On one class of nowhere non-monotonic functions with fractal properties that contains a subclass of singular functions

This paper defines and analyzes a class of continuous functions on $[0,1]$ constructed via an infinite stochastic matrix and specific parameters, establishing criteria for their monotonicity, differentiability, and singularity while investigating the properties of their level sets.

The Gist

Imagine you are an architect designing a very strange, winding road that stretches from point A (0) to point B (1). Usually, roads go up, go down, or stay flat. But the authors of this paper, Klymchuk and Pratsiovtyi, are designing a road that does something impossible in the normal world: it goes up, down, up, and down everywhere you look, no matter how closely you zoom in.

This paper is about a specific mathematical recipe for building such a "chaotic" road, and it turns out this road has some very special, fractal properties (like a snowflake or a coastline that looks the same whether you look at it from a plane or a microscope).

Here is the breakdown of their discovery in simple terms:

1. The Blueprint: A "Three-Color" Map

To build this road, the authors use a special way of numbering things, similar to how we use base-10 (0-9) or binary (0-1). They use Base-3 (digits 0, 1, and 2).

Think of the interval from 0 to 1 as a long strip of land.

• 0 is the left side.
• 2 is the right side.
• 1 is the middle.

The authors create a map where every point on the road is defined by a sequence of these three digits. But here's the twist: they don't treat these digits equally. They assign different "weights" or "probabilities" to them using a matrix (a grid of numbers). This is like saying, "If you take the left path, the road stretches a little; if you take the middle, it shrinks; if you take the right, it stretches differently."

2. The Secret Ingredient: The "Dial" ($\varepsilon$)

The most important part of their recipe is a control dial called $\varepsilon$ (epsilon). This dial controls how the road behaves.

• Dial set to "Low" ($\varepsilon < 0.5$): The road is a smooth, steady climb. It never goes down. It's a boring, straight-up hill.
• Dial set to "Medium" ($\varepsilon = 0.5$): The road has flat spots. It climbs, then stays perfectly flat for a while, then climbs again. This creates "singular" functions (math-speak for roads that climb but have zero slope almost everywhere).
• Dial set to "High" ($\varepsilon > 0.5$): This is the magic. The road becomes a "nowhere monotonic" function.
  • Imagine a mountain range where, no matter how small a patch of ground you look at, you will always find a peak and a valley right next to each other.
  • The road goes up, then immediately down, then up again, infinitely. It never settles into a straight line or a flat plateau.

3. The Fractal Nature: Zooming In Forever

The paper proves that when the dial is set to "High," the road is a fractal.

Think of a fern leaf. If you look at the whole leaf, it has a shape. If you zoom in on one tiny branch of that leaf, it looks exactly like the whole leaf. If you zoom in on a tiny part of that branch, it looks the same again.

The authors' function behaves the same way. If you zoom in on any tiny section of their chaotic road, you will see the same pattern of "up-down-up-down" repeating forever. It is infinitely complex.

4. The "Ghost" Road (Singular Functions)

The paper also discusses a special case where the road is "singular."
Imagine a road that goes from the bottom of a hill to the top, but 99.9% of the time, the road is perfectly flat.

• It only actually climbs at specific, invisible points (like dust motes in a sunbeam).
• If you try to measure the slope (derivative) of this road at almost any point, it is zero.
• Yet, somehow, it still manages to get from the bottom to the top.
  This sounds impossible, but the math proves it exists. These are called singular functions.

5. The "Level Sets": Where the Road Hits a Specific Height

Finally, the authors look at "level sets." Imagine you are flying a drone at a fixed altitude (say, 50 meters) and you want to see where the road crosses that altitude.

• If the road is smooth: It crosses your altitude exactly once.
• If the road is flat: It might run along your altitude for a long distance (a whole segment).
• If the road is the chaotic fractal (High $\varepsilon$): Your drone will cross the road, then cross it again, and again, and again. The paper proves that for this chaotic road, the set of points where it hits a specific height is a countable collection of scattered dots. It's like a sprinkling of salt on a table; you can count the grains, but they are everywhere.

Summary Analogy

Think of the function as a shapeshifting rollercoaster:

1. The Design: It's built using a 3-digit code system.
2. The Control Knob: A dial ($\varepsilon$) determines if the ride is a gentle slope, a flat plateau, or a wild, jagged mess.
3. The Result: When the knob is set to "wild," the ride is a fractal nightmare. It goes up and down so violently that you can never find a straight section, no matter how much you zoom in.
4. The Mystery: Even though it looks like a chaotic mess, it is perfectly continuous (no jumps) and follows a strict mathematical rule.

The paper is essentially the "construction manual" for building these impossible, infinitely jagged roads and proving exactly how they behave.

Technical Summary

Here is a detailed technical summary of the paper "On One Class of Nowhere Non-Monotonic Functions with Fractal Properties that Contains a Subclass of Singular Functions" by S. O. Klymchuk and M. V. Pratsiovtyi.

1. Problem Statement

The paper addresses the construction and analysis of a specific class of continuous functions defined on the unit interval $[0, 1]$. The primary objective is to define a function $f(x)$ using a generalized representation of real numbers (the $Q^*_3$-representation) and to investigate its analytical properties, specifically:

• Monotonicity: Determining conditions under which the function is strictly increasing, constant on intervals, or nowhere monotonic.
• Differentiability and Singularity: Analyzing where the function is differentiable and identifying conditions under which it becomes a singular function (continuous, derivative zero almost everywhere, but not constant).
• Fractal Structure: Examining the geometric properties of the function's graph and its level sets.

The study generalizes previous work on $s$-adic and $Q_s$-representations, extending them to infinite stochastic matrices and introducing a parameter sequence $(\varepsilon_k)$ to control the function's behavior.

2. Methodology

The authors employ a constructive approach based on the $Q^*_3$-representation of real numbers.

A. The $Q^*_3$-Representation

The domain $[0, 1]$ is represented using an infinite stochastic matrix $Q^*_3 = \|q_{ik}\|$ where $i \in \{0, 1, 2\}$ and $k \in \mathbb{N}$. Any $x \in [0, 1]$ is expressed as:
$$x = \beta_{\alpha_1(x)}(1) + \sum_{k=2}^{\infty} \left[ \beta_{\alpha_k(x)}(k) \prod_{j=1}^{k-1} q_{\alpha_j(x)}(j) \right]$$
where $\alpha_k(x) \in \{0, 1, 2\}$ are the digits of the expansion, and $\beta_{ik}$ are partial sums of the probabilities defined by the matrix columns.

B. Definition of the Function $f(x)$

The function $f: [0, 1] \to [0, 1]$ is defined by a series analogous to the $Q^*_3$-representation but using a different set of coefficients derived from a sequence $(\varepsilon_k)$ where $0 \le \varepsilon_k \le 1$:
$$f(x) = \delta_{\alpha_1(x)}(1) + \sum_{k=2}^{\infty} \left[ \delta_{\alpha_k(x)}(k) \prod_{j=1}^{k-1} g_{\alpha_j(x)}(j) \right]$$
The coefficients are defined as:

• $g_{0k} = g_{2k} = \frac{1 + \varepsilon_k}{3}$
• $g_{1k} = \frac{1 - 2\varepsilon_k}{3}$
• $\delta_{0k} = 0, \quad \delta_{1k} = g_{0k}, \quad \delta_{2k} = g_{0k} + g_{1k}$

C. Analytical Tools

The authors utilize:

• Series Convergence Analysis: To prove the function is well-defined and continuous.
• Induction: To establish bounds on partial sums.
• Cylinder Sets: Analyzing the function's behavior on intervals (cylinders) defined by fixed initial digits in the $Q^*_3$-expansion.
• Affine Transformations: To demonstrate the self-similar (fractal) nature of the graph.

3. Key Contributions and Results

A. Well-Definedness and Continuity

• Correctness: The authors prove that the series defining $f(x)$ converges absolutely for any sequence of digits. They also demonstrate that the function yields the same value for $Q^*_3$-rational numbers regardless of whether they are represented with a trailing sequence of 0s or 2s (handling non-unique representations).
• Range and Continuity: It is proven that $f(x)$ maps $[0, 1]$ onto $[0, 1]$. The function is continuous everywhere on $[0, 1]$.

B. Monotonicity Criteria

The behavior of the function is strictly determined by the sequence $(\varepsilon_k)$:

1. Strictly Increasing: If $0 \le \varepsilon_k < 1/2$for all$k$, then$g_{1k} > 0$. The function is strictly increasing on the entire domain.
2. Constant on Intervals: If $\varepsilon_k = 1/2$ for some $k$, then $g_{1k} = 0$. The function becomes constant on specific cylindrical intervals (intervals where the $k$-th digit is 1).
3. Nowhere Monotonic: If $1/2 < \varepsilon_k \le 1$for all$k$, then$g_{1k} < 0$. The function is nowhere monotonic. On any sub-interval (cylinder), the function exhibits both positive and negative increments, preventing monotonicity.

C. Singularity and Differentiability

• Singular Functions: In the case where $\varepsilon_k = 1/2$ for all $k$, the function is constant on a dense set of intervals whose total length sums to 1. Consequently, the derivative $f'(x) = 0$ almost everywhere (in the sense of Lebesgue measure), making $f$ a singular function.
• Nowhere Differentiability: The paper implies that under the nowhere monotonic condition ($\varepsilon_k > 1/2$), the function exhibits fractal properties that typically lead to non-differentiability almost everywhere, though the primary focus is on the lack of monotonicity.

D. Fractal Geometry and Level Sets

• Self-Similarity: The graph $\Gamma_f$ of the function is shown to be affinely equivalent to its subsets. Specifically, the graph restricted to a cylinder $\Delta_{c_1...c_m}$ is an affine transformation of the whole graph. This confirms the fractal nature of the function.
• Level Sets ($f^{-1}(y_0)$):
  • If strictly increasing ($\varepsilon_k < 1/2$): Level sets are single points.
  • If singular ($\varepsilon_k = 1/2$): Level sets are either points or segments (intervals).
  • If nowhere monotonic ($\varepsilon_k > 1/2$): Level sets are countable sets. The authors prove that for any $y_0$, the preimage consists of a countable number of points because the function oscillates infinitely often within any neighborhood.

4. Significance

This paper makes several significant contributions to the theory of real functions and fractal geometry:

1. Generalization: It extends the class of singular and nowhere monotonic functions beyond the classical Cantor-type constructions (like the Devil's Staircase) by utilizing a flexible matrix-based representation ($Q^*_3$) and a tunable parameter sequence $(\varepsilon_k)$.
2. Unified Framework: It provides a unified framework to generate functions with specific properties (strictly increasing, singular, or nowhere monotonic) simply by adjusting the parameter $\varepsilon_k$.
3. Level Set Analysis: The detailed characterization of level sets for the nowhere monotonic case (proving they are countable) adds depth to the understanding of the topological structure of such pathological functions.
4. Applications: These types of functions are relevant in probability theory (as distribution functions of singular measures), dynamical systems, and the study of fractal dimensions in analysis.

In summary, Klymchuk and Pratsiovtyi successfully construct a versatile class of continuous functions that bridge the gap between smooth monotonic functions, singular functions, and nowhere monotonic fractal functions, providing rigorous criteria for their classification.