Our first main result states that the spectral norm \(\gamma \) on \( \mathrm {Ham}(M, \omega ) \), introduced in the works of Viterbo, Schwarz and Oh, is continuous with respect to the \(C^0\) topology, when M is symplectically aspherical. This statement was previously proven only in the case of closed surfaces. As a corollary, using a recent result of Kislev-Shelukhin, we obtain the \(C^0\) continuity of barcodes on aspherical symplectic manifolds, and furthermore define barcodes for Hamiltonian homeomorphisms. We also present several applications to Hofer geometry and dynamics of Hamiltonian homeomorphisms. Our second main result is related to the Arnold conjecture about fixed points of Hamiltonian diffeomorphisms. The recent example of a Hamiltonian homeomorphism on any closed symplectic manifold of dimension greater than 2 having only one fixed point shows that the conjecture does not admit a direct generalization to the \( C^0 \) setting. However, in this paper we demonstrate that a reformulation of the conjecture in terms of fixed points as well as spectral invariants still holds for Hamiltonian homeomorphisms on symplectically aspherical manifolds.

我们的第一个主要结果表明，在Viterbo，Schwarz和Oh的著作中引入的\（\ mathrm {Ham}（M，\ omega）\）上的谱范数\（\ gamma \）关于\\是连续的。 （C ^ 0 \）拓扑，其中M是非球面的。以前仅在封闭表面的情况下证明了这一说法。作为推论，使用Kislev-Shelukhin的最新结果，我们获得\（C ^ 0 \）非球面辛流形上条形码的连续性，并进一步定义了汉密尔顿同胚性的条形码。我们还介绍了对哈弗几何和哈密顿同胚动力学的几种应用。我们的第二个主要结果与关于哈密顿微分定理的不动点的Arnold猜想有关。在尺寸大于2且仅具有一个固定点的任何闭合辛流形上的哈密顿同胚性的最新示例表明，该猜想不能直接推广到\（C ^ 0 \）设置。但是，在本文中，我们证明了在固定非球面流形上的哈密顿同胚性，在不动点和谱不变性方面的猜想重构仍然成立。