We consider the spherical mean operator on a complete Riemannian manifold with non-positive sectional curvature.
In this short paper we give a geometric proof for the fact that the operator regularized functions. Related results were previously obtained in a more general setting by Sunada and Tsujishita by other arguments.
We study a type of "development" mapping, which maps n-tuples of vectors at a point to the endpoint of a piecewise geodesic curve. We show that this mapping is a fold map, and deduce consequences for the iterated spherical mean operator and the associated geodesic random walk from this fact.
We think it should be possible to use this same approach to give lower bounds for the density of the n-th step of a geodesic random walk. And, therefore, obtain mixing rates for the geodesic random walk when the manifold is compact.
The article has been submitted to JMSJ.
On the speed of distance stationary sequences (2019, with Matías Carrasco and Elliot Paquette)This is an improved exposition of the "Furstenberg formula for speed" we obtained in a previous paper (which we no longer intend to publish) as well as a few applications.
One application in particular, showing dimension drop for the simple random walk on certain co-compact Fuchsian group, was recently re-obtained by Petr Kosenko as part of his Phd work under Giulio Tiozzo.
We also, give applications to the speed of random walks on weighted unimodular random trees (here the result is a slight generalization of an unpublished theorem which is in course notes by Nicolas Curien).
Also, the speed on percolation clusters of certain Cayley graphs can be estimated by this method.
We hope to use this paper in our (unpublished) work on Hyperbolic Poisson-Delaunay random graphs. It also seems possible that the method might be useful in the study of other well known random graph models.
The article has been submitted to ALEA.
On the discretness of states accessible via right-angled paths in hyperbolic space (2019, with Ernesto García)We answer the question of where a walker can get to in hyperbolic space if he is only allowed to advance a fixed distance and turn a right angle.
In the hyperbolic plane this is a discussion of the outcome of the Gilman-Maskit algorithm for a specific one parameter family of two-generator groups (but we seem to have found a mistake in the hyperbolic-hyperbolic case of the algorithm which we discuss in the paper).
In hyperbolic space, a result of Kapovich shows that there is no real number algorithm to determine discreteness. However, in our special case we obtain a complete classification in terms of the distance one is allowed to advance.
The article has been submitted to l'Enseignement mathematique.
Entropy and dimension of disintegrations of stationary measures (2019)An old result of Ledrappier, recently improved by Hochman and Solomyak, expresses the dimension of the stationary measure for products of i.i.d. 2x2 random matrices in terms of the Lyapunov exponents and a certain entropy.
Here we prove an analagous result for the disintegration of a stationary measure for larger matrices onto the foliation of the space of flags whose leaves are the sets of flags with the same i-dimensional subspace.
This can be seen as a baby step towards calculating the dimension of the full measure, or as a nuance on the old results of Goldsheid, Margulis, Guivarc'h and Raugi on the simplicity of the Lyapunov spectrum.
The article has been submitted to Transactions of the AMS.
On the existence of a common zero for two commuting vector fields on a surface (2018, Not for publication)An old theorem of Elon Lages Lima states that two commuting vector fields on a closed surface with non-zero Euler characteristic must have a common zero.
In this note I give what I consider to be an elegant proof of this result.
The idea is that the two commuting vector fields define on the connected components of the set where they are linearly independent a unique flat Riemannian metric for which the two fields are orthonormal. Hence on may foliate each such component by lines of constant slope.
Under the assumption of no common fixed points a minor argument is needed to pick a slope for which these foliations extend continuously to the entire surface (thus showing that the surface has zero Euler characteristic).
I was hoping to do a bit more with the idea of these "natural riemannian metrics". But a couple of years have past and I haven't managed to explore the idea further so I've decided to make this available here.
A few ideas are the following: Two linearly independent commuting vector fields on a 3-manifold define a foliation by flat leaves. It follows immediately that there are no vanishing cycles from which one re-obtains another theorem of Lima (from his 1965 Annals of Mathematics paper "Commuting vector fields on S^3"). The classification of Rosenberg, Roussarie, and Weil of 3-manifolds with rank 2 is also in this circle of ideas. Also, imposing the condition that the commutator of the two fields should be a fixed linear combination of the two one includes actions of the affine group (the natural metric on each orbit has constant negative curvature in this case) which were classified by Ghys under the assumtion of preserving a measure. Finally, relaxing the commutator condition slightly (so that the fields no longer define a group action) one might still be able to carry out some analysis.
A Furstenberg type formula for the speed of distance stationary sequences (2017, Not for publication, with Matías Carrasco and Elliot Paquette)We prove a formula for the speed of a distance stationary random sequence of points in a metric space. This is an incremental improvement over a previous chain of results starting with Furstenberg, Kaimanovich, Karlsson and Margulis, Karlsson and Ledrappier, and Karlsson and Gouezel.
We apply our result to study the simple random walk on a tiling associated to a random discrete set of points in hyperbolic space.
We are also able to estimate the dimension of the measure at infinity associated to the walk described above as well as the simple random walk on certain co-compact Fuchsian groups.
The article spent almost two years undergoing the peer review process and was rejected.
We've decided to split it in two parts, the current version will not be published.
The Teichmüller space of the Hirsch foliation (2015, with Sébastien Álvarez)We calculate the space of hyperbolic metrics on a particular minimal foliation by hyperbolic surfaces up to leaf-preserving isotopies. In particular we show that it is infinite dimensional (previously Deroin had established that foliations by hyperbolic surfaces containing a disk as a leaf have this property, however the foliation we study has no simply connected leaves).
The work is split essentially into two parts: constructing model metrics, and deforming any given metric via isotopy to a model metric. For the first part we rely on standard Teichmüller theory to construct certain sections of the Teichmüller space of the pair of pants. For the second we use several "hands on" deformations as well as the curve shortenning flow.
As a consequence we obtain the that identity component in the space of leaf-preserving diffeomorphisms is contractible.
The article was accepted for publication by Annales de l'Institut Fourier.
Equivalence of zero entropy and the Liouville property for stationary random graphs (2015, with Matías Carrasco)In 2012 Benjamini and Curien introduced the notion of a stationary random graph, which is a random rooted graph whose distribution is invariant under re-rooting by a simple random walk. In this context With Matías Carrasco we prove equivalence of zero entropy and the Liouville property for stationary random graphs answering a question in Benjamini and Curien's paper.
We also show that positive entropy is equivalent to the space of bounded harmonic functions on the random graph being almost surely infinite dimensional and positive speed for the simple random walk.
A nice example we apply our result to is the Delaunay graph associated to a homogenous Poisson point process on the hyperbolic plane. We obtain that the space of bounded harmonic functions on this random graph is almost surely infinite dimensional. This result was recently obtained by Benjamini, Paquette and Pfeffer.
The article was published in the Electronic Journal of probability.
Brownian motion on stationary random manifolds (2014)I introduce the notion of a stationary random manifold. Basically this is a random manifold with basepoint whose distribution is invariant under changing the basepoint by using a Brownian motion ("stationary random graphs" have been introduced a few years earlier by Benjamini and Curien). A typical example is to take the universal cover of a compact Riemannian manifold and put in a random basepoint whose distribution is uniform on a fundamental domain of the projection. Another intersting example is to consider the leaf of a random point in a foliation whose distribution is harmonic in the sense of Lucy Garnett.
I develop an entropy theory analogous to the well known Avez entropy theory which exists for Cayley graphs. The basic plan of what the definition of what the definition of entropy should be and what theorems should hold was layed out by Kaimanovich in two short publications from 1986 and 1988 (the first treating the case of the covering space of a compact manifold and the other treating the case of a random leaf in a foliation). I provide complete proofs of the expected results under suitable hypothesis in the general context of stationary random manifolds and also obtain some other properties of these random spaces (e.g. recurrence properties related to a theorem of Ghys for foliations by surfaces).
The article was published in Stochastics and Dynamics.
Reeb stability and the Gromov-Hausdorff limits of leaves in compact foliations (2013)A basic question in the geometry of foliations is to understand how the leaf varies when one moves a point slightly. To formalize this I study the regularity of the function which associates to each point in a compact foliation its leaf considered as a pointed proper metric space when the topology on the codomain is the so-called pointed Gromov-Hausdorff topology.
The simple example of a flow with an attracting orbit shows that the leaf function isn't continuous in general. However, I show that it is semicontinuous in the sense that all limit points of a sequence of leaves are covering spaces of the limiting leaf. An upper bound on these covering spaces is exhibited (the so-called holonomy-cover of the leaf). And it turns out that the classical stability theorems for simply connected leaves due to Reeb follows as a corollary. Other Reeb-stability-type results are also consequences of the semicontinuity of the leaf function.
The article was published in the Asian Journal of Mathematics.
Two fixed points can force positive entropy of a homeomorphism on a hyperbolic surface (2012, Not for publication)Using the Nielsen-Thurston classification of isotopy classes of homeomorphisms on surfaces I prove that a particular pattern of two fixed points can force positive entropy of a homeomorphism which is isotopic to the identity on a hyperbolic surface. I put special effort into the illustrations, which where generated using Asymptote.
The result turned out to be a consequence of previous work by Kra ("On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces." (1981)) and Imayoshi, Ito, and Yamamoto ("On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces with two specified points." (2003)).
Rotation vectors for homeomorphisms of non-positively curved manifolds (Published in 2011)I generalized rotation vectors as usually defined for torus homeomorphisms to other manifolds and proved they exist for many orbits. The techniques came from the works I studied for my masters thesis.
The article has been published in Nonlinearity.