相关论文: Dynamics of the scenery flow and geometry of measu…
We present an application of the recently developed ergodic theoretic machinery on scenery flows to a classical geometric measure theoretic problem in Euclidean spaces. We also review the enhancements to the theory required in our work. Our…
Conical density theorems are used in the geometric measure theory to derive geometric information from given metric information. The idea is to examine how a measure is distributed in small balls. Finding conditions that guarantee the…
We show that self-similar measures on $\mathbb{R}^d$ satisfying the weak separation condition are uniformly scaling. Our approach combines elementary ergodic theory with geometric analysis of the structure given by the weak separation…
Furstenberg, Katznelson and Weiss proved in the early 1980s that every measurable subset of the plane with positive density at infinity has the property that all sufficiently large real numbers are realised as the Euclidean distance between…
In the realm of Delone sets in locally compact, second countable, Hausdorff groups, we develop a dynamical systems approach in order to study the continuity behavior of measured quantities arising from point sets. A special focus is both on…
We continue our study of the dynamics of mappings with small topological degree on (projective) complex surfaces. Previously, under mild hypotheses, we have constructed an ergodic ``equilibrium'' measure for each such mapping. Here we study…
We expand the ergodic theory developed by Furstenberg and Hochman on dynamical systems that are obtained from magnifications of measures. We prove that any fractal distribution in the sense of Hochman is generated by a uniformly scaling…
In this article we prove that the Hausdorff dimension of geodesic directions that are recurrent and diverge on average coincides with the entropy at infinity of the geodesic flow for any complete, pinched negatively curved Riemannian…
We study how measures with finite lower density are distributed around $(n-m)$-planes in small balls in $\mathbb{R}^n$. We also discuss relations between conical upper density theorems and porosity. Our results may be applied to a large…
We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism $(x,y) \mapsto (mx \mod 1,ny \mod 1)$ that are Bernoulli measures for the natural Markov partition. We show that the statistics…
This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal…
This paper is partly an exposition, and partly an extension of our work [1] to the multiparameter case. We consider certain classes of parametrized dynamically defined measures. These are push-forwards, under the natural projection, of…
We prove the upper semicontinuity of the measure theoretic entropy for the geodesic flow on complete Riemannian manifolds without focal points and bounded sectional curvature. We then study the relationship between the escape of mass…
In a string of recent papers starting with (Transport in Porous Media, 125, 565 (2018)), a theory of immiscible two-phase flow in porous media based on Euler homogeneity of the total volumetric flow rate has been investigated. The…
We report a recent developement on the theory of upper conical densities. More precicely, we look at what can be said in this respect for other measures than just the Hausdorff measure. We illustrate the methods involved by proving a result…
We extend results of parametric geometry of numbers to a general diagonal flow on the space of lattices. Moreover, we compute the Hausdorff dimension of the set of trajectories with every given behavior, with respect to a nonstandard metric…
The inevitable noise in real measurements motivates the problem to continuously quantify the similarity between rigid objects such as periodic time series and proteins given by ordered points and considered up to isometry maintaining…
The conformal flow of metrics [2] has been used to successfully establish a special case of the Penrose inequality, which yields a lower bound for the total mass of a spacetime in terms of horizon area. Here we show how to adapt the…
The spectra of parallel flows (that is, flows governed by first-order differential operators parallel to one direction) are investigated, on both $L^2$ spaces and weighted-$L^2$ spaces. As a consequence, an example of a flow admitting a…
This paper is a review on recently found connection between geodesically equivalent metrics and integrable geodesic flows. Suppose two different metrics on one manifold have the same geodesics. We show that then the geodesic flows of these…