Mild mixing property for special flows under piecewise constant functions

We give a condition on a piecewise constant roof function and an irrational rotation by $\alpha$ on the circle to give rise to a special flow having the mild mixing property. Such flows will also satisfy Ratner's property. As a consequence we obtain a class of mildly mixing singular flows on the two-torus that arise from quasi-periodic Hamiltonians flows by velocity changes.Comment: Accepted for publication in Discrete Contin. Dyn. Sys

Generation of measures on the torus with good sequences of integers

Let $S= (s_1<s_2<\dots)$ be a strictly increasing sequence of positive integers and denote $\mathbf{e}(\beta)=\mathrm{e}^{2\pi i \beta}$. We say $S$ is good if for every real $\alpha$ the limit $\lim_N \frac1N\sum_{n\le N} \mathbf{e}(s_n\alpha)$ exists. By the Riesz representation theorem, a sequence $S$ is good iff for every real $\alpha$ the sequence $(s_n\alpha)$ possesses an asymptotic distribution modulo 1. Another characterization of a good sequence follows from the spectral theorem: the sequence $S$ is good iff in any probability measure preserving system $(X,\mathbf{m},T)$ the limit $\lim_N \frac1N\sum_{n\le N}f\left(T^{s_n}x\right)$ exists in $L^2$-norm for $f\in L^2(X)$. Of these three characterization of a good set, the one about limit measures is the most suitable for us, and we are interested in finding out what the limit measure $\mu_{S,\alpha}= \lim_N\frac1N\sum_{n\le N} \delta_{s_n\alpha}$ on the torus can be. In this first paper on the subject, we investigate the case of a single irrational $\alpha$. We show that if $S$ is a good set then for every irrational $\alpha$ the limit measure $\mu_{S,\alpha}$ must be a continuous Borel probability measure. Using random methods, we show that the limit measure $\mu_{S,\alpha}$ can be any measure which is absolutely continuous with respect to the Haar-Lebesgue probability measure on the torus. On the other hand, if $\nu$ is the uniform probability measure supported on the Cantor set, there are some irrational $\alpha$ so that for no good sequence $S$ can we have the limit measure $\mu_{S,\alpha}$ equal $\nu$. We leave open the question whether for any continuous Borel probability measure $\nu$ on the torus there is an irrational $\alpha$ and a good sequence $S$ so that $\mu_{S,\alpha}=\nu$.Comment: 44 page

Uniformity in the Wiener-Wintner theorem for nilsequences

We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.Comment: v3: 18 p., proof that the cube construction produces compact homogeneous spaces added, measurability issues in the proof of Theorem 1.5 addressed. We thank the anonymous referees for pointing out these gaps in v

Symbolic approach and induction in the Heisenberg group

We associate a homomorphism in the Heisenberg group to each hyperbolic unimodular automorphism of the free group on two generators. We show that the first return-time of some flows in "good" sections, are conjugate to niltranslations, which have the property of being self-induced.Comment: 18 page

Promoting ecological solutions for sustainable infrastructure

Linear infrastructure networks such as roads, railways, navigation and irrigation canals, and power lines have grown exponentially since the mid-20th century. Most of these networks built before the 1990s have a significant impact on the environment. While there is no doubt that humanity needs infrastructure to ensure safe, secure and sufficient access to food, water and energy, it is essential to prevent the loss of biodiversity and ecosystems which are also at the basis of the provision of such fundamental services. Those complex, interconnected issues cannot be tackled without research and innovation, both in the fields of biodiversity and of infrastructure.IENE (Infrastructure Ecology Network Europe) was set up in 1996 to meet this need. Its mission is to promote the exchange of knowledge, experience and best practice in safe and sustainable pan-European transport infrastructure. With a status of an association today, this independent network has more than 400 members consisting of researchers, engineers, decision makers and infrastructure operators. IENE functions as an international and interdisciplinary forum. It supports cross-border cooperation in research, mitigation, planning, design, construction and maintenance in the field of biodiversity and transport infrastructure.Every two years, IENE organises an international conference to present cutting-edge research, identify pressing issues and problems, discuss effective solutions and map out future activities in the field of transport ecology and infrastructure. We are very glad to present you in this special issue some of the best scientific outcomes of the IENE 2020 conference, hoping that it will contribute to further breakthroughs in science and uptake in policy-making and practices on the ground. We commend the organising team of the University of Evora, Portugal, for their excellent programming of the conference and for having gathered exceptional scientists on the topic of biodiversity and infrastructure. They managed to host a high-quality event, despite the many adjustments that had to be done because of the covid-19, including postponing the conference to January 2021 and holding it entirely online.The topic of IENE conference 2020 was “Linear Infrastructure Networks with Ecological Solutions” and the motto was “working together”. This means that every stakeholder has a role to play, and that biodiversity should be considered at all governance scales and during all phases of the set-up of infrastructure. The papers selected here are of particular interest to follow the path set forth in the conference’s final declaration, that is included in this issue

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