The Bounded Euler Class and the Symplectic Rotation Number

Ghys established the relationship between the bounded Euler class in $H_b^2({\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_{+}(S^1);\mathbb{Z})$ and the Poincar'{e} rotation number, that is, he proved that the pullback of the bounded Euler class under a homomorphism $\phi \text{\hspace{0.17em}⁣}:\mathbb{Z}\to {\mathrm{H}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{o}}_{+}(S^1)$ coincides with the Poincar'{e} rotation number of $\phi (1)$. In this paper, we extend the above result to the symplectic group in some sense, and clarify the relationship between the bounded Euler class in $H_b^2(Sp(2n;\mathbb{R});\mathbb{Z})$ and the symplectic rotation number investigated by Barge and Ghys.