Realizing arbitrary $d$-dimensional dynamics by renormalization of $C^d$-perturbations of identity

<p style='text-indent:20px;'>Any <inline-formula><tex-math id="M3">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> conservative map <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> of the <inline-formula><tex-math id="M5">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional unit ball <inline-formula><tex-math id="M6">\begin{document}$ {\mathbb B}^d $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ d\geq 2 $\end{document}</tex-math></inline-formula>, can be realized by renormalized iteration of a <inline-formula><tex-math id="M8">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> perturbation of identity: there exists a conservative diffeomorphism of <inline-formula><tex-math id="M9">\begin{document}$ {\mathbb B}^d $\end{document}</tex-math></inline-formula>, arbitrarily close to identity in the <inline-formula><tex-math id="M10">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> topology, that has a periodic disc on which the return dynamics after a <inline-formula><tex-math id="M11">\begin{document}$ C^d $\end{document}</tex-math></inline-formula> change of coordinates is exactly <inline-formula><tex-math id="M12">\begin{document}$ f $\end{document}</tex-math></inline-formula>.</p>

Otopy classification of gradient compact perturbations of identity in Hilbert space

We prove that the inclusion of the space of gradient local maps into the space of all local maps from Hilbert space to itself induces a bijection between the sets of the respective otopy classes of these maps, where by a local map we mean a compact perturbation of identity with a compact preimage of zero.

On the Second-Order Shape Derivative of the Kohn-Vogelius Objective Functional Using the Velocity Method

The exterior Bernoulli free boundary problem was studied via shape optimization technique. The problem was reformulated into the minimization of the so-called Kohn-Vogelius objective functional, where two state variables involved satisfy two boundary value problems, separately. The paper focused on solving the second-order shape derivative of the objective functional using the velocity method with nonautonomous velocity fields. This work confirms the classical results of Delfour and Zolésio in relating shape derivatives of functionals using velocity method and perturbation of identity technique.