arXiv subjects

Stéphane Tchuiaga

Publications and source records attributed to Stéphane Tchuiaga.

9 recordsLinked to original sources

arXiv2026

Hofer-Like Geometry Revisited

We prove that the inclusion of the Hamiltonian group $\Ham(M,ω)$ into the identity component $G_ω(M)$ of the symplectic diffeomorphism group is a bi-Lipschitz embedding with respect to the Hofer norm and the Hofer-like norm, and we identify geometric conditions under which this embedding is isometric: settling a conjecture of Banyaga. This conjecture was proved by Buss and Leclercq; our proof provides explicit equivalence constants. We also detail and simplify Banyaga's original proof of the non-degeneracy of the Hofer-like norm. We then extend the analysis to all of $G_ω(M)$: for $ϕ$ with flux class $γ$, the Hofer-like norm is given by a semidirect-product formula, the infimum over the harmonic locus $\Harm(γ)$ plus a Hofer residue. This yields a geometric condition for the two norms to agree on the Hamiltonian group. In particular, this geometric condition holds on all closed surfaces of genus $g\ge 2$.

math.SG↗

arXiv2026

$C^0$-Analogue of Ismagilov's Theorem

We establish a topological analogue of Ismagilov's theorem concerning the first continuous cohomology of volume-preserving diffeomorphisms. For a closed oriented manifold $M$ with volume form $Ω$, we consider the group $\mathbb{G}^Ω(M)$ of homeomorphisms obtained as uniform limits of smooth volume-preserving isotopies. We show that its first continuous cohomology with values in the Banach space of zero-mean continuous functions is isomorphic to the first de Rham cohomology of $M$. The proof develops a theory of transport for volume-preserving isotopies, producing a topological volume flux homomorphism and its associated transport cocycle. Under the additional hypothesis that $(M,Ω)$ satisfies the local $C^0$-generation property, and assuming the Müller-Sikorav approximation theorem (known for $n\neq 4$), the isomorphism extends to the full identity component $\operatorname{Homeo}_0^Ω(M)$ of the group of volume-preserving homeomorphisms, and the topological flux conjecture follows in that setting.

math.GT↗

arXiv2025

Isometric Splitting of Metrics Without Conjugate Points on $Σ\times S^1$

We establish a global rigidity theorem for Riemannian metrics without conjugate points on three-manifolds of the form $M = Σ\times S^1$, where $Σ$ is a compact orientable surface of genus at least 2. The main result states that any such metric must be a Riemannian product, with universal cover isometric to $(\mathbb{H}^2, g_0) \times (\mathbb{R}, dt^2)$. This extends the classical Hopf conjecture from tori to this natural class of manifolds with non-abelian fundamental group containing a central $\mathbb{Z}$ factor. We provide two independent proofs: one utilizing the regularity of Busemann functions and stability theory of the Riccati equation along Killing flows, and another based on a detailed analysis of the curvature operator acting on Jacobi fields. We derive sharp geometric inequalities, analyze the deformation space of such metrics, and discuss several geometric and dynamical consequences, including marked length spectrum rigidity, constraints on topological entropy, and incompatibility with non-trivial Sasakian structures.

math.DG↗

arXiv2025

A Hofer-like Metric on the Space of Anosov Flows

This paper develops a family of Hofer-like metrics ($\dAnV{V}$) on the space of Anosov vector fields $\An(M)$, providing dynamically relevant distances based on the cost of deformation paths using $\Ck{k}$ or Sobolev $\SobolevHk{k}$ norms. We establish fundamental properties, including completeness for $V= C^r (r \ge 1)$ or $H^k (k > \dim(M)/2+1)$, and naturality under diffeomorphisms. We show the utility of these metrics by proving quantitative stability results: proximity in $\dAnV{V}$ implies controlled variation of essential dynamical invariants, including topological entropy, Lyapunov exponents, SRB measures, thermodynamic pressure, spectral gaps (mixing rates), and zeta functions. Sufficient regularity ensures local Lipschitz continuity and Fréchet differentiability, connecting the metric structure to linear response formulas, particularly for pressure, exponents, and the spectral gap. While Sobolev metrics yield locally flat geometry with straight line geodesics, the framework is broadly applicable. We explore implications for the moduli space of Anosov flows, including stability of invariants and the framework for local slice theorems. Furthermore, we introduce \emph{Topological Anosov Flows}, defined via simultaneous uniform flow convergence and $\dAn$ metric convergence of the generating fields. This new class aims to capture essential hyperbolic features in non-smooth settings. Overall, the proposed metrics offer several geometric perspectives for analyzing the stability, classification, and possible extensions of Anosov dynamics.

math.DS↗

arXiv2025

Metrizability and Dynamics of Weil Bundles

This paper bridges synthetic and classical differential geometry by investigating the metrizability and dynamics of Weil bundles. For a smooth, compact manifold $M$ and a Weil algebra $\mathbf{A}$, we prove that the manifold $M^\mathbf{A}$ of $\mathbf{A}$-points admits a canonical, complete, weighted metric $\mathfrak{d}_w$ that encodes both base-manifold geometry and infinitesimal deformations. Key results include: (1) Metrization: $\mathfrak{d}_w$ induces a complete metric topology on $M^\mathbf{A}$. (2) Path Lifting: Curves lift from $M$ to $M^\mathbf{A}$ while preserving topological invariants. (3) Dynamics: Fixed-point theorems for diffeomorphisms on $M^\mathbf{A}$ connected to stability analysis. (4) Topological Equivalence: $H^*(M^\mathbf{A}) \cong H^*(M)$ and $π_\ast(M^\mathbf{A}) \cong π_\ast(M)$.

math.DG↗

arXiv2025

Extending the flux homomorphism to volume-preserving homeomorphisms

This paper extends the flux homomorphism to volume-preserving homeomorphisms. A surprising $(C^0, δ)-$rigidity result where the extended flux groups coincide with the standard flux group is proved. The introduced tools, which also include a Poincaré duality with Fathi's mass flow and a norm on the group of volume-preserving homeomorphisms, indicate a potential for new flexibility in the behavior of homeomorphisms. This flexibility could have implications for rigidity results in symplectic/cosymplectic geometry, particularly those concerning Lefschetz manifolds: Any finite energy symplectic homeomorphism of $(T^2, ω)$ with trivial flux, is a finite energy Hamiltonian homeomorphism of $(T^2, ω)$. We discuss the cohomology groups $H^\ast(Homeo_0(M,Ω), \mathcal{C}(M, \mathbb{R}) )$ of $ Homeo_0(M,Ω)$ with coefficients in $ \mathcal{C}(M, \mathbb{R})$.

math.SG↗

arXiv2019

Hofer-Like Geometry and Flux Theory

This paper meticulously revisit and study the flux geometry of any compact oriented manifold $(M; W)$. We generalize several well-known factorization results, exhibit some orbital conditions for the study of flux geometry, give a proof of the discreteness of the flux group for volume-preserving diffeomorphisms, show that any smooth path in the kernel of the flux is a vanishing flux path, and show that the kernel of the flux for volume-preserving diffeomorphisms is $C^1$closed inside the group of all volume-preserving diffeomorphisms isotopic to the identity map:This recovers several results from symplectic geometry. The fix-points theory does not resist to the above machinery: We prove a general contractibility result with respect to the orbits of the fix-points for volume-preserving diffeomorphisms isotopic to the identity map via vanishing-flux paths, generalize and solve the Arnold conjecture using the Thurston fragmentation property. In the sequel, we use fix-points to: Characterize the flux geometry of certain $C^0$limits of sequences of vanishing-flux paths and volumepreserving diffeomorphisms. Beside this, a $C^0$criterion for the existence of at least one fix-point is given, and a weak version of the generalized $C^0$flux conjecture is solved. Finally, we construct a pseudo right-invariant metric on the group of all volume-preserving diffeomorphisms isotopic to the identity map, prove several comparison results suitable to the study of the Hofer-like geometry of the group $Ham(N; w)$, of all Hamiltonian diffeomorphisms of a closed symplectic manifold $(N; w)$, derive the equivalence between the Hofer and the Hofer-like metrics on $Ham(N; w)$, and exhibit a computational proof of the non-degeneracy of the Hofer-like energies: Here, an outcome is that the Calabi group controls the Hofer-like geometry of the group $Ham(N; w)$ of any closed symplectic manifold $(N; w)$.

math.SG↗

arXiv2016

On Symplectic Dynamics

This paper continues to carry out a foundational study of Banyaga topologies of a closed symplectic manifold [3]. Our intension in writing this paper is to provide several symplectic analogues of some results found in the study of Hamiltonian dynamics. Especially, without appealing to the positivity of the symplectic displacement energy, we point out the impact of the $L^\infty$ version of Banyaga Hofer-like metric in the investigation of the symplectic nature of the $C^0-$limit of a sequence of symplectic maps. This result is the symplectic analogue of a result that was proved in Hofer-Zehnder [8] (for compactly supported Hamiltonian diffeomorphisms on $\mathbb{R}^{2n}$), and then reformulated in Oh-Müller [10] for Hamiltonian diffeomorphisms in general. Furthermore, we extend to symplectic isotopies the regularization procedure for Hamiltonian paths introduced in Polterovich [11], and then we use it to prove the equality between the two versions of Banyaga Hofer-like norms defined on the identity component in the group of symplectomorphisms. This result was announced in [2]. It shows the uniqueness of Banyaga Hofer-like geometry, and then yields the symplectic analogue of a result that was proved in Polterovich [11]. Finally, we elaborate the symplectic analogues of some approximation results found in Oh-Müller [10], and make some remarks on flux theory.

math.SG↗

arXiv2016

An enlargement of some symplectic objects

The study of algebraic properties of groups of transformations of a manifold gives rise to an interplay between different areas of mathemathics such as topology, geometry, and dynamical systems. Especially, in this paper, we point out some interplays between topology, geometry, and dynamical systems which are underlying to the group of symplectic homeomorphisms. The latter situation can occur when one thinks of the following question. Is there a flux geometry which is underlying to the group of strong symplectic homeomorphisms so that Fathi's Poincare duality theorem continues to hold? We discuss on some possible answers of the above preoccupation, and we elaborate various topological analogues of some well-known results found in the field of symplectic dynamics. We leave several open questions and conjectures.

math.SG↗