Related papers: Constructions in elliptic dynamics
On the dual space of \textit{extended structure}, equations governing the collective motion of two mutually interacting Lie-Poisson systems are derived. By including a twisted 2-cocycle term, this novel construction is providing the most…
Let $M$ be a smooth projective variety and $\mathbf{D}$ an ample normal crossings divisor. From topological data associated to the pair $(M, \mathbf{D})$, we construct, under assumptions on Gromov-Witten invariants, a series of…
The reconstruction theorem deals with dynamical systems that are given by a map $T:X\to X$ of a compact metric space $X$ together with an observable $f:X \to \R$ from $X$ to the real line $\R$. In 1981, by use of Whitney's embedding…
The main result of this paper is that two large collections of ergodic measure preserving systems, the Odometer Based and the Circular Systems have the same global structure with respect to joinings. The classes are canonically isomorphic…
In \cite{Boed}, C.-F. B\"odigheimer constructed a finite cell-complex $\mf{Par}_{g,n,m}$ and a bijective map $\cH: \mf{Dip}_{g,n,m} \to \mf{Par}_{g,n,m}$ (the Hilbert-uniformization) from the moduli space of dipole functions on Riemann…
We introduce a construction adding low-dimensional cells to a space that satisfies certain low-dimensional conditions; it preserves high-dimensional homology with appropriate coefficients. This includes as special cases Quillen's plus…
In a conservative and partially hyperbolic three-dimensional setting, we study three representative classes of diffeomorphisms: those homotopic to Anosov (or Derived from Anosov diffeomorphisms), diffeomorphisms in neighborhoods of the…
The goal of this paper is to generalize the main results of [KM] and subsequent papers on metric Diophantine approximation with dependent quantities to the set-up of systems of linear forms. In particular, we establish `joint strong…
These lecture notes focus on a recent result of Mike Hochman: an arbitrary standard Borel system can be embedded into a mixing Markov with equal entropy, respecting all invariant probability measures, with two exceptions: those carried by…
The theory of differential forms began with a discovery of Poincare who found conservation laws of a new type for Hamiltonian systems - The Integral Invariants. Even in the absence of non-trivial integrals of motion, there exist invariant…
It is well known that the Lagrangian and Hamiltonian descriptions of field theories are equivalent at the discrete time level when variational integrators are used. Besides the symplectic Hamiltonian structure, many physical systems exhibit…
Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic…
We introduce topological contact dynamics of a smooth manifold carrying a cooriented contact structure, generalizing previous work in the case of a symplectic structure [MO07] or a contact form [BS12]. A topological contact isotopy is not…
We study dynamical properties of automorphisms of compact nilmanifolds and prove that every ergodic automorphism is exponentially mixing and exponentially mixing of higher orders. This allows to establish probabilistic limit theorems and…
The geodesics of Kerr's metric are described by the four-dimensional Hamiltonian dynamics integrable in the Arnold--Liouville sense. It can be reduced to two-dimensional one by the use of Fermat's principle. The resulting Hamiltonian is,…
Link spectral invariants were introduced by Cristofaro-Gardiner, Humili\`ere, Mak, Seyfaddini, and Smith. They induce Hofer-Lipschitz quasimorphisms on the group of Hamiltonian diffeomorphisms of the two-dimensional sphere. We prove that…
This paper is a survey devoted to the study of probability and infinite ergodic invariant measures for aperiodic homeomorphisms of a Cantor set. We focus mostly on the cases when a homeomorphism has either a unique ergodic invariant measure…
We report on recent results and a new line of research at the crossroad of two major theories in the analysis of partial differential equations. The celebrated De Giorgi-Nash-Moser theory shows H{\"o}lder estimates and the Harnack…
All current formulations of thermodynamics invoke some form of coarse-graining or ensembles as the supposed link between their own laws and the microscopic laws of motion. They deal only with ensemble-averages, expectation values,…
This is a first investigation by the author of the similarity between Quillen's superconnection formalism, his constructions of (periodic) cyclic cocycles via algebra cochains on a bar construction, and Kasparov bimodules for KK-theory. In…