One one type of ud-preserving mapping

Based on the provided paper "On one type of ud-preserving mapping" by Milan Paštéka, here is a detailed technical summary covering the problem, methodology, key contributions, results, and significance.

1. Problem Statement

The paper addresses the construction and analysis of specific mappings on the unit interval $[0, 1]$ that preserve uniform distribution (ud). While the classical notion of uniform distribution (Weyl, 1946) relies on asymptotic density, this work focuses on Buck's uniform distribution, which is defined with respect to measure density (introduced by R.C. Buck in 1946).

The core problem is to construct a class of mappings $T_\pi$ such that:

1. The mapping itself preserves uniform distribution (i.e., if a sequence is uniformly distributed, its image under $T_\pi$ is also uniformly distributed).
2. The iterations of the mapping, $\{T_\pi^n(\alpha)\}$, form a sequence that is Buck uniformly distributed modulo 1 for any starting point $\alpha$.
3. To analyze the dynamical properties of these mappings, specifically their monotonicity and differentiability.

2. Methodology

The author employs a constructive approach based on Cantor series expansions and permutation groups.

• Cantor Series Representation:
  The unit interval is represented using a generalized Cantor series. Given a sequence of pairwise relatively prime natural numbers $\{m_i\}_{i=1}^\infty$, let $B_j = \prod_{i=1}^j m_i$ (with $B_0=1$). Any $\alpha \in [0, 1)$ is uniquely expressed as:
  $$\alpha = \sum_{j=0}^{\infty} \frac{b_j(\alpha)}{B_{j+1}}, \quad b_j(\alpha) \in \mathbb{Z}_{m_j}$$
  subject to a condition ensuring uniqueness (infinitely many digits are not maximal).

• Definition of the Mapping ($T_\pi$):
  The mapping is defined by applying a sequence of permutations $\pi = (\pi_1, \pi_2, \dots)$ to the digits of the Cantor expansion. If $\pi_n: \mathbb{Z}_{m_n} \to \mathbb{Z}_{m_n}$ are permutations, then:
  $$T_\pi(\alpha) = \sum_{j=0}^{\infty} \frac{\pi_j(b_j(\alpha))}{B_{j+1}}$$

• Analytical Tools:

  • Buck Measure Density ($\mu$): The paper utilizes the definition of Buck measure density for subsets of natural numbers to define Buck uniform distribution. A sequence $\{v(n)\}$ is Buck uniformly distributed if the preimage of any subinterval $I \subset [0, 1]$ is a Buck measurable set with measure equal to the length of $I$.
  • Chinese Remainder Theorem (CRT): This is used to analyze the behavior of cyclic permutations over iterations.
  • Interval Decomposition: The interval $[0, 1]$ is decomposed into subintervals $I_j^{(n)}$ of length $1/B_n$, corresponding to fixed initial digit sequences.

3. Key Contributions and Results

A. Construction of Buck Uniformly Distributed Iterations

Theorem 1: If the permutations $\pi_i$ are cyclic of length $m_i$, then for any $\alpha \in [0, 1)$, the sequence of iterations $\{v(n)\} = \{T_\pi^n(\alpha)\}$ is Buck uniformly distributed modulo 1.

• Proof Logic: The proof relies on showing that the preimage of any basic interval $I_j^{(k)}$ under the iteration map corresponds to a specific arithmetic progression (residue class) modulo $B_k$. Since arithmetic progressions are Buck measurable with density $1/B_k$, the condition for Buck uniform distribution is satisfied.

B. Preservation of Uniform Distribution

Theorem 2 (implied by Property 8): The mapping $T_\pi$ preserves uniform distribution.

• If a sequence $\{v(n)\}$ is uniformly distributed modulo 1, then $\{T_\pi(v(n))\}$ is also uniformly distributed modulo 1. This connects the constructed mapping to the broader class of "ud-preserving" mappings studied in previous literature (e.g., [10]).

C. Analysis of Monotonicity

Theorem 2: The function $T_\pi$ is nowhere monotone on any subinterval of positive length.

• Reasoning: Since the permutations $\pi_s$ are non-identity, within any small interval defined by the first $s$ digits, the permutation $\pi_{s+1}$ can be chosen to reverse the order of specific sub-intervals. This creates local increases and decreases arbitrarily close to any point, preventing global monotonicity.

D. Analysis of Differentiability

Theorem 3: The function $T_\pi$ is nowhere differentiable on $[0, 1]$.

• Reasoning: The derivative, if it existed, would have to be an integer (derived from the ratio of digit differences). However, Darboux's theorem (which states that derivatives satisfy the intermediate value property) contradicts the fact that the derivative can only take discrete integer values unless it is constant. Since the function is not monotone, the derivative cannot be constant.
• Condition for Non-differentiability: The paper notes that if the permutations are the identity (or specific cyclic shifts) for infinitely many $s$, the derivative might not exist. Specifically, if $\pi_s = (0, 1, \dots, m_s-1)$ for infinitely many $s$, the mapping is nowhere differentiable.

4. Significance

• Dynamical Systems: The paper provides a concrete class of dynamical systems on the unit interval that exhibit "chaotic" behavior in terms of distribution (Buck uniform distribution) and regularity (nowhere monotone/differentiable), yet are constructed via simple digit permutations.
• Number Theory: It bridges the gap between the arithmetic properties of residue classes (via the Chinese Remainder Theorem) and the analytic properties of functions on the continuum.
• Measure Theory: It contributes to the understanding of Buck measure density, demonstrating how specific arithmetic structures (residue classes) translate into measure-theoretic properties of dynamical orbits.
• Counterexamples: The constructed functions serve as examples of continuous, nowhere monotone, and nowhere differentiable functions that still preserve uniform distribution, enriching the library of pathological functions in analysis.

In summary, Paštéka constructs a family of mappings $T_\pi$ that act as "scramblers" of the unit interval. While they destroy local regularity (monotonicity and differentiability), they preserve and even generate global statistical regularity (Buck uniform distribution) in their iterations.