Common Index Jump Theorem in Symplectic Dynamics

• Common Index Jump Theorem is a result that defines synchronized jumps in Maslov-type indices over iterated symplectic paths, establishing a robust framework for index theory.
• It employs spectral arithmetic, normal form decomposition, and correction invariants to derive precise identities for indices and nullities in Hamiltonian systems.
• CIJT is pivotal in proving multiplicity results for closed geodesics and Reeb orbits, with applications spanning Finsler manifolds and star-shaped hypersurfaces.

The Common Index Jump Theorem (CIJT) is a fundamental result in symplectic dynamics, variational theory, and Hamiltonian systems. It characterizes the synchronized behavior and discrete "jumps" in the Maslov-type or Conley–Zehnder indices when symplectic paths or iterations of periodic orbits are taken to high powers. CIJT and its enhanced versions have become central tools for approaching multiplicity results for closed geodesics and Reeb orbits, and have shaped index theory for symplectic paths in both finite and infinite-dimensional settings.

1. Historical and Mathematical Origins

The CIJT was first established by Long and Zhu in 2002 for symplectic paths [LZ02]. The theorem asserts that for a given finite collection of symplectic paths $\{\gamma_k\}$, under appropriate mean index growth (specifically, $\hat{i}(\gamma_k, 1) > 0$ for all $k$), there exist iterations in which the Maslov-type indices of these paths synchronize and "jump" by a common amount. This established a new level of control for index theory in Hamiltonian systems and symplectic geometry. Subsequent refinements led to the enhanced CIJT (ECIJT) [DLW16], introducing more precise formulaic relationships for the indices of arbitrary iterates. The framework was later reformulated in the index recurrence theorem [CGG24], which captures the essence of these identities in a recurrent, sequence-based language.

2. Key Formulations and Fundamental Results

The core of CIJT, as enhanced in [DLW16], is expressed through exact identities for the Maslov-type index $i(\gamma_k, m)$ and nullity $\nu(\gamma_k, m)$ of iterated symplectic paths:

$$\begin{aligned} &\nu(\gamma_k, 2m_k - m) = \nu(\gamma_k, 2m_k + m) = \nu(\gamma_k, m), \quad (3.20) \ &i(\gamma_k, 2m_k + m) = 2N + i(\gamma_k, m), \quad (3.21) \ &i(\gamma_k, 2m_k - m) = 2N - i(\gamma_k, m) - 2\left(S_{M_k}^+(1)+Q_k(m)\right), \quad (3.22) \end{aligned}$$

for $1 \leq m \leq \bar{m}$ and with $N, m_k$ selected according to precise congruence and spectral conditions. Here $S_{M_k}^+(1)$ captures the contribution of the eigenvalue $1$ to the splitting numbers, and $Q_k(m)$ sums correction terms over relevant spectral data.

The index recurrence theorem in [CGG24] demonstrates that its first four assertions correspond exactly to these enhanced CIJT identities, establishing a rigorous equivalence and unifying the language across approaches.

3. Technical Structure and Spectral Ingredients

To facilitate these identities, CIJT and ECIJT utilize the normal form decomposition of symplectic matrices, the splitting numbers, and the spectral arithmetic on the unit circle via fractional parts. For instance, the formula

$$Q_k(m) = \sum_{\substack{\theta\in (0,2\pi),\, e^{i\theta}\in \sigma(M_k) \ \{m_k \theta/\pi\}=\{m/2\pi\}=0}} S^{-}_{M_k}(e^{i\theta})$$

quantifies the index contribution from crossing spectral thresholds as the iteration parameter varies.

The index recurrence theorem [CGG24] reformulates the iteration arithmetic as

$$\begin{aligned} &\mu_{\pm}(\Phi^{k+\ell}) = d + \mu_{\pm}(\Phi^{\ell}), \ &\mu_{+}(\Phi^{k-\ell}) = d - \mu_{-}(\Phi^{\ell}) + (\underline{+}(\Phi^{\ell})-\underline{-}(\Phi^{\ell})), \end{aligned}$$

for suitably chosen $k, d,$ and $\ell$, where $\mu_\pm$ are upper/lower semicontinuous indices.

These technical constructions enable delicate control over the synchronization of index jumps among sequences of iterated symplectic paths.

4. Applications in Hamiltonian Dynamics and Variational Theory

CIJT and ECIJT are pivotal in establishing lower bounds on the number of closed geodesics (or closed Reeb orbits) on Hamiltonian or Finsler manifolds. By controlling index jumps, one applies Morse inequalities and resonance identities to show that indices are "spread out" and cannot cluster, ensuring the existence of many distinct periodic orbits.

For example, in Finsler geometry, ECIJT yields multiplicity results for closed geodesics:

• On compact simply-connected manifolds $(M, F)$ with $H^*(M; \mathbf Q) \cong T_{d, n+1}(x)$, it is proved that there exist at least $\frac{dn(n+1)}{2}$ distinct non-hyperbolic closed geodesics with odd Morse indices, whenever certain non-degeneracy holds (Duan et al., 2015).
• On star-shaped hypersurfaces in $\mathbb R^{2n}$, the generalized CIJT ensures at least $n$ geometrically distinct closed characteristics when every prime closed characteristic has nonzero mean index (Duan et al., 2022).

These results are sharp in many cases, matching the topological predictions by Katok and Anosov.

5. Generalizations and Contemporary Extensions

Recent work [CGG24], [GGM18], [GG20] generalizes CIJT in several directions:

• The index recurrence theorem applies CIJT machinery to more general symplectic paths, including those with mixed sign mean indices.
• Formulas and assertions accommodate higher-dimensional and more degenerate spectral configurations, extending the applicability to non-hyperbolic orbits and complex spectral flows.
• The explicit identification of correction invariants (splitting numbers, spectral sums) is formalized and shown to coincide across formulations.

A substantial implication is the ability to treat cases where closed characteristics or Reeb orbits possess negative or zero mean index, as in the treatment of closed characteristics on star-shaped hypersurfaces in $\mathbb R^{2n}$ (Duan et al., 2022).

6. Significance and Relevance for Symplectic Geometry

The CIJT, ECIJT, and their equivalent index recurrence versions have become essential frameworks for analyzing periodic orbit growth, bifurcation, and stability in modern symplectic geometry and Hamiltonian dynamics. They establish the principle that the discrete invariants (indices and nullities) of iterated objects in symplectic path spaces exhibit highly controlled and synchronized jump phenomena, which translate directly into multiplicity and rigidity results for periodic solutions.

Their precise, spectral-based technical structure allows for applications in areas ranging from algebraic geometry—via the paper of multiplier ideals and jumping numbers (Schwede, 2010, Graf, 2014)—to variational calculus and the computation of invariants in infinite-dimensional Morse theory. The recent synthesis and identification of their equivalence across different mathematical regimes fortifies their role as a cornerstone in the field.

7. Illustrative Summary Table

Theorem/Version	Core Formulas for Index Jumps	Applications
CIJT (Long–Zhu, 2002)	$i(\gamma_k, 2m_k + m) = 2N + i(\gamma_k, m)$	Closed geodesic multiplicity
ECIJT (Duan–Long–Wang, 2016)	Plus full correction terms $S_{M_k}^+(1), Q_k(m)$	Finsler manifolds, Reeb flows
Index Recurrence ([CGG24])	$(IR2), (IR3)$: Recursion for $\mu_\pm$, with correction invariants	Broader classes, star-shaped hypersurfaces

The equivalence and generalization of these formulations ensure that the theory of index jumps remains robust and applicable across diverse analytic and geometric settings.