Deautonomisation of Birational Maps

• Deautonomisation of birational maps is the process of replacing constant parameters with functions while preserving key singularity structures such as confinement and anticonfinement.
• It provides a framework for classifying discrete integrable systems, including discrete Painlevé equations, by analyzing the growth patterns in singularities using algebraic entropy.
• Ultradiscrete (max-plus) methods are employed to efficiently compute multiplicity growth, serving as practical diagnostics for assessing integrability in both low- and high-order mappings.

A deautonomisation of a birational map refers to the procedure whereby a mapping, originally defined with constant parameters (“autonomous”), is generalized by promoting these parameters to functions of the discrete independent variable(s) in such a way that its singularity structure (specifically, confinement and anticonfinement properties) is preserved. This process plays a central role in the theory and classification of discrete integrable systems, especially in the context of discrete Painlevé equations, the analysis of birational mappings, and their singularity patterns.

1. Definitions and Singularity Structures in Birational Maps

A birational mapping $f : \mathbb{P}^n \dashrightarrow \mathbb{P}^n$ is well defined almost everywhere on projective space, except on singular hypersurfaces (“singular loci”) where some of its rational expressions become ill-defined (vanishing denominator, $0/0$ form, or poles). The local behavior of an iterate at or near these loci gives rise to various singularity types (Willox et al., 4 Feb 2026):

• Confined singularity: A loss of freedom at a singular locus is recovered after a finite number of further iterates that return to generic (finite, nonzero) values.
• Non-confined (unconfined) singularity: Iterates fail to recover genericity, with the singularity propagating indefinitely without closure.
• Cyclic singularity: Iterates cycle through a fixed, finite set of singular values (e.g., $7$-cycle of $0$ and $\infty$).
• Anticonfined singularity: A finite block of generic values is surrounded by infinite sequences of singular values (typically $0$ or $\infty$) in both forward and backward directions; only a finite “window” of regular values separates two semi-infinite blocks of singularities (Mase et al., 2015).

The singularity structure—encompassing the full ensemble of confined, cyclic, non-confined, and anticonfined patterns—is a primary diagnostic of potential integrability.

2. Deautonomisation Procedure and Its Mathematical Foundations

The classic deautonomisation procedure begins with an autonomous birational difference equation and promotes its constant parameters to nonautonomous sequences/functions of the iteration index, $n$. The goal is to preserve the singularity structure—most importantly, the confinement of singularities—across all $n$.

In the canonical context, this procedure is responsible for the discrete Painlevé equations and integrable nonautonomous recurrences. For a mapping

$x_{n+1} = F(x_n, x_{n-1}; a_1, \ldots, a_k),$

deautonomising promotes $a_i \mapsto a_i(n)$. Functional constraints on $a_i(n)$ are determined by demanding recurrence or preservation of (anti)confinement properties in the mapping's singularity analysis (Mase et al., 2015, Willox et al., 4 Feb 2026).

At second order, this often leads to integrable hierarchies parameterized by discrete “time”, and it is conjectured—and partially established—that suitable deautonomisations uniquely specify integrable nonautonomous extensions. In higher-order cases, the landscape is more intricate, as integrable maps may exhibit non-confined or even anticonfined singularities, complicating the identification of appropriate constraints (Willox et al., 4 Feb 2026).

3. Anticonfined Patterns and Their Classification

Anticonfined singularity patterns, introduced in (Mase et al., 2015), are formalized as follows: for a discrete mapping, an initial condition is chosen so that almost every iterate (in both forward and backward directions) is singular (e.g., diverges as $\epsilon^{-d_k}$ for some integer exponent $d_k > 0$), save for a finite central window of generic (finite, nonzero) values. The sequence of exponents $\{d_k\}$ is termed the anticonfined pattern.

Anticonfined patterns are empirically classified according to the growth behavior of $\{d_k\}$:

Growth Type	Exponent Behavior	Associated Degree Growth	Integrability Diagnostic
Zero/bounded	$d_k \equiv D$ (constant)	$O(1)$	Inconclusive; may be integrable
Linear	$d_k \sim C|k|$	$O(n)$	Linearisable (degree growth 1)
Exponential	$d_k \sim \lambda^{|k|}$, $\lambda>1$	$O(\lambda^n)$	Non-integrable (positive algebraic entropy)

For second-order mappings, exponential anticonfined growth unambiguously signals non-integrability, while linear growth implies linearisability but not QRT-type integrability. For higher-order mappings, the correspondence is preserved, but cubic and quadratic growths appear depending on the order of the coupling (Mase et al., 2015, Willox et al., 4 Feb 2026).

4. Ultradiscrete (Max-Plus) Methods for Multiplicity Growth

To efficiently compute the growth of multiplicities in anticonfined patterns without laborious Laurent expansions, the ultradiscretisation (max-plus) technique is employed (Willox et al., 4 Feb 2026). The variable substitutions $x_n = \exp(X_n/\epsilon)$ as $\epsilon \to 0^+$ lead to a “tropical” recurrence: $$\exp(U/\epsilon) + \exp(V/\epsilon) \to \exp(\max(U,V)/\epsilon),$$ so the original rational maps translate into max-plus recurrences for the exponents $X_n$ that track multiplicities directly.

For example, a third-order map

$y_{n+1} = y_{n-2} + \max(y_n, y_{n-1}) - y_{n-1}$

gives rise to a linear-max recurrence: $X_{n+1} = X_{n-2} + \max(X_n, X_{n-1}) - X_{n-1}.$ By seeding the “entry” into a singular block (e.g., $X_2 = -1$), the max-plus recurrence iterates reveal the multiplicity growth type—linear, quadratic, or cubic—directly correlated with the algebraic degree growth of the underlying mapping (Willox et al., 4 Feb 2026).

5. Deautonomisation in the Context of Integrability Detection

The growth behavior of multiplicities in anticonfined patterns provides necessary conditions for integrability in birational mappings:

• Exponential growth: Implies dynamical degree $\delta > 1$; mapping is non-integrable.
• Linear growth: Mapping is linearisable (dynamical degree 1), but not generically integrable in the QRT sense.
• Zero/bounded growth: No conclusion; further tests involving other singularity patterns and/or a full deautonomisation procedure are required (Mase et al., 2015).

A general lower bound theorem states: if the pole orders in an anticonfined singularity pattern obey $\limsup_{k\to\infty} d_k^{1/|k|} = \lambda_*$, then the dynamical degree $\delta$ of the mapping satisfies $\delta \ge \lambda_*$ (Mase et al., 2015). Anticonfined patterns thus provide only necessary (not sufficient) conditions for integrability; comprehensive singularity analysis, including all singularity types, is required.

Notably, the preservation of the multiplicity growth of anticonfined patterns under parameter deformation—i.e., as part of the deautonomisation process—serves as a stringent constraint. Demanding that no new multiplicity transitions occur across steps generates functional equations for the time-dependence of parameters, repairing or obstructing integrability in nonautonomous settings (Willox et al., 4 Feb 2026).

6. Higher-Order Mappings, Codimension, and Subtleties

In higher-order birational recurrences, anticonfined (and other) singularities can exhibit more intricate behavior due to the possible involvement of higher-codimension singular loci (Willox et al., 4 Feb 2026). While the regularisation of codimension-one singularities (hypersurfaces) via blow-ups remains a primary analytic tool, certain integrable cases rely on specific cancellations that may actually occur only at codimension-two loci.

Nevertheless, in a broad class of studied higher-order mappings, the observed anticonfined patterns in codimension one suffice to explain polynomial (e.g., linear, quadratic, cubic) degree growth when the mapping is integrable or properly deautonomised. When no such pattern exists, or multiplicities grow exponentially, non-integrability (positive algebraic entropy) persists.

Cyclic and fixed (anticonfined) singularity patterns arising from non-generic initial conditions (e.g., corner zeros in the discrete KdV equation) do not contribute to the calculation of dynamical degree, since only “movable” (i.e., generic initial data–dependent) singularities are relevant for entropy computations (Um et al., 2019). The existence of fixed anticonfined patterns does not contradict integrability.

7. Applications and Conclusions

Deautonomisation of birational maps, informed by a detailed taxonomy and growth analysis of their singularity structures—including anticonfined patterns—lies at the core of discrete integrability theory. This framework underpins the classification of integrable and non-integrable discrete systems, the construction of discrete Painlevé equations, and the development of rigorous integrability detectors using algebraic entropy and degree growth rates (Mase et al., 2015, Willox et al., 4 Feb 2026, Um et al., 2019).

Anticonfined singularity analysis extends the classical confinement criterion, providing both a practical diagnostic for integrability (via dynamical degree lower bounds and multiplicity growth) and a set of constraints for admissible deautonomisations. The incorporation of max-plus ultradiscrete methods greatly enhances computational tractability, while subtle issues related to codimension and movable vs. fixed singularities continue to drive progress in the field.

The interplay between deautonomisation, singularity structure, and algebraic dynamics remains a vigorous area of research, especially in the context of higher-order and multidimensional birational mappings.