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Transformations and functions that preserve the asymptotic mean of digits in the ternary representation of a number

M. V. Pratsiovytyi, S. O. Klymchuk, O. P. Makarchuk

Abstract

In the paper we study transformations of the interval $[0;1)$ and functions that preserve the asymptotic mean $r$ of the digits in the $s$--adic representation of a number $x$, $$r(x)=\lim\limits_{n\to\infty}\frac{1}{n}\sum\limits^{n}_{i=1}α_i(x)$$ specify the necessary and sufficient conditions for a transformation to belong to this class

Transformations and functions that preserve the asymptotic mean of digits in the ternary representation of a number

Abstract

In the paper we study transformations of the interval and functions that preserve the asymptotic mean of the digits in the --adic representation of a number , specify the necessary and sufficient conditions for a transformation to belong to this class
Paper Structure (6 sections, 9 theorems, 48 equations)

This paper contains 6 sections, 9 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Transformations of interval $[0,1)$ and functions that preserve digit frequencies
  3. Connection with functions that preserve fractal dimension
  4. Transformations of $[0,1)$ and functions that preserve the asymptotic mean of digits but do not preserve digit frequencies
  5. Preservation of the asymptotic mean and digit frequencies almost everywhere
  6. Preservation of the asymptotic digit mean by a function when digit frequencies do not exist

Key Result

Lemma 1

The set $V$ of all transformations of the interval $[0;1)$ that preserve the frequencies of all digits in the $s$--adic representation of a number, with respect to the composition operation $\circ$ (superposition) of transformations, forms a noncommutative group.

Theorems & Definitions (22)

  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Theorem 1
  • proof
  • Definition 3
  • Definition 4
  • Lemma 2
  • proof
  • ...and 12 more