New domains of discontinuity found for Anosov representations.
problem Understanding Anosov representations acting on homogeneous spaces.
method Constructing open domains of discontinuity for Anosov representations acting on specific homogeneous spaces.
result Describes the largest possible open domains of discontinuity for Zariski dense Anosov representations.
Anosov subgroup equidistributes geodesics and holonomies on homogeneous spaces.
problem Equidistribution of geodesics and holonomies in Anosov homogeneous spaces.
method Analyzes maximal flat cylinders and their holonomies for Anosov subgroups.
result Joint equidistribution of maximal flat cylinders and holonomies as circumference tends to infinity.
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.
The paper describes decompositions of geometric measures on Anosov homogeneous spaces.
problem Decomposing geometric measures on Anosov homogeneous spaces.
method Ergodic decompositions of Burger-Roblin and Bowen-Margulis-Sullivan measures.
result The space of non-trivial invariant ergodic measures is homeomorphic to a product space.
Classifies measures for Anosov subgroups in higher ranks.
problem Classifying horospherical invariant measures for Anosov subgroups.
method Geometric approach, not relying on flows or ergodic theorems.
result Extends results from rank one to higher ranks, solving open problems.
The paper studies ergodicity of flows on subspaces, generalizing earlier work.
problem Ergodicity of flows on subspaces of higher rank groups.
method Analyzes one-parameter diagonalizable subgroups of connected semisimple groups acting on homogeneous spaces.
result Obtains an ergodicity criterion similar to Hopf-Tsuji-Sullivan for general Anosov subgroups.
Study proves projective Anosov subgroups lead to mixing flows in specific spaces.
problem Understanding mixing properties of flows on specific geometric spaces.
method Constructing non-empty domain of discontinuity in homogeneous space, using spectral estimates for transfer operators.
result Exponential mixing, spectral gap, and meromorphic continuation of zeta functions established.
We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi, we derive applications to proper actions on homogeneous spaces of red…
The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher Teichmueller spaces, and for lattices in SO(1,n). In this article we extend the…
Study on horospheres in higher rank homogeneous spaces, proving density properties.
problem Density of horospheres in higher rank homogeneous spaces.
method Analyzing maximal horospherical subgroups and their minimal subsets in the context of Furstenberg boundary.
result Equivalence of horospherical limit points and density properties in higher rank homogeneous spaces.
We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous sp…
Sharpness of actions on reductive homogeneous spaces proven for various groups.
problem Proving proper and cocompact actions on reductive homogeneous spaces.
method Using quasi-isometric embedding and Anosov representations.
result Characterization and proof of non-compactness for certain homogeneous spaces.
Study shows mixing of flows on specific geometric spaces.
problem Mixing of one-parameter diagonal flows on Anosov homogeneous spaces.
method Proves local mixing for flows on $Γackslash G$ with deviations in transverse subspaces.
result Local mixing of flows on $Γackslash G$ for various directions.
Develops measures for non-Borel Anosov groups on Furstenberg boundary.
problem Measuring non-Borel Anosov groups on the Furstenberg boundary.
method Theory of Patterson--Sullivan measures, strict convexity, entropy rigidity.
result Existence, uniqueness, and ergodicity of measures on Furstenberg boundary.
We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the 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.
Proves transitivity of real Anosov diffeomorphisms with specific properties.
problem Transitivity of real Anosov diffeomorphisms with specific properties.
method Proves transitivity using specific properties of real Anosov diffeomorphisms.
result Proves transitivity of real Anosov diffeomorphisms.
Characterizes Anosov flows via contact geometry.
problem Understanding Anosov 3-flows through contact geometry.
method Investigates interactions with Reeb dynamics and proves a technical theorem.
result Space of adapted geometries homotopy equivalent to Anosov flows.
The paper counts conjugacy classes of pseudo-Anosov homeomorphisms in Teichmüller space.
problem Counting conjugacy classes of pseudo-Anosov homeomorphisms in Teichmüller space.
method Analyzes the asymptotic behavior of conjugacy classes as the radius of a ball in Teichmüller space increases.
result Asymptotics for the number of pseudo-Anosov homeomorphisms conjugate to a given homeomorphism within a ball of radius R centered at X.
Characterizes pseudo-Anosov orbit spaces via bifoliated planes
problem Characterizing actions on bifoliated planes arising from pseudo-Anosov flows
method Using branched covers, veering triangulations, and a compactness criterion
result Extends previous work on the special case with no odd-prong singularities
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.
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.
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.
Uniformizes compact complex manifolds via Anosov representations.
problem Uniformization of compact complex manifolds.
method Anosov homomorphisms with small limit sets.
result Local homeomorphism of character variety to Teichmüller space.
This paper classifies expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.
problem Classifying expanding attractors and non-transitive Anosov flows on specific knot and manifold spaces.
method Using the derived Anosov (DA) expanding attractor and the Franks-Williams manifold, the paper proves the uniqueness of these structures.
result The DA expanding attractor and the Franks-Williams non-transitive Anosov flow are the unique structures supported by N0 and M0 respectively. Anosov maps study with new Banach space and foliation method.
problem Understanding statistical properties of Anosov maps.
method Constructing a new Banach space and using a new foliation method.
result New Banach space provides insights into foliation absolute continuity.
New growth rate for pseudo-Anosov conjugacy classes in Teichmüller space.
problem Understanding growth rates of conjugacy classes in Teichmüller space.
method Analyzing pseudo-Anosov mapping classes and their conjugacy classes in Teichmüller space.
result The number of lattice points of pseudo-Anosov conjugacy classes intersecting a closed ball of radius R is coarsely asymptotic to \(e^{\frac{h}{2}R}\).
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.
In hyperbolic L-spaces, we find multiple pseudo-Anosov flows with unique properties.
problem Finding multiple pseudo-Anosov flows in hyperbolic L-spaces.
method Used cylindrical contact homology and reconstruction of pseudo-Anosov flows from their closed orbits.
result Proved the existence of n pseudo-Anosov flows in hyperbolic L-spaces, each with unique properties. The paper introduces cataclysm deformations for Anosov representations.
problem Deforming Anosov representations in Lie groups.
method Constructing cataclysm deformations for θ-Anosov representations into semisimple Lie groups. result Cataclysm deformations are injective for Hitchin representations but not for all θ-Anosov representations. Characterizes Anosov reducible representations in terms of eigenvalues.
problem Understanding Anosov representations in reducible settings.
method Characterizes Anosov representations using eigenvalue magnitudes of irreducible block factors.
result Connected components of character varieties do not contain reducible representations for many non-elementary hyperbolic groups.
In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any …
Study non-transitive pseudo-Anosov flows using group actions.
problem Characterize pseudo-Anosov flows in 3-manifolds.
method Extend pseudo-Anosov action to non-transitive flows, use group actions on orbit spaces and boundary at infinity.
result Pseudo-Anosov flows in 3-manifolds are determined by their group actions on boundary at infinity.
Theory of relatively Anosov representations using flow methods.
problem Developing a theory for relatively Anosov representations.
method Using the contracting flow on a bundle to define and study relatively Anosov representations.
result Definition and study of uniformly relatively Anosov representations and a stability result.
Anosov deformations created for surfaces with specific properties.
problem Creating proper slices in character varieties for specific surface groups.
method Constructing proper slices in character varieties for surface groups into SL(3,R).
result Representations are Borel Anosov.
Identifies Anosov representations of hyperbolic triangle groups in SL(3,R).
problem Classifying Anosov representations of hyperbolic triangle groups into SL(3,R).
method Proving representations are Anosov if they lie in the Hitchin component or the Barbot component, with specific conditions for eigenvalues.
result Anosov representations in SL(3,R) have non-convex boundary maps.
Anosov groups limit sets are Ahlfors regular, with applications in Teichmüller spaces.
problem Understanding Ahlfors regularity of limit sets for Anosov groups.
method Proving Ahlfors regularity for limit sets and Patterson-Sullivan measures.
result Patterson-Sullivan measures are Ahlfors regular if and only if associated linear forms are symmetric.
Study of laminations for pseudo-Anosov flows on three-manifolds.
problem Understanding laminations for pseudo-Anosov flows on three-manifolds.
method Analyzing laminations Λu± for pseudo-Anosov orbit space universal circles, using prelaminations and results from Barthelmé, Bonatti, and Mann. result Laminations Λu+ and Λu− are completely determined by prelaminations on the boundary of the orbit space. In this paper we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a differentiable submanifold of projective space with Holder continuous derivatives. We also calculate the optimal value of the Holder constant in terms of the eigenvalue data of the Anosov repre…
Legendrian arcs connect veering triangulations to Anosov flows.
problem Connecting veering triangulations to Anosov flows for study.
method Realizing edges as Legendrian arcs with a bicontact structure.
result Veering triangulations can be placed in steady position.
An outsider's overview of Anosov flows in 3D.
problem Understanding Anosov flows in 3D and their interactions.
method Overview and discussion of main results and questions.
result Highlighting the interaction with topology and symplectic geometry.
Fixed pseudo-Anosov homeomorphisms detect cinquefoil knot.
problem Detecting the cinquefoil knot using knot Floer homology.
method Using pseudo-Anosov homeomorphisms and Floer homology.
result The cinquefoil knot is the only genus-two L-space knot.
Study shows universal circle isomorphic to flow space ideal boundary.
problem Understanding foliations and pseudo-Anosov flows in 3-manifolds.
method Analyzing depth-one foliations and pseudo-Anosov flows in atoroidal 3-manifolds.
result Universal circle is isomorphic to flow space ideal boundary.
New insights into Anosov representations of hyperbolic groups.
problem Understanding Anosov representations of relatively hyperbolic groups.
method Proving representations can be interpreted as restricted Anosov representations over flow spaces and showing stability under deformations.
result Representations of certain types are divergent, extended geometrically finite and stable under small deformations.
We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any…
Simplified approach to pseudo-Anosov flows on 3-manifolds.
problem Complexity in understanding pseudo-Anosov flows on 3-manifolds.
method Streamlined framework called Anosov-like group actions.
result Unified and simplified presentation of pseudo-Anosov flows.
Biholomorphisms of transport twistor spaces are rigid under certain conditions.
problem Rigidity of biholomorphisms in transport twistor spaces.
method Proof of rigidity for biholomorphisms between transport twistor spaces of simple or Anosov surfaces.
result Biholomorphisms are rigid, up to constant rescaling and the antipodal map, being lifts of orientation-preserving isometries.
Classifies pseudo-Anosov flows on 3-manifolds up to orbit equivalence.
problem Classifying pseudo-Anosov flows on 3-manifolds up to orbit equivalence.
method Generalized Anosov-like actions on bifoliated planes, ideal boundary analysis.
result Pseudo-Anosov flows on 3-manifolds are determined up to orbit equivalence by their ideal boundary actions.