Gromov–Yomdin-like Equality

• Gromov–Yomdin-like equality is a framework that equates dynamical or geometric invariants (like entropy and Milnor numbers) with spectral or combinatorial data using optimal decompositions.
• It is applied across singularity theory, categorical dynamics, and mirror symmetry, providing explicit resolutions and parametrizations to compute complex invariants.
• The methodology leverages controlled stratifications and decompositions to transform intricate geometric or categorical data into accessible linear-algebraic measures, guiding further research.

A Gromov–Yomdin-like equality refers to structural results—often taking the form of precise equalities—relating topological, homological, or dynamical invariants of geometric or algebraic objects to spectral, combinatorial, or linear-algebraic data extracted from their resolutions, symmetries, or parametrizations. Originating in the context of dynamics and complex geometry, such equalities have re-emerged across singularity theory, arithmetic geometry, derived categories, and algebraic geometry, extending the original themes introduced by Gromov and Yomdin: the computation of entropy, Milnor number, or analogous complexity measures via optimal geometric decompositions or categorical and arithmetic data.

1. Fundamental Principles and Classical Sources

The classical Gromov–Yomdin theorem equates the topological entropy $h_{\mathrm{top}}(f)$ of a smooth automorphism $f$ of a compact Kähler manifold with the logarithm of the spectral radius $p(f^*)$ of the induced map on cohomology: $h_{\mathrm{top}}(f) = \log p(f^*).$ This equality connects dynamical growth rates to linear algebraic invariants and has inspired analogous results in singularity theory, algebraic geometry, and categorical dynamics.

In the context of singularities, Gromov–Yomdin-like equalities typically express an invariant of a complex analytic variety (e.g., the Milnor number or monodromy zeta function) in terms of explicit contributions from a resolution process (as in (Martín-Morales, 2012)).

In categorical dynamics, these equalities equate the categorical entropy $h(\Phi)$ of an exact autoequivalence $\Phi$ of a triangulated category with the logarithm of the spectral radius of its induced action on the numerical Grothendieck group: $h(\Phi) = \log \rho([\Phi])$ (see (Kikuta et al., 2017, Ouchi, 2017, Barbacovi et al., 2021, Yoshida, 5 Jul 2025)).

2. Gromov–Yomdin-like Equalities in Singularity Theory

For singular hypersurfaces, especially Yomdin–Lê surface singularities of the form $f = f_m + f_{m+k} + \dots$, embedded $\mathbf{Q}$-resolutions enable explicit determination of topological invariants via weighted blow-ups, abelian quotient singularities, and a generalized A'Campo formula (Martín-Morales, 2012). The resolution process is made "optimal" in the sense that every exceptional divisor—excluding a possible initial divisor—contributes to the monodromy if and only if the corresponding divisor in the resolution of the tangent cone does. This yields precise formulas: $$\Delta_{(V,0)}(t) = \frac{(t^m - 1)^{\chi(\mathbb{P}^2 \setminus C)}}{t - 1} \prod_p \Delta_{(C,p)}(t^{m+1}),$$ and

$\mu(V, 0) = (m-1)^3 + \sum_p \mu(C, p),$

where $\Delta_{(V,0)}(t)$ is the characteristic polynomial of monodromy and $\mu(V,0)$ the Milnor number.

This type of equality directly parallels the Gromov–Yomdin principle, transferring complexity from a high-dimensional singularity to its (simpler) tangent cone data.

3. Real and Non-Archimedean Yomdin–Gromov Algebraic Lemmas and Parametrizations

In tame geometry and real/complex semialgebraic and subanalytic sets, the Yomdin–Gromov algebraic lemma provides polynomial upper bounds for the number and complexity of $C^r$-smooth charts required to cover a set, depending polynomially on the degree and the smoothness order (Binyamini et al., 2020, Binyamini et al., 2018, Novikov et al., 2023). For instance: $$C(n, \mu, r, \beta) = \mathrm{poly}_n(\beta) \cdot r^\mu,$$ where $\beta$ is the complexity of the defining equations and $\mu$ the dimension.

These optimal decompositions underlie Gromov–Yomdin-type equalities, most notably in dynamics—where, for $C^\infty$ maps, tail entropy and local volume growth can be made negligible: $$v^0_l(f) = 0,\qquad v_l(f,\delta) \leq O_m(\log L)\cdot \frac{\log|\log \delta|}{|\log \delta|}.$$ The topological entropy equals the spectral radius defined via derivatives and local volume growth.

In the non-Archimedean setting, uniform Yomdin–Gromov parametrizations extend the principle to $p$-adic and more general valued fields, where definable sets can be covered by fibred charts with uniform Taylor approximation: $|f(x) - T_{f, a}(x)| \leq |x - a|^r.$ This uniform control is essential for point-counting results and analogues of the Pila–Wilkie theorem (Cluckers et al., 2014, Cluckers et al., 2019, Nowak, 26 May 2025).

4. Categorical Gromov–Yomdin Equalities and Entropy

In the theory of triangulated categories, especially the bounded derived category $D^b(X)$ of coherent sheaves, categorical entropy $h(\Phi)$ is compared to the spectral radius of the induced endomorphism $[\Phi]$ on the numerical Grothendieck group $N(X)_\mathbb{R}$ (Kikuta et al., 2017, Yoshida, 5 Jul 2025).

• General lower bound: $h(\Phi) \geq \log \rho([\Phi])$. This holds for smooth proper dg categories, under mild conditions.
• Equality cases: Full equality $h(\Phi) = \log \rho([\Phi])$ is established for orbifold projective lines (Kikuta et al., 2017), abelian surfaces, bielliptic surfaces (Yoshida, 5 Jul 2025), and for spherical twists (Ouchi, 2017) and certain holomorphic autoequivalences (Barbacovi et al., 2021).
• Counterexamples: For some autoequivalences on K3 and Enriques surfaces, strict inequality $h(\Phi) > \log \rho([\Phi])$ is observed (Ouchi, 2017, Yoshida, 5 Jul 2025). This demonstrates that categorical entropy can in some cases encode information not detectable by the linearized action on $N(X)_\mathbb{R}$—often due to the presence of spherical objects or complex derived category structure.

5. Model-Theoretic and Stratification Results Supporting Gromov–Yomdin Equalities

Powerful stratification and desingularization methods—quantifier elimination, term descriptions, and Lipschitz cell decompositions—enable the construction of Yomdin–Gromov-type charts in o-minimal or Henselian analytic settings (Nowak, 26 May 2025). A smooth stratification is constructed via multi-blowup maps, ensuring definable functions exhibit controlled, quantitative approximation (e.g., $T_r$-approximation).

These model-theoretic approaches yield piecewise analytic or algebraic parametrizations necessary for both point-counting and for transferring analytic or arithmetic invariants between geometric objects, mirroring the Gromov–Yomdin reduction of complexity via cell decomposition.

6. Gromov–Yomdin-like Equalities in Enumerative Geometry and Mirror Symmetry

In relative and orbifold Gromov–Witten theory, polynomiality and stabilization phenomena encode Gromov–Yomdin-like behavior (Tseng et al., 2018, You, 2019). For smooth pairs $(X, D)$, one has: $\langle \cdots \rangle_{(X, D)}^{g, k, n, d} = \text{[r}^0\text{-coefficient of %%%%24%%%% for large %%%%25%%%%]},$ where the left side is a relative invariant, the right an orbifold invariant of an $r$-th root stack, and the equality is in the constant term after polynomial expansion in $r$. Such relationships recover classical invariants by "stabilization," exemplifying the transfer principle at the heart of Gromov–Yomdin-like statements.

In mirror symmetry for toric Calabi–Yau orbifolds (You, 2019), the inverse mirror map is explicitly determined by generating series of relative or open Gromov–Witten invariants: $$1 + \sum_{d} N^{rel}_d q^d = \exp(-g_{i_0}(y(q,T))),$$ establishing an exact equality between enumerative invariants and the analytic structure of the mirror moduli.

7. Implications, Limitations, and Future Directions

Gromov–Yomdin-like equalities provide a robust conceptual and computational bridge between analytic/dynamical complexity, algebraic invariants, and categorical growth rates. The phenomena are robust in settings where optimal decompositions, stratifications, or categorical symmetries exist, but counterexamples show that the identification is sensitive to deeper geometric and categorical features (notably, presence of spherical objects or nontrivial derived equivalence structure).

Future investigations aim to sharpen criteria for equality, to classify the potential gaps or "defects" in the Gromov–Yomdin correspondence, and to refine these principles in arithmetic and motivic contexts, with increasing emphasis on uniformity across parameter spaces, families, or fields of definition.

Table: Summary of Gromov–Yomdin-like Equalities Across Domains

Domain	Main Invariant	Equality Form
Dynamics	$h_{\mathrm{top}}(f)$	$h_{\mathrm{top}}(f) = \log p(f^*)$
Singularity theory	Monodromy, Milnor number	$\Delta_{(V, 0)}(t)$ via tangent cone data
Tame/cell geometry	Chart/volume complexity	Poly($\text{degree}$)/volume growth matches
Derived categories	$h(\Phi)$ (categorical entropy)	$h(\Phi) = \log \rho([\Phi])$ under conditions
Gromov–Witten theory	Relative/orbifold invariants	const-term or full match by stabilization
Mirror symmetry	Mirror map / disk invariants	Mirror = disk generating series

These Gromov–Yomdin-like equalities continue to structure current research at the intersection of geometry, dynamics, categorification, and arithmetic, systematically exposing the ways in which geometric or categorical complexity can be “transferred” to optimal linear or combinatorial data.