<abstract xmlns="http://www.w3.org/1999/xhtml">Given a countably infinite group <italic>G</italic> acting on some space <italic>X</italic>, an increasing family of finite subsets <italic>Gn</italic>, <italic>x</italic>∈ <italic>X</italic> and a function <italic>f</italic> over <italic>X</italic> we consider the sums <italic>Sn</italic>(<italic>f, x</italic>) = ∑<italic>g∈Gnf</italic>(<italic>gx</italic>). The asymptotic behaviour of <italic>Sn</italic>(<italic>f, x</italic>) is a delicate problem that was studied under various settings. In the following paper we study this problem when <italic>G</italic> is a specific lattice in SL (2, ℤ ) acting on the projective line and <italic>Gn</italic> are chosen using the word metric. The asymptotic distribution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on <italic>G</italic> defined by a specific measure <italic>µ</italic>. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over <italic>X</italic> pushed forward by the convolution power <italic>µ∗n</italic>.</abstract>

Given a countably infinite group G acting on some space X, an increasing family of finite subsets Gn, x∈ X and a function f over X we consider the sums Sn(f, x) = ∑g∈Gnf(gx). The asymptotic behaviour of Sn(f, x) is a delicate problem that was studied under various settings. In the following paper we study this problem when G is a specific lattice in SL (2, ℤ ) acting on the projective line and Gn are chosen using the word metric. The asymptotic distribution is calculated and shown to be tightly connected to Minkowski’s question mark function. We proceed to show that the limit distribution is stationary with respect to a random walk on G defined by a specific measure µ. We further prove a stronger result stating that the asymptotic distribution is the limit point for any probability measure over X pushed forward by the convolution power µ∗n.

Word Metric, Stationary Measure and Minkowski’s Question Mark Function

Uniform distribution theory

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

{"article-title":"Word Metric, Stationary Measure and Minkowski\u2019s Question Mark Function"}

Given a countably infinite group G acting on some space X, an increasing family of finite subsets Gn, x∈ X and a function f over X we consider the sums Sn(f,...

Word Metric, Stationary Measure and Minkowski’s Question Mark Function

Published Online: Dec 25, 2020

Page range: 23 - 38

Received: Jun 20, 2020

Accepted: Jul 14, 2020

DOI: https://doi.org/10.2478/udt-2020-0009

Keywords
Stationary measure, Minkowski’s question matk function, Word metric, Lattice orbits

© 2020 Uriya Pumerantz, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Word Metric, Stationary Measure and Minkowski’s Question Mark Function

Published Online: Dec 25, 2020

Page range: 23 - 38

Received: Jun 20, 2020

Accepted: Jul 14, 2020

DOI: https://doi.org/10.2478/udt-2020-0009

KeywordsStationary measure, Minkowski’s question matk function, Word metric, Lattice orbits

© 2020 Uriya Pumerantz, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
Stationary measure, Minkowski’s question matk function, Word metric, Lattice orbits