skip to main content

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

Any \begin{document}$C^d$\end{document} conservative map \begin{document}$f$\end{document} of the \begin{document}$d$\end{document}-dimensional unit ball \begin{document}${\mathbb B}^d$\end{document}, \begin{document}$d\geq 2$\end{document}, can be realized by renormalized iteration of a \begin{document}$C^d$\end{document} perturbation of identity: there exists a conservative diffeomorphism of \begin{document}${\mathbb B}^d$\end{document}, arbitrarily close to identity in the \begin{document}$C^d$\end{document} topology, that has a periodic disc on which the return dynamics after a \begin{document}$C^d$\end{document} change of coordinates is exactly \begin{document}$f$\end{document}.

more » « less
Award ID(s):
NSF-PAR ID:
10351433
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Discrete & Continuous Dynamical Systems
Volume:
42
Issue:
2
ISSN:
1078-0947
Page Range / eLocation ID:
597
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
1. Consider the linear transport equation in 1D under an external confining potential \begin{document}$\Phi$\end{document}:

For \begin{document}$\Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2$\end{document} (with \begin{document}$\varepsilon >0$\end{document} small), we prove phase mixing and quantitative decay estimates for \begin{document}${\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v$\end{document}, with an inverse polynomial decay rate \begin{document}$O({\langle} t{\rangle}^{-2})$\end{document}. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasovâ€“Poisson system in \begin{document}$1$\end{document}D under the external potential \begin{document}$\Phi$\end{document}.

more » « less
2. Let \begin{document}$f_c(z) = z^2+c$\end{document} for \begin{document}$c \in {\mathbb C}$\end{document}. We show there exists a uniform upper bound on the number of points in \begin{document}${\mathbb P}^1( {\mathbb C})$\end{document} that can be preperiodic for both \begin{document}$f_{c_1}$\end{document} and \begin{document}$f_{c_2}$\end{document}, for any pair \begin{document}$c_1\not = c_2$\end{document} in \begin{document}${\mathbb C}$\end{document}. The proof combines arithmetic ingredients with complex-analytic: we estimate an adelic energy pairing when the parameters lie in \begin{document}$\overline{\mathbb{Q}}$\end{document}, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proofs are effective, and we provide explicit constants for each of the results.

more » « less
3. This paper investigates the global existence of weak solutions for the incompressible \begin{document}$p$\end{document}-Navier-Stokes equations in \begin{document}$\mathbb{R}^d$\end{document} \begin{document}$(2\leq d\leq p)$\end{document}. The \begin{document}$p$\end{document}-Navier-Stokes equations are obtained by adding viscosity term to the \begin{document}$p$\end{document}-Euler equations. The diffusion added is represented by the \begin{document}$p$\end{document}-Laplacian of velocity and the \begin{document}$p$\end{document}-Euler equations are derived as the Euler-Lagrange equations for the action represented by the Benamou-Brenier characterization of Wasserstein-\begin{document}$p$\end{document} distances with constraint density to be characteristic functions.

more » « less
4. We study the convergence rate of a continuous-time simulated annealing process \begin{document}$(X_t; \, t \ge 0)$\end{document} for approximating the global optimum of a given function \begin{document}$f$\end{document}. We prove that the tail probability \begin{document}$\mathbb{P}(f(X_t) > \min f +\delta)$\end{document} decays polynomial in time with an appropriately chosen cooling schedule of temperature, and provide an explicit convergence rate through a non-asymptotic bound. Our argument applies recent development of the Eyring-Kramers law on functional inequalities for the Gibbs measure at low temperatures.

more » « less
5. In this paper, we propose a new class of operator factorization methods to discretize the integral fractional Laplacian \begin{document}$(- \Delta)^\frac{{ \alpha}}{{2}}$\end{document} for \begin{document}$\alpha \in (0, 2)$\end{document}. One main advantage is that our method can easily increase numerical accuracy by using high-degree Lagrange basis functions, but remain its scheme structure and computer implementation unchanged. Moreover, it results in a symmetric (multilevel) Toeplitz differentiation matrix, enabling efficient computation via the fast Fourier transforms. If constant or linear basis functions are used, our method has an accuracy of \begin{document}${\mathcal O}(h^2)$\end{document}, while \begin{document}${\mathcal O}(h^4)$\end{document} for quadratic basis functions with \begin{document}$h$\end{document} a small mesh size. This accuracy can be achieved for any \begin{document}$\alpha \in (0, 2)$\end{document} and can be further increased if higher-degree basis functions are chosen. Numerical experiments are provided to approximate the fractional Laplacian and solve the fractional Poisson problems. It shows that if the solution of fractional Poisson problem satisfies \begin{document}$u \in C^{m, l}(\bar{ \Omega})$\end{document} for \begin{document}$m \in {\mathbb N}$\end{document} and \begin{document}$0 < l < 1$\end{document}, our method has an accuracy of \begin{document}${\mathcal O}(h^{\min\{m+l, \, 2\}})$\end{document} for constant and linear basis functions, while \begin{document}${\mathcal O}(h^{\min\{m+l, \, 4\}})$\end{document} for quadratic basis functions. Additionally, our method can be readily applied to approximate the generalized fractional Laplacians with symmetric kernel function, and numerical study on the tempered fractional Poisson problem demonstrates its efficiency.

more » « less