New metrics for Anosov representations defined from Thurston's asymmetric metrics.
problem Defining metrics for Anosov representations.
method Generalizing Thurston's asymmetric metric to Anosov representations.
result Provides a (possibly asymmetric) Finsler distance in some cases.
Metric WPD for pseudo-Anosov maps shows many unbounded quasi-morphisms.
problem Understanding quasi-morphisms on pseudo-Anosov maps.
method Metric weak proper discontinuity (WPD) for pseudo-Anosov maps.
result Existence of many unbounded quasi-morphisms on homeomorphisms with large fixed sets.
New metrics connect surfaces with Anosov flows to those with negative curvature.
problem Creating metrics with Anosov flows on surfaces of positive curvature.
method Constructing metrics with Anosov geodesic flows and positive curvature regions, connecting to negative curvature metrics via smooth conformal deformations.
result Existence of a smooth curve of conformal deformations connecting Anosov metrics to metrics of negative curvature.
The paper connects geodesic flows, hyperbolic geodesics, and stable ergodicity.
problem Understanding conditions for geodesic flows to be Anosov and ergodic.
method Analyzing Finsler and Riemannian metrics on surfaces, using recent results.
result Geodesic flows on surfaces are C2 stably ergodic if and only if they are Anosov. Metrics are isometric for certain Anosov magnetic systems.
problem Isometry of metrics for Anosov magnetic systems.
method Conjugacy isotopic to the identity, volume-preserving conjugacy, cohomology class.
result Isometric metrics for conjugate Anosov magnetic systems.
Bi-contact surgery operations can be applied to Anosov flows.
problem Characterizing Anosov flows and their properties.
method Metric and contact geometric characterizations, Liouville geometry, Reeb dynamics.
result Bi-contact surgery operations can be applied to Anosov flows.
We provide obstructions to the existence of conformally Anosov Reeb flows on a 3-manifold that partially generalize similar obstructions to Anosov Reeb flows. In particular, we show S3 does not admit conformally Anosov Reeb flows. We also give a Riemannian geometric condition on a metric compatible with a c…
New separation concepts for Anosov representations help bound Thurston asymmetric metric.
problem Understanding diverging families of Anosov representations.
method Introducing separation concepts and analyzing combinatorial invariants.
result Critical exponent asymptotic to a graph invariant.
Two Anosov metrics with same boundary distance are isometric.
problem Boundary rigidity for surfaces of Anosov type.
method Transfer principle linking marked length spectrum rigidity to marked boundary distance rigidity.
result Two metrics of Anosov type with the same marked boundary distance are isometric.
The paper classifies contact 3-manifolds with critical metrics and connects entropy to optimization.
problem Classifying contact 3-manifolds with critical metrics and understanding their entropy.
method Critical metrics optimization and entropy analysis.
result Anosov contact metrics' optimization is linked to Reeb dynamics and entropy.
The paper defines Dirichlet domains for Anosov subgroups in Lie groups.
problem Finding finite-sided Dirichlet domains for Anosov subgroups in Lie groups.
method Introducing a sufficient condition for finite-sided Dirichlet domains in semisimple Lie groups for polyhedral Finsler metrics.
result Finitely generated subgroups of semisimple Lie groups can have finite-sided Dirichlet domains under certain conditions.
Anosov subgroups' deformations affect limit cones and growth indicators continuously.
problem Understanding continuous changes in Anosov subgroups' effects on limit cones and growth indicators.
method Continuous variation of limit cones and growth indicators under deformations of Anosov subgroups, with convexity assumptions.
result Limit cones and growth indicators vary continuously under deformations of Anosov subgroups.
We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit…
Anosov surfaces with same length spectrum are isometric.
problem Identifying metrics on surfaces based on their length spectrum.
method Combining microlocal tools with complex curve geometry.
result Metrics with the same length spectrum on Anosov surfaces are isometric.
Exponential proportion of pseudo-Anosovs in mapping class groups.
problem Proportion of non-pseudo-Anosov mapping classes in a ball of radius R.
method Using word metric and finite generating sets, we show that the proportion decreases exponentially.
result Proportion of non-pseudo-Anosov mapping classes decreases exponentially.
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 completely new stabilit…
Study shows exact dimensionality and regularity of manifolds for specific groups.
problem Exact dimensionality and regularity of manifolds for relatively Anosov groups.
method Dynamical methods, including finite and mixing of Bowen–Margulis–Sullivan measures.
result Manifolds are C1-regular and growth indicator is strictly concave. Paper constructs new non-Anosov Partially Hyperbolic Geodesic flows using conformal deformations.
problem Creating new non-Anosov Partially Hyperbolic Geodesic flows.
method Using conformal deformations to produce examples of partially hyperbolic geodesic flows.
result Proves ergodicity for the Liouville measure and uniqueness of the measure of maximal entropy.
Using the thermodynamics formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce a Out(Γ)-invariant Riemannian metric on the smooth …
Generalizes surgery techniques for projectively Anosov flows.
problem Creating new projectively Anosov flows from existing ones.
method Introducing a generalized Goodman surgery technique for projectively Anosov flows.
result Generates new examples of projectively Anosov flows on hyperbolic 3-manifolds.
In this note we formulate a condition for complete, connected and non-compact Riemannian manifolds which implies no conjugate points in case that the geodesic flow is Anosov with respect to the Sasaki metric.
Rigidity theorem for curved manifolds with boundary.
problem Proving that certain curved manifolds are uniquely determined by their lens data.
method Analyzing lens data and using properties of Anosov type metrics.
result Local lens rigidity holds for negatively-curved manifolds and metrics of Anosov type.
New insights into the geometry of flows on 3-manifolds.
problem Understanding the geometry of flows on 3-manifolds.
method Analyzing the action of pseudo-Anosov flows on Gromov-hyperbolic spaces.
result Genericity of non-periodic elements in the fundamental group.
Quasimorphisms and pseudo-Anosov flows
problem connections between quasimorphisms and pseudo-Anosov flows
method showing connections between quasimorphisms and pseudo-Anosov flows
result upper bounds on exponential growth rate of closed orbits in pseudo-Anosov flows
We consider a billiard in the sphere S^2 with circular obstacles, and give a sufficient condition for its flow to be uniformly hyperbolic. We show that the billiard flow in this case is approximated by an Anosov geodesic flow on a surface in the ambiant space S^3. As an application, we show that every orientable surfac…
Proves stability of geodesic flows on closed surfaces.
problem Stability of geodesic flows on closed surfaces.
method Generic Riemannian metrics and Reeb flows.
result Proves C2-stability conjecture for geodesic flows. The paper explores how magnetic systems' spectra can identify metrics and 1-forms.
problem Can the marked magnetic action spectrum of magnetic systems with Anosov flow determine the metric and 1-form?
method The paper addresses this question in two settings: locally for systems with close metrics and 1-forms, and for metrics in the same conformal class.
result The paper answers the question affirmatively in both settings.
We refine the recent local rigidity result for the marked length spectrum obtained by the first and third author in \cite{Guillarmou-Lefeuvre-18} and give an alternative proof using the geodesic stretch between two Anosov flows and some uniform estimate on the variance appearing in the central limit theorem for Anosov …
Word metrics from large balls in injective spaces are exponentially generic and growth tight.
problem Understanding genericity and growth in word metrics from injective spaces.
method Geometric arguments and large ball considerations.
result Exponential genericity and growth tightness for word metrics.
In the first part of this dissertation, we give a new definition of a Laplace operator for Finsler metric as an average, with regard to an angle measure, of the second directional derivatives. This operator is elliptic, symmetric with respect to the Holmes-Thompson volume, and coincides with the usual Laplace--Beltrami…
We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space o…
The paper studies pseudo-Anosov maps from typical Thurston constructions.
problem Estimating the entropy of pseudo-Anosov maps from Thurston's constructions.
method Developed a method to extract information about random walks associated with Thurston's construction.
result Random walks eventually become pseudo-Anosov under certain conditions.
Smooth orbit equivalence proves metric equivalence for geodesic flows.
problem Proving metric equivalence for geodesic flows under orbit equivalence.
method Proving metric equivalence for geodesic flows under orbit equivalence.
result Smooth orbit equivalence implies conformal equivalence of metrics.
The paper develops techniques to study dynamical systems with Carnot metrics.
problem Understanding smooth dynamical systems in the presence of Carnot metrics.
method Employing techniques from Margulis-Mostow, Métivier, Mitchell, and Pansu on tangent cones, the paper establishes resonances between Lyapunov exponents.
result Local rigidity properties of higher hyperbolic rank metrics and uniform lattice actions on quaternionic and octonionic symmetric spaces.
Study of centralizer elements preserving geodesic flow foliations on covers.
problem Rigidity properties of centralizer elements in geodesic flows.
method Definition and study of foliated centralizer, proving rigidity properties.
result Foliated centralizer is a finite-dimensional Lie group, discrete unless metric is homothetic to real hyperbolic.
This paper contains two main results. The first is the existence of an equivariant Weil-Petersson geodesic in Teichmueller space for any choice of pseudo-Anosov mapping class. As a consequence one obtains a classification of the elements of the mapping class group as Weil-Petersson isometries which is parallel to the T…
The paper solves the Steklov spectral inverse problem for conformal metrics.
problem Recovering a metric from its Steklov spectrum in dimension n≥3.
method Combines wave trace formula techniques with geodesic X-ray transform.
result Steklov isospectral metrics must coincide under real-analyticity assumption.
We introduce a new family of thermostat flows on the unit tangent bundle of an oriented Riemannian 2-manifold. Suitably reparametrised, these flows include the geodesic flow of metrics of negative Gauss curvature and the geodesic flow induced by the Hilbert metric on the quotient surface of divisible convex sets. We …
Study Anosov representations of reducible suspensions of hyperbolic groups.
problem Characterize dynamical properties of reducible suspensions of Anosov representations.
method Analyzing linear representations of non-elementary hyperbolic groups, focusing on weak unipotent actions on subspaces.
result Characterize when reducible suspensions are discrete and faithful, quasi-isometrically embedded, and Anosov.
Constructs flows on quotients of Lie groups for Anosov subgroups.
problem Understanding dynamics on quotients of Lie groups by Anosov subgroups.
method Utilizes geometric structures and Lie group theory to construct and analyze flows.
result Establishes that all refraction flows arise from this construction.
The paper proves a conjecture about the minimum number of closed geodesics on a Finsler 3-sphere.
problem Proving the Anosov conjecture for bumpy Finsler 3-spheres.
method Analyzing the Morse index of prime closed geodesics.
result Established the conjecture for Finsler 3-spheres with nonzero Morse index.
Constructs Anosov flows in hyperbolic 3-manifolds, disproving a conjecture.
problem Proving existence of infinitely many distinct Anosov flows in certain 4-manifolds.
method Using Cannon-Thurston maps and pseudo-Anosov quasigeodesic flows.
result Infinitely many distinct Anosov flows in some 4-manifolds.
Develops theory of Anosov representations for Fuchsian groups, showing stability and analytical properties.
problem Understanding geometrically finite Fuchsian groups and their representations.
method Theory of Anosov representations, type-preserving deformations, limit maps, relative Anosov and dominated representations.
result Cusped Hitchin representations are Borel Anosov, stable under deformations, and limit maps vary analytically.
The paper studies Hamiltonian flows for pseudo-Anosov mapping classes on surfaces.
problem Understanding the dynamics of pseudo-Anosov mapping classes on Teichmüller spaces.
method Explicit formulae for Hamiltonian flows generated by invariant functions.
result Hamiltonian flows coincide with the action of pseudo-Anosov homeomorphisms at time one.
This paper is devoted to higher dimensional Anosov flows and consists of two parts. In the first part, we investigate fiberwise Anosov flows on affine torus bundles which fiber over 3-dimensional Anosov flows. We provide a dichotomy result for such flows --- they are either suspensions of Anosov diffeomorphisms or the …
New method shows pseudo-Anosov flows on graph manifolds can be simplified.
problem Understanding pseudo-Anosov flows on graph manifolds.
method Constructing a partial Birkhoff section with genus one components that misses finitely many closed orbits.
result Every pseudo-Anosov flow on a graph manifold is almost equivalent to a totally periodic flow or a suspension Anosov flow.
Develops theory of relatively Anosov representations using flow examples.
problem Understanding relatively Anosov representations.
method Uses a contracting flow on a bundle to define Anosov representations and builds examples.
result Builds families of examples of relatively Anosov representations.
We study nilmanifolds admitting Anosov automorphisms by applying elementary properties of algebraic units in number fields to the associated Anosov Lie algebras. We identify obstructions to the existence of Anosov Lie algebras. The case of 13-dimensional Anosov Lie algebras is worked out as an illustration of the techn…