Related papers: On the Shroer-Sauer-Ott-Yorke predictability conje…

It is well known that given two probability measures $\mu$ and $\nu$ on $\mathbb{R}$ in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod…

We consider the geodesic flow for a rank one non-positive curvature closed manifold. We prove an asymptotic version of the Central Limit Theorem for families of measures constructed from regular closed geodesics converging to the…

We construct the entropic measure $\mathbb{P}^\beta$ on compact manifolds of any dimension. It is defined as the push forward of the Dirichlet process (another random probability measure, well-known to exist on spaces of any dimension)…

Let $f:M\to M$ be a $C^{1+\epsilon}$-map on a smooth Riemannian manifold $M$ and let $\Lambda\subset M$ be a compact $f$-invariant locally maximal set. In this paper we obtain several results concerning the distribution of the periodic…

Solomonoff's central result on induction is that the posterior of a universal semimeasure M converges rapidly and with probability 1 to the true sequence generating posterior mu, if the latter is computable. Hence, M is eligible as a…

In 1989 H. Tverberg proposed a quite general conjecture in Discrete geometry, which could be considered as the common basis for many results in Combinatorial geometry and at the same time as a discrete analogue of the common transversal…

We propose a dimension reduction framework for feature extraction and moment reconstruction in dynamical systems that operates on spaces of probability measures induced by observables of the system rather than directly in the original data…

For continuous maps on a compact manifold M, particularly for those that do not preserve the Lebesgue measure m, we define the observable invariant probability measures as a generalization of the physical measures. We prove that any…

This article is an exposition of recent results and methods on the prevalence of normal numbers in the support of self-similar measures on the line. We also provide an essentially self-contained proof of a recent Theorem that the Rajchman…

A conjecture of Erd\H{o}s states that for any infinite set $A \subseteq \mathbb R$, there exists $E \subseteq \mathbb R$ of positive Lebesgue measure that does not contain any nontrivial affine copy of $A$. The conjecture remains open for…

Let $A$ be a compact set in ${\mathbb R}^p$ of Hausdorff dimension $d$. For $s\in(0,d)$, the Riesz $s$-equilibrium measure $\mu^s$ is the unique Borel probability measure with support in $A$ that minimizes $$…

The definition of probabilities in eternally inflating universes requires a measure to regulate the infinite spacetime volume, and much of the current literature uses a global time cutoff for this purpose. Such measures have been found to…

In this paper we prove that for an ergodic hyperbolic measure $\omega$ of a $C^{1+\alpha}$ diffeomorphism $f$ on a Riemannian manifold $M$, there is an $\omega$-full measured set $\widetilde{\Lambda}$ such that for every invariant…

We obtain improved local well-posedness results for the Lorentzian timelike minimal surface equation. In dimension $d=3$, for a surface of arbitrary co-dimension, we show a gain of $1/3$ derivative regularity compared to a generic equation…

We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, equivalently, Schreier graphs of quasi-pmp actions of countable groups. For ergodic graphs, the theorem gives an…

Making accurate forecasts for a complex system is a challenge in various practical applications. The major difficulty in solving such a problem concerns nonlinear spatiotemporal dynamics with time-varying characteristics. Takens' delay…

Diffusion models have achieved state-of-the-art performance, demonstrating remarkable generalisation capabilities across diverse domains. However, the mechanisms underpinning these strong capabilities remain only partially understood. A…

In this work, we are concerned with existence and uniqueness of invariant measures for path-dependent random diffusions and their time discretizations. The random diffusion here means a diffusion process living in a random environment…

We suggest an algorithm for determining the proper delay time and the minimum embedding dimension for Takens' delay-time embedding procedure. This method resorts to the rate of change of the spatial distribution of points on a reconstructed…

In \cite{NaTa} Naor and Tao extended to the metric setting the $O(d \log d)$ bounds given by Stein and Str\"omberg for Lebesgue measure in $\mathbb{R}^d$, deriving these bounds first from a localization result, and second, from a random…