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We show that the linear drift of the Brownian motion on the universal cover of a closed connected smooth Riemannian manifold is

In all dimensions, we prove that the marked length spectrum of a Riemannian manifold (M, g) with Anosov geodesic flow and non-positive curvature locally determines the metric in the sense that two close enough metrics with the same marked length spectrum are isometric. In addition, we provide a new stability estimate quantifying how the marked length spectrum controls the distance between the isometry classes of metrics. In dimension 2 we obtain similar results for general metrics with Anosov geodesic flows. We also solve locally a rigidity conjecture of Croke relating volume and marked length spectrum for the same category of metrics. Finally, by a compactness argument, we show that the set of negatively curved metrics (up to isometry) with the same marked length spectrum and with curvature in a bounded set of C ∞ is finite.

The paper surveys open problems and questions related to geodesics defined by Riemannian, Finsler, semi-Riemannian and magnetic structures on manifolds. It is an extended report on problem sessions held during the International Workshop on Geodesics in August 2010 at the Chern Institute of Mathematics in Tianjin.

In this paper, we consider the geodesic tree in exponential last passage percolation. We show that for a large class of initial conditions around the origin, the line-to-point geodesic that terminates in a cylinder located around the point (N, N), and whose width and length are o(N 2/3) and o(N), respectively, agrees in the cylinder, with the stationary geodesic sharing the same end-point. In the case of the point-to-point model where the geodesic starts from the origin, we consider width δN 2/3, length up to δ 3/2 N/(log(δ −1))³, and provide lower and upper bounds for the probability that the geodesics agree in that cylinder.

We develop cohomological interpretations for several types of automorphic forms for Hecke triangle groups of infinite covolume. We then use these interpretations to establish explicit isomorphisms between spaces of automorphic forms, cohomology spaces and spaces of eigenfunctions of transfer operators. These results show a deep relation between spectral entities of Hecke surfaces of infinite volume and the dynamics of their geodesic flows.

In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact Anosov flows. The ultimate aim is to establish exponential decay of correlations for Hölder observables with respect to a very general class of Gibbs measures. The approach invented in 1997 by Dolgopyat in “On decay of correlations in Anosov flows” and further developed in Stoyanov (2011) is substantially refined here, allowing to deal with much more general situations than before, although we still restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which produces the desired estimates whenever the Gibbs measure admits a Pesin set with exponentially small tails, that is a Pesin set whose preimages along the flow have measures decaying exponentially fast. We call such Gibbs measures regular. Recent results in Gouëzel and Stoyanov (2019) prove existence of such Pesin sets for hyperbolic diffeomorphisms and flows for a large variety of Gibbs measures determined by Hölder continuous potentials. The strong spectral estimates for Ruelle operators and well-established techniques lead to exponential decay of correlations for Hölder continuous observables, as well as to some other consequences such as: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error.

Existing Source-Free Domain Adaptation (SFDA) methods typically adopt the feature distribution alignment paradigm via mining auxiliary information (eg., pseudo-labelling, source domain data generation). However, they are largely limited due to that the auxiliary information is usually error-prone whilst lacking effective error-mitigation mechanisms. To overcome this fundamental limitation, in this paper we propose a novel Target Prediction Distribution Searching (TPDS) paradigm. Theoretically, we prove that in case of sufficient small distribution shift, the domain transfer error could be well bounded. To satisfy this condition, we introduce a flow of proxy distributions that facilitates the bridging of typically large distribution shift from the source domain to the target domain. This results in a progressive searching on the geodesic path where adjacent proxy distributions are regularized to have small shift so that the overall errors can be minimized. To account for the sequential correlation between proxy distributions, we develop a new pairwise alignment with category consistency algorithm for minimizing the adaptation errors. Specifically, a manifold geometry guided cross-distribution neighbour search is designed to detect the data pairs supporting the Wasserstein distance based shift measurement. Mutual information maximization is then adopted over these pairs for shift regularization. Extensive experiments on five challenging SFDA benchmarks show that our TPDS achieves new state-of-the-art performance. The code and datasets are available at https://github.com/tntek/TPDS.

Given a homogeneous pseudo-Riemannian space (G/H,⟨,⟩), a geodesic γ:I→G/H is said to be two-step homogeneous if it admits a parametrization t=ϕ(s) (s affine parameter) and vectors X, Y in the Lie algebra g, such that γ(t)=exp(tX)exp(tY)·o, for all t∈ϕ(I). As such, two-step homogeneous geodesics are a natural generalization of homogeneous geodesics (i.e., geodesics which are orbits of a one-parameter group of isometries). We obtain characterizations of two-step homogeneous geodesics, both for reductive homogeneous spaces and in the general case, and undertake the study of two-step g.o. spaces, that is, homogeneous pseudo-Riemannian manifolds all of whose geodesics are two-step homogeneous. We also completely determine the left-invariant metrics ⟨,⟩ on the unimodular Lie group SL(2,R) such that (SL(2,R),⟨,⟩) is a two-step g.o. space.