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Entropy rates in the dimension theory of self-similar measures

This survey paper explores the dimension theory of self-similar measures on the real line, with a specific focus on the significance and application of entropy rates.

Original authors: Péter P. Varjú

Published 2026-03-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Péter P. Varjú

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Measuring the "Fuzziness" of Fractals

Imagine you have a magical machine that takes a line and shrinks it, then copies it, then shrinks those copies again, and repeats this forever. This creates a fractal—a shape that looks the same no matter how much you zoom in.

In mathematics, we want to know: How "thick" or "dense" is this shape?

If it's a solid line, its "thickness" (dimension) is 1.
If it's a single point, its thickness is 0.
If it's a messy, cloud-like fractal, its thickness might be 0.7 or 0.9.

The paper asks: How do we calculate this thickness?

Usually, there's a simple formula based on how much the machine shrinks the line and how likely it is to pick each copy. This formula gives us a "predicted" thickness. However, this prediction only works perfectly if the copies don't overlap too much. If the copies pile on top of each other (like stacking pancakes), the shape becomes "thinner" than the formula predicts.

The paper is a detective story about figuring out exactly when and why these copies overlap, and how to fix the formula to get the right answer every time.

Key Concepts & Analogies

1. The "Exact Overlap" Problem (The Duplicate Recipe)

Imagine you are making a soup. You have a recipe that says: "Take a pot, add salt, then add salt again."

Scenario A: You add salt, then add pepper. (Two distinct steps).
Scenario B: You add salt, then add salt again. (Redundant steps).

In the math world, if your machine does the exact same thing twice in a row, it's a waste of effort. It's like having two identical instructions in a computer program.

The Problem: If the machine has "exact overlaps" (duplicate instructions), the predicted thickness formula breaks because it thinks there are more unique paths than there actually are.
The Goal: We need a way to measure how much "wasted effort" (overlap) is happening.

2. The "Entropy Rate" (The Measure of Confusion)

To fix the broken formula, the paper introduces a new tool called Entropy Rate.

The Analogy: Imagine a game of "Telephone." You whisper a message to a friend, they whisper it to another, and so on.
- If everyone whispers clearly, the message stays unique. This is high entropy (lots of information, lots of variety).
- If everyone starts saying the same thing or repeating the same phrase, the message gets boring and repetitive. This is low entropy.
In the Paper: The "Entropy Rate" measures how many unique paths the fractal machine actually takes after many steps. If the machine is full of duplicates (overlaps), the entropy rate drops.
The Big Conjecture: The paper suggests a new "Golden Rule":

Actual Thickness = The smaller of (1) or (Predicted Thickness based on Entropy).

This rule works even if the copies overlap, as long as we use the "Entropy" version of the formula instead of the simple one.

3. The "Magic Numbers" (Algebraic vs. Transcendental)

The paper explores how this rule works with different types of numbers used in the machine's settings.

Algebraic Numbers (The "Rational" Crowd): These are numbers that are solutions to simple polynomial equations (like $\sqrt{2}$ or $1/3$ ).
- The Discovery: For these numbers, the "Golden Rule" works perfectly! If the machine doesn't have exact duplicates, the thickness is exactly what the formula says. If it does have duplicates, the entropy rate fixes the calculation.
- The Breakthrough: A mathematician named Hochman proved that for these numbers, the "wasted effort" (overlaps) is so rare that we can ignore the messy details and just trust the entropy formula.
Bernoulli Convolutions (The Famous Case): This is a specific, famous type of fractal machine where the settings are just $\lambda$ (a shrink factor) and $\pm 1$ .
- The Mystery: For a long time, mathematicians didn't know the thickness of these shapes for certain "weird" numbers (like the inverse of a Pisot number).
- The Result: The paper confirms that for these famous cases, the thickness is indeed determined by the entropy rate. If the entropy is low (lots of overlaps), the shape is thin. If the entropy is high, the shape fills the space (thickness = 1).
Transcendental Numbers (The "Wild" Crowd): These are numbers like $\pi$ or $e$ that can't be solved by simple equations.
- The Challenge: When you have two wild numbers controlling the machine (like a shrink factor and a shift factor), the "duplicates" don't just happen at specific points; they happen along curves.
- The Difficulty: It's like trying to find a needle in a haystack, but the haystack is a moving, twisting snake. The paper shows that while we are getting closer to solving this, there are still some "curves" of duplicates we haven't fully mapped out yet.

4. The "Mahler Measure" (The Fingerprint of a Number)

The paper introduces a concept called Mahler Measure.

The Analogy: Think of a number as a fingerprint. The Mahler Measure is a way to quantify how "complex" or "wild" that fingerprint is.
The Connection: The paper finds a link between how complex a number is (Mahler Measure) and how much the fractal overlaps (Entropy).
- If a number is "simple" (like a Pisot number), the fractal overlaps a lot, and the shape is thin.
- If a number is "wild" (high Mahler Measure), the overlaps are rare, and the shape is thick.
- The authors use this to prove that for most numbers, the fractal is actually as thick as a full line (dimension 1).

The Takeaway: What Did They Actually Solve?

The Formula is Fixed: They confirmed that if you replace the old "simple" formula with the "Entropy Rate" formula, you get the correct thickness for almost all fractal machines, even the messy ones with overlaps.
The "Algebraic" Case is Solved: If the machine uses "rational" numbers (algebraic), we know the answer 100%.
The "Transcendental" Case is Half-Solved: If the machine uses "wild" numbers (transcendental), we know the answer for many cases, but there are still some tricky curves of duplicates we need to understand better.
The "Why" Matters: The paper explains that the reason these shapes behave the way they do is deeply connected to the hidden arithmetic properties of the numbers used to build them.

In a Nutshell

Imagine you are trying to guess how much paint is needed to cover a fractal wall.

Old Way: You guess based on the size of the wall. (Works if the wall is smooth).
New Way (This Paper): You count how many times the painter accidentally painted over the same spot (Entropy).
Conclusion: By counting the "paint-over" mistakes, you can predict exactly how much paint is needed, even if the wall is a chaotic, overlapping mess. The paper proves this works for almost all types of walls, except for a few very weird, twisting ones that are still being studied.

1. Problem Statement and Context

The paper addresses the fundamental problem in fractal geometry and dynamical systems: determining the Hausdorff dimension ( $\dim \mu$ ) of a self-similar measure $\mu$ on the real line.

Setup: Let $\Phi = \{\phi_i : i \in \Lambda\}$ be an Iterated Function System (IFS) of contractive similarities $\phi_i(x) = \lambda_i x + \tau_i$ , and let $p = \{p_i\}$ be a probability vector. The self-similar measure $\mu$ satisfies $\mu = \sum p_i \phi_i(\mu)$ .
The Challenge: While the dimension is well-understood when the IFS satisfies strong separation conditions (e.g., Open Set Condition), the general case allows for exact overlaps (where distinct compositions of maps result in the same function).
The Conjecture: The central open problem is Conjecture 3, which posits that the dimension is determined by the minimum of the ambient dimension (1) and a ratio involving the entropy rate ( $h(\Phi, p)$ ) and the average logarithmic contraction rate:
$\dim \mu = \min\left(1, \frac{h(\Phi, p)}{\sum p_i \log |\lambda_i|^{-1}}\right)$
Here, $h(\Phi, p)$ is the Shannon entropy rate of the random walk on the semigroup generated by the IFS. In the absence of exact overlaps, $h(\Phi, p)$ equals the Shannon entropy $H(p)$ . The gap $H(p) - h(\Phi, p)$ quantifies the "amount" of exact overlaps.

2. Methodology

The paper surveys a progression of techniques used to resolve Conjecture 3, moving from algebraic settings to transcendental ones. The core methodology relies on:

Entropy Rates: Using the entropy rate as a proxy for the "effective" number of distinct maps in the system, accounting for overlaps.
Exponential Separation: Utilizing Hochman's Exponential Separation Condition (ESC), which requires that the distance between distinct compositions of maps does not decay too rapidly (specifically, not faster than exponential).
Diophantine Analysis and Algebraic Number Theory:
- Mahler Measure ( $M(\lambda)$ ): Relating the entropy rate to the Mahler measure of the scaling parameter $\lambda$ .
- Pisot and Salem Numbers: Identifying specific algebraic integers (Pisot and Salem numbers) as the only candidates for parameters where the dimension might drop below the similarity dimension.
- Approximation by Algebraic Numbers: Proving that if a transcendental parameter yields a low dimension, it must be extremely well-approximated by algebraic numbers with specific properties (low Mahler measure).
Transversality Arguments: Using geometric arguments to show that "bad" parameters (where separation fails) are sparse or repel each other in the parameter space.

3. Key Contributions and Results

A. The Algebraic Case (Section 2)

Hochman's Breakthrough (Theorem 5): Proved Conjecture 3 for IFSs satisfying the Weak Exponential Separation Condition. This condition is much weaker than the Open Set Condition and holds for almost all parameters in algebraic families.
Rapaport's Improvement (Theorem 6): Extended the result to IFSs where scaling parameters are algebraic numbers, even if the exponential separation condition fails dramatically (i.e., distances between compositions can decay arbitrarily fast).
- Key Insight: While distances may decay fast, the number of such close pairs is small enough that the entropy loss is negligible.
- Result: For algebraic scaling parameters without exact overlaps, $\dim \mu = \min(1, H(p)/\sum p_i \log |\lambda_i|^{-1})$ .

B. Bernoulli Convolutions (Section 3)

Bernoulli convolutions $\nu_\lambda$ are self-similar measures generated by $\{x \mapsto \lambda x \pm 1\}$ with weights $(1/2, 1/2)$ .

Theorem 7 (Hochman, Varj´u): Confirms Conjecture 3 for all $\lambda \in (0, 1)$ .
$\dim \nu_\lambda = \min(1, h(\lambda) \log \lambda)$
Exceptional Parameters: The dimension is strictly less than 1 if and only if $h(\lambda) < \log \lambda^{-1}$ $h (λ) < lo g λ^{- 1}$ . This occurs only if $\lambda^{-1}$ $λ^{- 1}$ is a Pisot number (or potentially a Salem number).
- Connection to Mahler Measure: The entropy rate is bounded by the Mahler measure: $h(\lambda) \le \log M(\lambda)$ .
- Lower Bound: $h(\lambda) \ge 0.44 \log \min(2, M(\lambda))$ .
Open Questions:
- Question 8: Are there any $\lambda$ where $\dim \nu_\lambda < 1$ but $\lambda^{-1}$ is not a Pisot number? (Currently, only Pisot numbers are known to cause dimension drops).
- Question 9: Is the map $\lambda \mapsto h(\lambda)$ lower semi-continuous? This is linked to the closedness of the set of Pisot/Salem numbers (Boyd's Conjecture).

C. Two-Parameter Families (Section 4)

The paper extends the theory to families with two parameters, specifically $\Phi_{\lambda, \tau} = \{x \mapsto \lambda x \pm 1 \pm \tau\}$ .

The Difficulty: Unlike the single-parameter case (where exact overlaps occur at discrete points), in two-parameter families, exact overlaps occur along curves in the $(\lambda, \tau)$ plane. This destroys the minimal separation between "bad" parameters.
Generalized Entropy Rate: Varj´u introduces a definition of entropy rate for subsets of the parameter space (Definition 10) to quantify overlaps along curves.
Theorem 12 (Streck): A major step toward resolving the two-parameter case. It states that if the entropy rate is significantly below the maximum, then either the Mahler measure of $\lambda$ is bounded, or the parameter lies on a specific curve (specifically $\tau=0$ ).
Theorem 13: Combining Streck's result with previous work, it is proven that for $\lambda > 2^{-3/2}$ , if there are no exact overlaps, the dimension is 1. This covers a new range $\lambda \in (2^{-3/2}, 1/2)$ for convolutions of Bernoulli measures.

4. Significance and Impact

Unification of Dimension Theory: The paper demonstrates that entropy rates are the correct invariant to describe the dimension of self-similar measures, unifying cases with and without exact overlaps.
Resolution of Long-standing Conjectures: It confirms the dimension formula for all algebraic scaling parameters and for all Bernoulli convolutions, resolving decades-old problems.
Bridging Fields: The work creates a deep bridge between ergodic theory (entropy, random walks), fractal geometry (Hausdorff dimension), and Diophantine approximation (Mahler measures, Pisot numbers).
Future Directions: The paper highlights that the remaining obstacles are purely arithmetic:
1. Determining if non-Pisot/Salem numbers can cause dimension drops.
2. Proving the lower semi-continuity of entropy rates.
3. Extending the "entropy drop implies bounded Mahler measure" result to multi-parameter families with non-linear constraints.

In summary, Varj´u's survey illustrates how the introduction of entropy rates and the analysis of their relationship with algebraic number theory have transformed the dimension theory of self-similar measures from a field dominated by separation conditions to one governed by the arithmetic properties of the parameters.