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For strictly ergodic systems, we introduce the class of CF-Nil($k$) systems: systems for which the maximal measurable and maximal topological $k$-step pronilfactors coincide as measure-preserving systems. Weiss theorem implies that such systems are abundant in a precise sense. We show that the CF-Nil($k$) systems are precisely the class of minimal systems for which the $k$-step nilsequence version of the Wiener-Wintner average converges everywhere. As part of the proof we establish that pronilsystems are $coalescent$. In addition, we characterize a CF-Nil($k$) system in terms of its $(k+1)$-$th dynamical cubespace$. In particular, for $k=1$, this provides for strictly ergodic systems a new condition equivalent to the property that every measurable eigenfunction has a continuous version.
We prove that every $mathbb{Z}^{k}$-action $(X,mathbb{Z}^{k},T)$ of mean dimension less than $D/2$ admitting a factor $(Y,mathbb{Z}^{k},S)$ of Rokhlin dimension not greater than $L$ embeds in $(([0,1]^{(L+1)D})^{mathbb{Z}^{k}}times Y,sigmatimes S)$,
Given a compact topological dynamical system (X, f) with positive entropy and upper semi-continuous entropy map, and any closed invariant subset $Y subset X$ with positive entropy, we show that there exists a continuous roof function such that the se
We study a mechanical system that was considered by Boltzmann in 1868 in the context of the derivation of the canonical and microcanonical ensembles. This system was introduced as an example of ergodic dynamics, which was central to Boltzmanns deriva
We prove unique continuation principles for solutions of evolution Schrodinger equations with time dependent potentials. These correspond to uncertainly principles of Paley-Wiener type for the Fourier transform. Our results extends to a large class of semi-linear Schrodinger equation.
The classical Wiener-Khinchin theorem (WKT), which can extract spectral information by classical interferometers through Fourier transform, is a fundamental theorem used in many disciplines. However, there is still need for a quantum version of WKT,