Dynamics of subgroups of Aut(X_1) x Aut(X_2) on spaces of holomorphic maps between Hermitian symmetric spaces

Gekhtman (2019) analyzed the translation flow, i.e. the conjugation action of a unipotent subgroup of Aut(H) on the class of holomorphic maps H^n -> H that restrict to the identity on the diagonal, proving that the translates of any such map converge to a linear map in probability (density-one recurrence), that the flow has a unique invariant measure, and that its periodic points are exactly the linear maps. The paper then poses the open question this task addresses: what can be said about the dynamics of subgroups of Aut(X_1) x Aut(X_2) acting by conjugation on spaces of holomorphic maps O(X_1, X_2) between arbitrary Hermitian symmetric spaces X_1, X_2? No general theory exists: it is unknown which subgroup actions rigidify or linearize holomorphic maps, which produce chaotic behavior, for which source/target pairs the phenomena extend, and even in the already-treated flow the pointwise convergence of translates for every map remains open.

Background

A Hermitian symmetric space is a simply connected complex manifold carrying a Riemannian metric for which the geodesic symmetry at every point is holomorphic; the noncompact examples of primary relevance here include the bounded symmetric domains, of which the simplest are the upper half-plane $\mathbb{H}$, the poly-plane $\mathbb{H}^n$ (the $n$-fold product of $\mathbb{H}$ with itself), and the unit ball. Their biholomorphism groups $\operatorname{Aut}(X)$ are real semisimple Lie groups, e.g. $\operatorname{Aut}(\mathbb{H}) \cong \operatorname{PSL}_2(\mathbb{R})$. Given two such spaces $X_1, X_2$, the product $\operatorname{Aut}(X_1) \times \operatorname{Aut}(X_2)$ acts on the space $\mathcal{O}(X_1, X_2)$ of holomorphic maps $X_1 \to X_2$ by conjugation, $(\gamma_1, \gamma_2) \cdot f = \gamma_2 \circ f \circ \gamma_1^{-1}$, and one can study the dynamics of a subgroup $\Gamma$ of this product on a $\Gamma$-invariant class of maps: orbit closures, invariant probability measures, periodic points, and whether the dynamics rigidifies maps, meaning that orbit-accumulation points, periodic points, or invariant maps are forced into a small finite-dimensional family such as the linear maps.

The motivation and the single understood case come from Gekhtman's study of the translation flow. Let $\mathcal{D}$ be the family of holomorphic $f : \mathbb{H}^n \to \mathbb{H}$ that restrict to the identity on the diagonal, $f(\lambda, \dots, \lambda) = \lambda$. This class arises in Teichmüller theory, the study of the Teichmüller space $\mathcal{T}$ of a surface (the complex manifold parametrizing its marked conformal structures): the Carathéodory metric (the smallest pseudometric on a complex manifold $X$ for which every holomorphic map $X \to \mathbb{H}$ is distance-nonincreasing) and the Kobayashi metric (the largest pseudometric on $X$ for which every holomorphic map $\mathbb{H} \to X$ is distance-nonincreasing) need not agree on a Teichmüller disk (a holomorphic embedding $\mathbb{H} \to \mathcal{T}$, i.e. a complex geodesic for the Kobayashi metric) generated by a Jenkins–Strebel differential (a meromorphic quadratic differential on the surface whose regular horizontal trajectories are closed curves); when the two metrics do agree on such a disk, Markovic's construction yields a holomorphic embedding $E^\varphi : \mathbb{H}^n \to \mathcal{T}$ such that the composition of $E^\varphi$ with a holomorphic retraction $\mathcal{T} \to \mathbb{H}$ onto the disk lies in $\mathcal{D}$. Markovic used the resulting rigidity statements to prove that the two metrics differ on the Teichmüller space of any closed orientable surface of genus at least 2, and Gekhtman–Markovic later classified the complex geodesics of low-dimensional Teichmüller spaces. The translation flow on $\mathcal{D}$ is $f_t(z_1, \dots, z_n) = f(z_1 - t, \dots, z_n - t) + t$; it is exactly the conjugation action of the subgroup of translations in $\operatorname{PSL}_2(\mathbb{R})$, a unipotent subgroup (a one-parameter subgroup all of whose non-identity elements are parabolic, i.e. fix exactly one point of the boundary $\partial\mathbb{H}$), and every unipotent one-parameter subgroup of $\operatorname{PSL}_2(\mathbb{R})$ is conjugate to it. Writing $\alpha_j = \partial f / \partial z_j(i, \dots, i)$, so that $\sum_j \alpha_j = 1$, and $g(z) = \sum_j \alpha_j z_j$, Gekhtman proved: the translates $f_t$ converge to $g$ in probability, in the precise sense that for every compact-open neighborhood $U$ of $g$ the set $\{ t : f_t \in U \}$ has density one in $\mathbb{R}$ (equivalently, the pushforwards of uniform measure on $[-r, r]$ under $t \mapsto f_t$ converge weakly to $\delta_g$); the only translation-invariant probability measure on the normalized class $\mathcal{C} \subset \mathcal{D}$ (where $\partial f / \partial z_j(\lambda, \dots, \lambda) = 1/n$ on the diagonal) is $\delta_g$; and every periodic point of the flow is linear, $f = g$ (a rigidity result recovering and generalizing Markovic's $n = 2$ theorem). The proof rests on two special facts: the normalized class $\mathcal{C}$ is compact in the compact-open topology (a Montel-type phenomenon — normal-family compactness of holomorphic maps into the bounded hyperbolic disk — special to the one-dimensional target), and a multivariate Schwarz lemma in the spirit of Knese's classification of extremal maps $\mathbb{D}^n \to \mathbb{D}$.

Two limitations of this picture motivate the general question. First, the convergence is only in probability: the paper states explicitly that it is not known whether $\lim_{t \to \infty} f_t = g$ for every $f \in \mathcal{C}$. Second, the phenomenon is fragile: for the analogous class of maps $\mathbb{C}^k \to \mathbb{C}$ fixing the diagonal, the same translation flow is chaotic in the sense of linear dynamics (the study of iterates of continuous linear operators on topological vector spaces) — by results of Godefroy–Shapiro and Bonilla–Grosse-Erdmann on continuous linear operators commuting with the partial derivatives, it admits a full-support ergodic probability measure and a dense set of periodic points (the target $\mathbb{C}$ destroys the compactness that governed the $\mathbb{H}^n \to \mathbb{H}$ case). Against this background the source poses, as an explicitly open direction with (to the author's knowledge) no previous literature, the question of the dynamics of subgroups of $\operatorname{Aut}(X_1) \times \operatorname{Aut}(X_2)$ acting on subsets of $\mathcal{O}(X_1, X_2)$ for general Hermitian symmetric spaces.

Problem Statement

Following the question posed in Gekhtman (2019), the problem is to develop a theory of the dynamics of subgroups $\Gamma \subseteq \operatorname{Aut}(X_1) \times \operatorname{Aut}(X_2)$ acting by conjugation, $(\gamma_1, \gamma_2) \cdot f = \gamma_2 \circ f \circ \gamma_1^{-1}$, on natural $\Gamma$-invariant subsets of the space $\mathcal{O}(X_1, X_2)$ of holomorphic maps between Hermitian symmetric spaces $X_1$ and $X_2$. The target phenomena to be understood, generalizing what is known for the unipotent translation flow on the class $\mathcal{D} \subset \mathcal{O}(\mathbb{H}^n, \mathbb{H})$ of maps restricting to the identity on the diagonal, are: orbit closures and recurrence of translates; invariant probability measures; periodic points; and rigidification or linearization — the extent to which the group action forces invariant maps, periodic maps, or accumulation points of orbits to lie in a finite-dimensional family such as the linear maps $\sum_j \alpha_j z_j$. Concretely, the problem asks: (1) for which subgroups $\Gamma$ (e.g. unipotent, semisimple, diagonal, or higher-rank subgroups) and which invariant classes of maps does the dynamics rigidify or linearize holomorphic maps, and for which source/target pairs $(X_1, X_2)$ of Hermitian symmetric spaces do the phenomena persist or fail — including whether chaotic behavior, as in the $\mathbb{C}^k \to \mathbb{C}$ analogue, is the generic alternative; and (2) as the first concrete open case inside the already-studied flow — explicitly declared open in the source — decide whether the translates $f_t(z_1, \dots, z_n) = f(z_1 - t, \dots, z_n - t) + t$ converge, as $t \to \infty$, locally uniformly on $\mathbb{H}^n$ to $g = \frac{1}{n} \sum_j z_j$ for every $f$ in the normalized class $\mathcal{C}$ of holomorphic maps $\mathbb{H}^n \to \mathbb{H}$ satisfying $f(\lambda, \dots, \lambda) = \lambda$ and $\partial f / \partial z_j(\lambda, \dots, \lambda) = 1/n$ for every $j$ on the diagonal $\{(\lambda, \dots, \lambda) : \lambda \in \mathbb{H}\}$, or exhibit a counterexample map whose translates fail to converge. These two parts have independent standing. Part (2) is a self-contained decide-or-counterexample question, resolved by either a proof that the translates $f_t$ converge locally uniformly to $g$ for every $f \in \mathcal{C}$, or a single counterexample $f \in \mathcal{C}$ whose translates fail to converge. Part (1) is the broader program; it asks, for at least one concrete choice of a nontrivial infinite subgroup $\Gamma$ acting with non-closed orbits on a nondegenerate infinite $\Gamma$-invariant class of maps, and a source/target pair $(X_1, X_2)$ of Hermitian symmetric spaces other than the unipotent translation flow on $\mathcal{O}(\mathbb{H}^n, \mathbb{H})$ treated by (2), which of the target phenomena listed above — orbit closures and recurrence of translates, invariant probability measures, periodic points, and rigidification or linearization — hold and which fail; a complete classification over all subgroups and all source/target pairs is the long-range goal.

References

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