Inconstancy of finite and infinite sequences

In order to study large variations or fluctuations of finite or infinite sequences (time series), we bring to light an 1868 paper of Crofton and the (Cauchy-)Crofton theorem. After surveying occurrences of this result in the literature, we introduce the inconstancy of a sequence and we show why it seems more pertinent than other criteria for measuring its variational complexity. We also compute the inconstancy of classical binary sequences including some automatic sequences and Sturmian sequences.Comment: Accepted by Theoretical Computer Scienc

Hausdorff dimension, projections, and the Fourier transform

This is a survey on transformation of fractal type sets and measures under orthogonal projections and more general mappings

Invisibility via reflecting coating

We construct a subset $A$ of the unit disc with the following properties. (i) The set $A$ is the finite union of disjoint line segments. (ii) The shadow of $A$ is arbitrarily close to the shadow of the unit disc in "most" directions. (iii) If the line segments are considered to be mirrors reflecting light according to the classical law of specular reflection then most light rays hitting the set emerge on the other side of the disc moving along a parallel line and shifted by an arbitrarily small amount. We also construct a set which reflects almost all light rays coming from one direction to another direction but its shadow is arbitrarily small in other directions, except for an arbitrarily small family of directions

An uncountable union of line segments with null two-dimensional measure

In this paper we construct an uncountable union of line segments $T$ which has full intersection with the sets $(\{ 0\} \times [0, 1]) \cup (\{ 1\} \times [0, 1])\subset\mathbb{R}^2$ but has null two-dimensional measure. Further results are proved on the decay rate of $\mu (T)$ if the line segments comprising $T$ are replaced with increasingly fine approximations by parallelograms

Optimal transport and dynamics of expanding circle maps acting on measures

Using optimal transport we study some dynamical properties of expanding circle maps acting on measures by push-forward. Using the definition of the tangent space to the space of measures introduced by Gigli, their derivative at the unique absolutely continuous invariant measure is computed. In particular it is shown that 1 is an eigenvalue of infinite multiplicity, so that the invariant measure admits many deformations into nearly invariant ones. As a consequence, we obtain counter-examples to an infinitesimal version of Furstenberg's conjecture. We also prove that this action has positive metric mean dimension with respect to the Wasserstein metric.Comment: 35 pages; v4 includes a corrigendum (Lemma 4.2 statement and proofs are corrected without influence on the main results) and an addendum (application to an infinitesimal version of Furstenberg Conjecture, Theorem 1.7 and Corollary 1.8