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Aihua Fan

Publications and source records attributed to Aihua Fan.

29 records · Page 2Linked to original sources
arXiv2018

On uniformity of $q$-multiplicative sequences

We show that any $q$-multiplicative sequence which is \emph{oscillating} of order $1$, i.e.\ does not correlate with linear phase functions $e^{2πi nα}$ ($α\in \mathbb{R})$, is Gowers uniform of all orders, and hence in particular does not correlate with polynomial phase functions $e^{2πi p(n)}$ ($p \in \mathbb{R}[x]$). Quantitatively, we show that any $q$-multiplicative sequence which is of Gelfond type of order 1 is automatically of Gelfond type of all orders. Consequently, any such $q$-multiplicative sequence is a good weight for ergodic theorems. We also obtain combinatorial corollaries concerning linear patterns in sets which are described in terms of sums of digits.

math.NT↗
arXiv2017

Topological Wiener-Wintner ergodic theorem with polynomial weights

For a totally uniquely ergodic dynamical system, we prove a topological Wiener-Wintner ergodic theorem with polynomial weights under the coincidence of the quasi discrete spectrums of the system in both senses of Abramov and of Hahn-Parry. The result applies to ergodic nilsystems. Fully oscillating sequences can then be constructed on nilmanifolds.

math.DS↗
arXiv2017

Weighted Birkhoff ergodic theorem with oscillating weights

We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than $\frac{1}{2}$ are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of strong Gelfond property is well controlled almost everywhere. We prove that for any $q$-multiplicative sequence, the Gelfond property implies the strong Gelfond property and that sequences realized by dynamical systems can be fully oscillating and have the Gelfond property.

math.DS↗
arXiv2016

Compact Open Spectral Sets In $\mathbb{Q}_p$

In this article, we prove that a compact open set in the field $\mathbb{Q}_p$ of $p$-adic numbers is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation, and also if and only if it is $p$-homogeneous which is easy to check. We also characterize spectral sets in $\mathbb{Z}/p^n \mathbb{Z}$ ($p\ge 2$ prime, $n\ge 1$ integer) by tiling property and also by homogeneity. Moreover, we construct a class of singular spectral measures in $\mathbb{Q}_p$, some of which are self-similar measures.

math.FA↗
arXiv2015

Fuglede's conjecture holds in $\mathbb{Q}_p$

Fuglede's conjecture in $\mathbb{Q}_p$ is proved. That is to say, a Borel set of positive and finite Haar measure in $\mathbb{Q}_p$ is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation.

math.CA↗
arXiv2015

Oscillating Sequences, Minimal Mean Attractability and Minimal Mean-Lyapunov-Stability

We define oscillating sequences which include the Möbius function in the number theory. We also define minimally mean attractable flows and minimally mean-L-stable flows. It is proved that all oscillating sequences are linearly disjoint from minimally mean attractable and minimally mean-L-stable flows. In particular, that is the case for the Möbius function. Several minimally mean attractable and minimally mean-L-stable flows are examined. These flows include the ones defined by all $p$-adic polynomials, all $p$-adic rational maps with good reduction, all automorphisms of $2$-torus with zero topological entropy, all diagonalized affine maps of $2$-torus with zero topological entropy, all Feigenbaum zero topological entropy flows, and all orientation-preserving circle homeomorphisms.

math.DS↗
arXiv2014

Almost everywhere convergence of ergodic series

We consider ergodic series of the form $\sum_{n=0}^\infty a_n f(T^n x)$ where $f$ is an integrable function with zero mean value with respect to a $T$-invariant measure $μ$. Under certain conditions on the dynamical system $T$, the invariant measure $μ$ and the function $f$, we prove that the series converges $μ$-almost everywhere if and only if $\sum_{n=0}^\infty |a_n|^2<\infty$, and that in this case the sum of the convergent series is exponentially integrable and satisfies a Khintchine type inequality. We also prove that the system $\{f\circ T^n\}$ is a Riesz system if and only if the spectral measure of $f$ is absolutely continuous with respect to the Lebesgue measure and the Radon-Nikodym derivative is bounded from above as well as from below by a constant. We check the conditions for Gibbs measures $μ$ relative to hyperbolic dynamics $T$ and for Hölder functions $f$. An application is given to the study of differentiability of the Weierstrass type functions $\sum_{n=0}^\infty a_n f(3^n x)$.

math.DS↗
arXiv2014

Some Aspects of Multifractal analysis

The aim of this survey is to present some aspects of multifractal analysis around the recently developed subject of multiple ergodic averages. Related topics include dimensions of measures, oriented walks, Riesz products etc.

math.DS↗
arXiv2013

On minimal decomposition of $p$-adic homographic dynamical systems

A homographic map in the field of $p$-adic numbers $\mathbb{Q}_p}$ is studied as a dynamical system on $\mathbb{P}^{1}(\mathbb{Q}_p)$, the projective line over $\mathbb{Q}_p$. If such a system admits one or two fixed points in $\mathbb{Q}_p$, then it is conjugate to an affine dynamics whose dynamical structure has been investigated by Fan and Fares. In this paper, we shall mainly solve the remaining case that the system admits no fixed point. We shall prove that this system can be decomposed into a finite number of minimal subsystems which are topologically conjugate to each other. All the minimal subsystems are exhibited and the unique invariant measure for each minimal subsystem is determined.

math.DS↗