1. bookVolume 15 (2020): Issue 2 (December 2020)
Journal Details
License
Format
Journal
eISSN
2309-5377
First Published
30 Dec 2013
Publication timeframe
2 times per year
Languages
English
access type Open Access

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

Published Online: 25 Dec 2020
Volume & Issue: Volume 15 (2020) - Issue 2 (December 2020)
Page range: 23 - 38
Received: 20 Jun 2020
Accepted: 14 Jul 2020
Journal Details
License
Format
Journal
eISSN
2309-5377
First Published
30 Dec 2013
Publication timeframe
2 times per year
Languages
English
Abstract

Given a countably infinite group G acting on some space X, an increasing family of finite subsets Gn, xX 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.

Keywords

MSC 2010

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