Given a finite number of isometries of R^d you want to compose them so as to send the origin to infinity as quickly as possible (in terms of the number of isometries used).
This simple-looking problem gives rise to a very interesting case-study (in my opinion) in ergodic optimization. The "strategies" are codified by shift invariant ergodic measures, and the quantity you want to maximize makes the problem fit into the field of "subadditive ergodic optimization". Very early on you can tell that usually the best strategy does not come from repeating some finite combination of isometries over and over (in fact you can set things up so that doing that gives speed zero).
We show how to calculate the best strategy if the linear parts of the isometries generate a finite group. We also prove a number of general results. But the best part, in my opinion, of the paper is a very detailed study of a particular two parameter family of isometries in the plane. The best strategy is identified as "Sturmian" for some parameters but shown to be more complicated in another, both sets of parameters have positive measure. We show that the maximal speed (joint maximal drift) is discontinuous exactly at parameters for which the linear parts generate a finite group.
Some of the arguments are quite technical involving, for exmample, Diophantine conditions to guarantee a continuous solution to a cohomological equation, or rather sublte cancelation properties when one perturbs the linear part of all isometries by adding a small turn.
The article was submitted for publication, however we did not upload it to the arxiv. This decision was made "in mode of protest" against the indiscriminate use of mathematical work to train large language models. Jairo has strong opinions on this matter, some of which I share. In any case, if someone wants the full paper they need to write us.
On the distribution of the angle between Oseledets spaces (2025, with Jairo Bochi)This paper adresses a minor "bug" that had annoyed me in ergodic theory for some time. I first heard about it from François Ledrappier.
When you prove Oseledet's theorem for an i.i.d. random matrix product you show by a certain argument with "ad-hoc" feel that the angle between distinct Oseledet's subspaces do not go to zero exponentially fast along an orbit. This factoid turns out to be important in applications.
The factoid would follow immediately if logarithm of the angle between subspaces was an integrable function for the invariant measure under consideration. There was neither a proof of this nor a counter-example until we wrote this paper with Jairo.
In our paper we show that there are i.i.d. matrices for which the log-angle is not integrable. Around the same time as part of his thesis Axel Peneau proved that for measures with strongly irreducible support log-angle is integrable, so things are rather subtle.
For general, non i.i.d., stationary products we show with Jairo that you can pick the joint distribution of the Oseledets splitting more or less at well. This falls into a larger panorama of "flexibility" of Katok, Jairo, and others.
The paper was published in Annales Henri Lebesgue.
Quasi-isometric free group representations into SL3(R) (2025, with León Carvajales and Rafael Potrie)Rafael asked the very interesting question of whether a robustly quasi-isometric representation is necessarily Anosov. For SL_2(C) this was proved by Sullivan. In general it turns out to be false (in particular Konstantinos Tsouvalas has constructed several families of examples).
The counter-examples somehow have "very few" perturbations. So we decided to investigate if robustly quasi-isometric representations of the free group (where the space of perturbations is as large as possible in terms of the number of generators) are necessarily Anosov.
Even this turned out to be too much for us but we obtained some partial results for representations of free groups into SL_3(R).
I consider this result a "data point", I think there is a statement which includes Sullivan's results for SL_2(C) and should be of the form "if there is a sufficiently large space of deformations then robustly quasi-isometric representations are Anosov". I also believe that even a counter-example to this formulation would be quite interesting.
The paper was published in Annales Henri Lebesgue.
Dimension drop of harmonic measure for some finite range random walks on Fuchsian Schottky groups (2025, with Ernesto García)We proposed a generalization of the conjecture that the stationary measure of a finite range random walk on a Fuchsian group should be singular with respect to Lebesgue (the full conjecture has since been solved by Timothée Bénard, and for symmetric random walks by Giulio Tiozzo and Petr Kosenko).
Our conjecture is that for a Schottky group, the stationary measure of a finite range random walk should have dimension strictly less than the dimension of the limit set.
We prove the result under rather unsatisfying restrictive hypothesis.
The paper was published in Probability and Related Fields.
Dimension gap and variational principle for Anosov representations (2025, with François Ledrappier)We discovered that the limit set of representation into SL_d(R) which is Anosov with respect to at least two roots cannot have full dimension in a partial flag space coarser than the one corresponding to these two roots.
For example for a Zariski-dense Anosov subgroup of SL_3(R), the limit set in the space of full flags has dimension at most 2.5, whereas the flag manifold has dimension 3.
This result was very surprising to me (it's my fault that the paper includes a phrase to this effect which I think we both now regret :)). I think the first time I saw something similar was when we considered the limit sets of Hitchin representations in our first collaboration. We observed that stationary measures had at most dimension 1 on all flag manifold. It was later explained to me by Andrés Sambarino that the full limit set has dimension 1 because in each Grasmannian it is homeomorphic to a circle (the boundary of the group), and on larger partial flag manifolds you can see it as the graph of a homeomorphism of a circle.
In this paper things come full circle. We prove that the limit set does not have maximal dimension because the stationary measures cannot approach maximal dimension.
I like many things about this paper. We used the nice coordinates of our previous work to establish the "obvious" upper bound for the Hausdorff dimension of the limit set which comes from coverings by ellipses (the Falconer dimension). And also, we have to construct probability measures on the group with controlled Lyapunov exponents and entropy to prove our "variational principle".
On this last point our work overlapped with that of Jialun Li, Wenyu Pan, and Disheng Xu, who also were working on some version of the variational principle at the time. I discussed with Jialun at the "Beyond hyperbolicity" conference in Poland, and later we discussed over zoom with the whole group but at the time their version of the variational principle did not fit our needs so each group ended up with its own little variant. I'm not sure if their final version would work for our purposes.
The article was published in Journal de l'École Polytechnique.
Volume growth of horospheres in diagonalizable Heintze groups (2025, with Gilles Courtois and Emiliano Sequeira)Gilles has been interested in the geometry of horospheres of negatively curved manifolds for a while now. In this paper we consdider the horospheres of so-called "Heintze spaces" which are negatively curved homogeneous spaces which usually do not admit a compact quotient.
The spaces are constructed by taking the product of a nilpotent group (which is the "model horosphere") and R. There are two isometry classes of horospheres, we are interested in the geometry of the ones which are not modelled by the nilpotent group.
We failed to treat the general case, but in the case the model horospheres are flat under some conditions we show that the other horospheres have exact polynomial volume group of the form radius^k for some explicit k strictly larger than their topological dimension.
The article was submitted for publication.
Exact dimension of dynamical stationary measures (2024, with François Ledrappier)We removed the restriction on simplicity of the spectrum from our previous work on exact dimensionality of Furstenberg measure. The point is that Oseledet's theorem tells you what the finest flag manifold you should consder is, and we prove exact dimension on this and all coarser flag manifolds.
Also, the fiber bundles which we introduced in our previous work, so-called "configuration spaces", have much cleaner description and coordinates in this version. In fact they became vector bundles with explicit fiberwise affine action of the group.
This paper made me very happy :).
The article was published in Journal of Modern Dynamics.
Geodesic tracking and the shape of ergodic rotation sets (2023, with Alejo García-Sassi and Pierre-Antoine Guihéneuf)We are interested in the dynamics of homeomorphisms isotopic to the identity on a closed surface which have zero topological entropy. Here there is an amazing preprint of Handel from the 80s which I have been interested in for several years which gives a conjectural picture. Some of the results of the preprint (the fixed point theorem, results for anulii and sphere) have been proved by Handel, Le Calvez, Fabio Tal and others (my collegue Juliana Xavier for example gave an alternate proof of the fixed point theorem as part of her Phd thesis some while ago) mostly in the case of area-preserving maps as far as I know.
Each ergodic invariant-measure has an associated "drift" in the universal cover, and typical orbits of measures with positive drift turn around the surface following a geodesic lamination. We show that the laminations thus-obtained cannot cross each other. This gives a strong restriction on what the homological rotation set can look like.
We also analyze along the same vein the homological rotation set of positive entropy maps.
The main tool is the "forcing" theory of Le Calvez and Tal, on which Pierre-Antoine is an expert.
The article was submitted for publication.
Random walk speed is a proper function on Teichmüller space (2023, with Azemar, Gadre, Gouëzel, Haettel, and Uyanik)This was the outgrowth of an AIM conference, whose specific objective is to foster collaboration.
If you fix a finite support probability measure on a cocompact Fuchsian group each point in Teichmüller space gives a corresponding stationary measure on the boundary of the hyperbolic plane. We prove that the dimension of this measure goes to zero when you go to the boundary of Teichmüller space.
We did this by looking at the random walk speed associated to a limiting action of the group on a real tree. To show continuity we used some new at the time techniques of Gouezel called the "pivoting technique". Related results had been obtained earlier for holomorphic families of representations by Dujardin and Favre.
The article was published in Journal of Modern Dynamics.
Exact dimension of Furstenberg measure (2021, with François Ledrappier)We prove that the stationary measures on spaces of partial flags associated to a countably supported probability measure in SL_d(R) whose Lyapunov exponents are simple are exact dimensional.
This is an extension of the result of De-Jun Feng which applied to self-affine measures in R^d.
Our work overlapped with that of Ariel Rapaport who proved exact dimensionality and a corresponding Entropy-Exponent formula in the case of projective stationary measures.
The article was published in Geometric and Functional Analysis.
Fold maps associated to geodesic random walks on non-positively curved manifolds (2020, with Lucas Oliveira)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 was published by Hokkaido Journal of Mathematics.
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 was pusblished by 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 was published in 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 was published in 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.