav V. Chistyakov, Svetlana A. Chistyakova, Journal of Mathematical Analysis and Applications 2017 Vol. 452 No. 2 P. 970-989

We introduce a pseudometric TV on the set M^X of all functions mapping a rectangle X on the plane R^2 into a metric space M, called the total joint variation. We prove that if two sequences {fj} and {gj} of functions from M^X are such that {fj} is pointwise precompact on X, {gj} is pointwise ...

Added: April 13, 2017

Vyacheslav V. Chistyakov, Springer, 2021

This book addresses the minimization of special lower semicontinuous functionals
over (closed) balls in metric spaces, called the approximate variation. The new
notion of approximate variation contains more information about the bounded
variation functional and has the following features: the infimum in the definition
of approximate variation is not attained in general and the total Jordan variation
of a function ...

Added: October 29, 2021

Vyacheslav V. Chistyakov, / Cornell University Library, NY, USA. Series arXiv [math.FA] "Functional Analysis". 2019. No. arXiv: 1910.08490.

Let $T\subset\mathbb{R}$, $M$ be a metric space with metric $d$, and $M^T$ be the set of all functions mapping $T$ into $M$. Given $f\in M^T$, we study the properties of the approximate variation $\{V_\varepsilon(f)\}_{\varepsilon>0}$, where $V_\varepsilon(f)$ is the greatest lower bound of Jordan variations $V(g)$ of functions $g\in M^T$ such that $d(f(t),g(t))\le\varepsilon$ for all $t\in T$. The notion of $\varepsilon$-variation ...

Added: October 21, 2019

Schwarzman O., Функциональный анализ и его приложения 2009 Т. 43 № 2 С. 64-72

Let Γ ⊂ U(1, 1) be the subgroup generated by the complex reflections. Suppose that Γ acts discretely on the domain K = {(z1, z2) ∈ C2 | |z1|2 − |z2|2 < 0} and that the projective group PΓ acts on the unit disk B = {|z1/z2| < 1} as a Fuchsian group of signature ...

Added: January 25, 2013

Blank M., Nonlinearity 2017 Vol. 30 No. 12 P. 4649-4664

The classical Birkhoff ergodic theorem in its most popular version says that the
time average along a single typical trajectory of a dynamical system is equal
to the space average with respect to the ergodic invariant distribution. This
result is one of the cornerstones of the entire ergodic theory and its numerous
applications. Two questions related to this subject ...

Added: July 16, 2018

Minabutdinov A., Лодкин А. А., Манаев И. Е., / Санкт-Петербургское математическое общество. Серия MSP "Preprints of the St. Petersburg Mathematical Society". 2014. № 11.

The paper generalizes the results by \'E. Janvresse, T. de la Rue and Y. Velenik [22] and the work [9] of the third author on fluctuations in ergodic sums for the Pascal adic in the case of arbitrary ergodic invariant measures. In particular, we answer several questions from [22] and [9]. ...

Added: March 28, 2015

Chistyakov V., Tretyachenko Y., Journal of Mathematical Analysis and Applications 2010 Vol. 370 No. 2 P. 672-686

Given two points a=(a1,…,an) and b=(b1,…,bn) from Rn with a<b componentwise and a map f from the rectangle <img src="http://ars.els-cdn.com/content/image/1-s2.0-S0022247X10003574-si5.gif" /> into a metric semigroup M=(M,d,+), we study properties of the total variation<img src="http://ars.els-cdn.com/content/image/1-s2.0-S0022247X10003574-si7.gif" /> of f on <img src="http://ars.els-cdn.com/content/image/1-s2.0-S0022247X10003574-si8.gif" /> introduced by the first author in [V.V. Chistyakov, A selection principle for mappings of ...

Added: November 22, 2012

Minabutdinov A., Записки научных семинаров ПОМИ РАН 2015 Т. 432 С. 224-260

The paper generalizes the results by E. Janvresse, T. de la Rue and Y. Velenik [18] on fluctuations in ergodic sums for the Pascal adic transformation in the case of Lebesgue measure for a wide class of functions. In particular, we answer several questions from [18]. ...

Added: March 15, 2015

Minabutdinov A., Лодкин А. А., Записки научных семинаров ПОМИ РАН 2015 Т. 437 С. 145-183

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 any ergodic invariant measure and cylindric function. The result on non-discreteness of the spectrum of the Pascal adic is stated. ...

Added: October 14, 2015

V'yugin V., Theory of Computing Systems 2016 P. 403-423

We study a stability property of probability laws with respect to small violations of algorithmic randomness. Some sufficient
condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like the
strong law of large numbers, the law of iterated logarithm, and even Birkhoff's pointwise ergodic theorem for ergodic
transformations, are stable in ...

Added: May 16, 2015

Bufetov A. I., Romaskevich O. L., Bowen L., Geometriae Dedicata 2016 Vol. 181 No. 1 P. 293-306

Mean convergence of Markovian spherical averages is established for a measure-preserving action of a finitely-generated free group on a probability space. We endow the set of generators with a generalized Markov chain and establish the mean convergence of resulting spherical averages in this case under mild nondegeneracy assumptions on the stochastic matrix (Formula presented.) defining ...

Added: July 7, 2016

С.А.Чистякова, В.В.Чистяков, В кн. : Труды Математического центра имени Н.И.Лобачевского. Т. 54: Теория функций, ее приложения и смежные вопросы.: Каз. : Издательство Казанского математического общества и Академии наук РТ, 2017. С. 399-402.

Given a closed interval $I=[a,b]$ and a metric space $(M,d)$, we introduce a
nondecreasing sequence $\{\nu_n\}$ of pseudometrics on $M^I$ (the set of all
functions from $I$ into $M$), called the {\it joint modulus of variation}. We show that
if two sequences of functions $\{f_j\}$ and $\{g_j\}$ from $M^I$ are such that
$\{f_j\}$ is pointwise relatively compact on $I$, ...

Added: August 29, 2017

Tretyachenko Y., Известия высших учебных заведений. Математика 2010 № 5 С. 41-54

In this paper we consider sequences of functions that are defined on a subset of the real line with values in a uniform Hausdorff space. For such sequences we obtain a sufficient condition for the existence of pointwise convergent subsequences. We prove that this generalization of the Helly theorem includes many results of the recent ...

Added: September 28, 2012

Tretyachenko Y., Russian Mathematics 2010 Vol. 54 No. 5 P. 35-46

In this paper we consider sequences of functions that are defined on a subset of the real line and take on values in a uniform Hausdorff space. For such sequences we obtain a sufficient condition for the existence of pointwise convergent subsequences. We prove that this generalization of the Helly theorem includes many results of ...

Added: November 14, 2012

Chistyakov V., Tretyachenko Y., Journal of Mathematical Analysis and Applications 2010 Vol. 369 No. 1 P. 82-93

Given a=(a1,…,an), b=(b1,…,bn)∈Rn with ab componentwise and a map f from the rectangle Iab=[a1,b1]×⋯×[an,bn] into a metric semigroup M=(M,d,+), denote by TV(f,Iab) the Hildebrandt–Leonov total variation of f on Iab, which has been recently studied in [V.V. Chistyakov, Yu.V. Tretyachenko, Maps of several variables of finite total variation. I, J. Math. Anal. Appl. (2010), submitted ...

Added: January 9, 2013

Blank M., [б.и.], 2017

We study typical points with respect to ergofic averaging of a general dynamical system. ...

Added: February 10, 2018