Maximal Escape of Mass in Dynamics

• Maximal escape of mass is a phenomenon in dynamical systems where invariant measures asymptotically concentrate in divergent, cuspidal regions.
• It refines classical views by distinguishing minimal, maximal, and generic escape, providing a nuanced framework for measure distribution.
• Studies employ combinatorial models, number walls, and automatic tilings to quantify mass loss and challenge traditional rigidity conjectures.

Maximal escape of mass refers to a phenomenon in dynamical systems—especially in noncompact homogeneous spaces and Diophantine approximation over function fields—in which a sequence of probability measures, invariant under a group or flow, loses an asymptotically maximal proportion of its “mass” to the noncompact “cuspidal” regions or to infinity. The rigorous definition and paper of maximal escape of mass, as well as related notions such as generic escape of mass, have become central to recent advances across dynamics, ergodic theory, and number theory, providing a refined framework to analyze how finite, invariant, or periodic measures distribute or concentrate in the limit.

1. Definitions and Rigorous Formulation

Maximal escape of mass occurs when, for a family or sequence of measures or approximants, the limiting configuration allocates all or nearly all of the normalized mass to the divergent or “cusp” region as certain parameters go to infinity. In function field Diophantine approximation, for a Laurent series $\Theta(t) \in \mathbb{F}_p((t^{-1}))$, the key quantity is the degree profile of the partial quotients $A_i^{(\Theta \cdot P^k)}(t)$ in its continued fraction expansion after applying a scaling by powers of an irreducible polynomial $P(t)$. One considers the escape function: $$e_{k,n}(\Theta) = \frac{\sum_{i=1}^{\ell_k} \max\{\deg(A_i^{(\Theta \cdot P^k)}(t)) - n, 0\}}{\sum_{i=1}^{\ell_k} \deg(A_i^{(\Theta \cdot P^k)}(t))}$$ The traditional “full escape of mass” holds if

$$\lim_{n \to \infty} \liminf_{k \to \infty} e_{k,n}(\Theta) = 1$$

However, in light of counterexamples, the notion is refined into:

• Maximal escape of mass: replacing liminf by limsup,

$$\lim_{n \to \infty} \limsup_{k \to \infty} e_{k,n}(\Theta) = 1$$

• Generic escape of mass: requiring that for every $\epsilon>0$,

$$\lim_{n \to \infty} d(\left\{k : e_{k,n}(\Theta) > 1-\epsilon\right\}) = 1$$

where $d(\cdot)$ denotes natural density in $\mathbb{N}$.

These refinements capture different aspects: maximal escape of mass reflects that there exists some sequence or subsequence along which escape is full, while generic escape asserts that most times (in a density-one set), the escape is nearly full.

2. Historical Development and Motivating Conjectures

The concept traces to rigidity phenomena observed for homogeneous flows and Diophantine approximation, as in Ratner’s theorems and later in function-field analogues (Shapira, 2015, Kemarsky et al., 2015). Kemarsky, Paulin, and Shapira (KPS) conjectured that for any Laurent series over $\mathbb{F}_p$, and any irreducible polynomial $P(t)$, one would see full escape of mass. This was based on observed unboundedness in the degree sequences, as well as the analogy with measure rigidity and entropy escape in homogeneous dynamics. Early counterexamples, such as Laurent series constructed via the Thue–Morse sequence (Nesharim et al., 24 Mar 2025), and later the $p$-Cantor sequence (Aranov et al., 22 Oct 2025), established rigorously that the naive full escape conjecture fails, at least in the “liminf” formulation.

3. Variants and Quantitative Descriptions

The $p$-Cantor sequence and Thue–Morse sequence elucidate the need for more nuanced notions:

• For the $p$-Cantor sequence and odd $p$, the minimal escape is shown to equal $2/p < 1$, but the maximal escape of mass is full: for any $\epsilon > 0$, the set of times $k$ where $e_{k,n}(\Theta_p) > 1-\epsilon$ has density tending to one as $n \to \infty$.
• In the Thue–Morse case ($p=2$), the minimal escape is $2/3$ (Nesharim et al., 24 Mar 2025), contradicting the original KPS conjecture, while maximal and generic escape notions still hold.

These differences reflect deep combinatorial properties of the underlying sequence (as encoded in the “number wall” or Hankel/Toeplitz determinant tilings) and interactions with the automorphism group and morphisms defining the sequence.

Mathematical Formulas

For a Laurent series $\Theta(t)$, and scaling sequence ${P(t)^k}$: $$\text{Minimal escape:}\quad \lim_{n \to \infty} \liminf_{k \to \infty} e_{k,n}(\Theta)$$

$$\text{Maximal escape:}\quad \lim_{n \to \infty} \limsup_{k \to \infty} e_{k,n}(\Theta)$$

$$\text{Generic escape:}\quad \forall \epsilon > 0,\ \lim_{n \to \infty} d(\left\{k : e_{k,n}(\Theta) > 1-\epsilon\right\}) = 1$$

4. Connections to Dynamics, Rigidity, and Number Walls

Analyses of maximal escape of mass are intertwined with structure results on flows on homogeneous spaces—both in zero and positive characteristic. In positive characteristic, escape phenomena can be more pronounced due to slower orbits growth (only linear rather than exponential) (Kemarsky et al., 2015). The tools for studying these include:

• Number walls: Two-dimensional arrays encoding vanishing and nonvanishing determinants, which are linked to the size of partial quotients in the continued fraction expansion.
• Automatic substitution tilings and group equivariance: For example, the Thue–Morse and $p$-Cantor sequences yield explicit, self-similar block structures with symmetry, leading to recursively constructed formulas for escape rates along wall diagonals.
• Frame relations: Algebraic and combinatorial relations ensuring propagation of zero-blocks in the number wall, central for determining limiting escape properties.

For the $p$-Cantor sequence, it is precisely the block structure and group invariance of the number wall under the morphism and the dihedral group $G$ that forces generic and maximal escape, even when the minimal escape is strictly less than one (Aranov et al., 22 Oct 2025).

5. Implications and Variations

The construction of explicit counterexamples to full escape (in the “liminf” sense) revealed that the landscape is far richer: leading to orthogonal distinctions among minimal, maximal, and generic escape concepts. For instance, for the $p$-Cantor sequence:

• Minimal escape = $2/p$,
• Maximal escape = $1$,
• Generic escape = $1$.

Despite the presence of large degree partial quotients along a density-zero set of times, most times, almost all the degree (mass) escapes, demonstrating a dichotomy between worst-case and typical behavior. This is mirrored in the analysis of number wall diagonals, where “holes” with large zero blocks (corresponding to unbounded partial quotient degrees) become dense along certain sequences but not universally dominant.

Beyond function fields, analogues exist in entropy escape for geodesic flows in negative curvature (Riquelme et al., 2016), entropy at infinity for Markov shifts (Iommi et al., 2019), or measures on spaces of lattices under diagonal or unipotent flows (Shapira, 2015), often parametrized by upper or lower limits of entropy or proportion of measure escaping to infinity.

6. Broader Context and Future Directions

The refined understanding and variants of maximal escape of mass have had several consequences:

• Disproving naive rigidity or equidistribution conjectures in positive characteristic;
• Providing combinatorial models (substitution systems and automatic tilings) that govern the subtle spectral and geometric features of escaping orbits;
• Suggesting generalizations involving genericity and maximal escape as new benchmarks for Diophantine systems, ergodic flows, and symbolic dynamics.

Open directions include extending these methods to broader classes of automatic and substitutive sequences, higher rank dynamics, and further exploring the translation to metric Diophantine results in positive characteristic, as well as connections to entropy at infinity and variational principles for more general noncompact spaces.

7. Summary Table: Minimal, Maximal, and Generic Escape of Mass

Variant	Definition / Formula	Attainable Value (Example: $p$-Cantor)
Minimal escape	$$\lim_{n\to\infty}\liminf_{k\to\infty} e_{k,n}(\Theta)$$	$2/p$
Maximal escape	$$\lim_{n\to\infty}\limsup_{k\to\infty} e_{k,n}(\Theta)$$	$1$
Generic escape	$\forall \epsilon>0,$ density-one set of large escape	$1$

This table illustrates the orthogonality and necessity of distinguishing these variants in rigorous applications.

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In conclusion, maximal escape of mass constitutes a central mechanism in understanding measure dynamics and Diophantine approximation over noncompact or positive-characteristic settings, demanding variant definitions and delicate structural analyses, notably as illustrated in the paper of the $p$-Cantor sequence (Aranov et al., 22 Oct 2025) and the Thue–Morse case (Nesharim et al., 24 Mar 2025).