Algebraic Entropy of Birational Maps

• Algebraic entropy is a quantitative invariant defined by the asymptotic growth rate of degrees of iterates, distinguishing periodic, integrable, and chaotic regimes.
• It bridges algebraic geometry and dynamical systems by using dynamical degrees and the spectral radius of induced actions to measure complexity.
• Its computation, involving recursive degree sequences and characteristic polynomials, aids in classifying birational transformations across various algebraic structures.

Algebraic entropy of birational transformations is a quantitative invariant that captures the complexity of a birational map through the asymptotic growth rate of the degrees of its iterates. For a birational transformation $f$ of an algebraic variety $X$, the algebraic entropy is conventionally defined as $$h_{\text{alg}}(f) = \lim_{n\to\infty} \frac{1}{n} \log(\deg(f^n))$$, linking algebraic geometry, dynamical systems, and spectral theory. Algebraic entropy provides a rigorous framework for distinguishing integrable, periodic, and chaotic regimes via the behavior of dynamical degrees and the growth of recursive formulae, underlying ergodic and geometric aspects of the Cremona group, projective surfaces, and higher-dimensional models.

1. Fundamental Definition and Computation

The algebraic entropy for a birational transformation $f$ is determined by the exponential rate of degree growth under iteration: $$h_{\text{alg}}(f) = \lim_{n\to\infty} \frac{1}{n} \log(\deg(f^n))$$ For surfaces, this coincides with the first dynamical degree $\lambda_1(f)$, given by

$\lambda_1(f) = \lim_{n\to\infty} (\deg(f^n))^{1/n}$

and thus $h_{\text{alg}}(f) = \log \lambda_1(f)$ (Bedford et al., 2011, Blanc et al., 2013). Algebraic entropy vanishes if $\lambda_1(f)=1$, which occurs for periodic, integrable, or polynomial growth cases (bounded/linear/quadratic degree sequences), and is positive for exponentially growing sequences signaling complex or chaotic dynamics.

Birational transformations on higher-dimensional varieties or compact Kähler manifolds are quantified using dynamical degrees $d_p(f)$, defined for each $0 \leq p \leq \dim X$ by the spectral radius of the pullback map $f^*$ on cohomology classes. The maximal logarithm among these yields the topological entropy: $$h_{\text{top}}(f) = \max_{0\leq p \leq \dim X} \log d_p(f)$$ This identification is justified by results of Gromov and Yomdin (Oguiso, 2014).

2. Spectral and Geometric Interpretation

Algebraic entropy is inherently related to the spectral radius of the induced action on the appropriate lattice (Néron–Severi group, Picard group, or cohomology): $$h_{\text{alg}}(f) = \log (\text{spectral radius of } f^* \text{ on } H^{1,1}(X))$$ For automorphisms of projective surfaces or hyperkähler manifolds, this framework extends to the Beauville–Bogomolov lattice (Grossi, 2019, Beri et al., 2020).

Values of the dynamical degree greater than unity (i.e., $\lambda_1(f) > 1$) are discrete and well-ordered for the Cremona group of $\mathbb{P}^2$ (Blanc et al., 2013); they are typically Salem or Pisot numbers for monomial maps or automorphisms on surfaces. The translation length in the Picard–Manin hyperbolic space induced by $f$ is given by $L(f_*) = \log \lambda(f)$.

Periodicity ($f^p=\text{id}$) results in $\lambda_1(f)=1$ and vanishing entropy. Integrable cases display polynomial degree growth, whereas loxodromic/hyperbolic maps exhibit exponentially increasing degrees and positive entropy, manifesting more intricate dynamics (Bedford et al., 2011, Cima et al., 2017).

3. Classification via Degree Growth

The growth type of the degree sequence $\{d_n\}$ of iterates categorizes birational transformations:

• Periodic and bounded growth: $d_n$ constant or repeating (periodic maps); entropy $0$ (Cima et al., 2017, Cima et al., 2017).
• Linear/quadratic/polynomial growth: $d_n = 1 + n$, $d_n \sim n^2$ or similar (integrable or rational fibration-preserving maps); entropy $0$ (Cima et al., 2017, Hone et al., 2015).
• Exponential growth: $d_n \sim \lambda^n$ for $\lambda>1$ (chaotic or nonintegrable maps); entropy $>0$ (Bedford et al., 2011, Blanc et al., 2013). This leads to a dynamic classification: integrable and periodic systems with algebraic entropy zero, chaotic systems with positive entropy. Recurrence relations and spectral properties of the induced matrix (by $f^*$ on Picard group) determine the precise type (Cima et al., 2017).

Explicit formulas for the dynamical degree include characteristic polynomials arising from the matrix action on cohomology or Picard group, with the entropy given by the logarithm of the maximal real root (Bedford et al., 2011, Cima et al., 2017).

4. Algebraic Structures and Families

The group of birational transformations, $\text{Bir}(X)$, admits various algebraic structures: ind–schemes, Zariski topology, and algebraic subgroups, often with scheme-theoretic or functorial descriptions (Blanc, 2015, Regeta et al., 10 Sep 2024). Families of birational transformations are parametrized by locally closed subsets in projective spaces of homogeneous polynomials (parameter spaces $H_{(d)}$), with Chevalley-type results ensuring constructibility of images and precise control over fiber dimensions (Regeta et al., 10 Sep 2024).

Algebraic subgroups of $\text{Bir}(X)$ are those that are closed, finite-dimensional, and have finitely many connected components, and they inherit an algebraic group structure by universality properties. If a birational transformation belongs to such a subgroup and has bounded degree, its entropy vanishes (Blanc, 2015, Regeta et al., 10 Sep 2024). Preservation of fibrations facilitates decomposition of entropy into base and fiber components.

For the Cremona group, the asymptotic algebraic growth (number of irreducible components $N_d$) of birational transformations of degree $d$ is intermediate: neither polynomial nor exponential, but double-logarithmic, with bounds

$$A\sqrt{\ln d} \leq \ln(\ln(\sum_{e\leq d} N_e)) \leq B \sqrt{\ln d}$$

for suitable constants $A$, $B$ (Calabri et al., 6 Mar 2025). This indicates deep combinatorial structure in moduli spaces of birational maps, distinct from classical entropy for individual transformations.

5. Extensions and Generalizations: Algebraic Maps, Systems, and Noncommutative Geometry

The concept of algebraic entropy extends to algebraic maps (possibly multivalued, with field extensions defined by auxiliary variables), provided the iterates remain in a fixed field extension and can be expanded in a fixed basis (Hone et al., 2015). Degree growth is defined by the maximal degree of coefficients in the expansion; entropy vanishes for integrable cases and is positive for exponential growth, such as trace maps governed by linear recurrences with Fibonacci growth rates.

For systems of quad equations, especially in discrete integrable systems, algebraic entropy is computed by tracking the degree growth along admissible (birational) directions on a lattice. Only directions yielding a birational evolution are suitable for entropy calculation, with the entropy extracted from the generating function of the degree sequence via the minimal modulus of its poles (Gubbiotti, 2023). Integrable systems display polynomial degree growth and entropy zero, while nonintegrable cases exhibit exponential growth.

In noncommutative algebraic geometry, intersection theory for line modules replaces classical set-theoretic intersection, connecting the calculation of intersection multiplicities with birational transformations akin to noncommutative analogues of Bezout’s theorem (Rogalski et al., 2021). The structure of birational maps in this context mirrors the classical theory but is enriched by Ext-group computations and module-theoretic invariants.

6. Periodicity, Integrability, and Special Transformations

Examples such as special birational transformations of type $(2,1)$ reveal that maps defined by quadratic systems with linear inverses, or de Jonquières transformations arising from Galois points with cyclic groups of order three, often have highly constrained degree growth and zero algebraic entropy (Fu et al., 2015, Miura, 2023). Classification theorems—identifying rigid geometric types (hyperquadrics, Segre varieties, Grassmannians, etc.)—enable inference of preemptively low dynamical complexity.

For birational maps on Hilbert schemes of points on K3 surfaces, such as involutive automorphisms classified via lattice-theoretic criteria (solutions to Pell’s equations with congruence conditions), the induced action on cohomology is of finite order, leading to entropy zero (Beri et al., 2020). Similarly, induced birational transformations on O'Grady's sixfolds are governed by eigenvalues of lattice isometries, with zero entropy for automorphisms of abelian surfaces of finite order (Grossi, 2019).

7. Connections to Geometry, Dynamical Systems, and Moduli Spaces

Algebraic entropy encapsulates the complexity of birational maps in moduli problems, dynamical systems, and geometry. The identification of invariant K3 surfaces, elliptic fibrations, and connections with minimal model theory deepen its relevance (Bedford et al., 2011, Oguiso, 2014). The dynamical spectrum in Cremona dynamics (discreteness, well-ordered, spectral gaps) reveals rigidity reminiscent of mapping class group growth rates and hyperbolic geometry (Blanc et al., 2013, Calabri et al., 6 Mar 2025).

Entropy sensitivity to real vs. complex dynamics, as seen in rational surface automorphisms with invariant cubics, demonstrates that computation of entropy in homology or Picard group provides powerful quantitative information regarding orbit structure and periodicity (Diller et al., 2017).

Algebraic entropy remains a pivotal tool both in classifying the dynamical behavior of explicit birational transformations and in understanding the growth structure of transformation groups, tracking the transition from ordered, integrable, or periodic regimes to chaotic or complex behaviors through sharply defined spectral and geometric invariants.