Study obstructions to conformally Anosov Reeb flows on 3-manifolds.
problem Existence of conformally Anosov Reeb flows on 3-manifolds.
method Geometric and topological obstructions, Riemannian conditions.
result Obtained obstructions and conditions for the existence of conformally Anosov Reeb flows.
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.
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 proves a unique conformal measure for Anosov groups and shows local mixing.
problem Proving the uniqueness of conformal measures for Anosov groups.
method Analogue of Sullivan's theorem for Anosov subgroups of semisimple groups.
result Uniqueness of conformal measures and local mixing for Anosov groups.
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.
Anosov groups' measures on limit sets are uniquely determined by their dimension.
problem Characterizing measures on limit sets of Anosov groups.
method Higher rank Hopf-Tsuji-Sullivan dichotomy for maximal diagonal actions.
result Uniqueness of Γ-conformal measures for critical dimensions. Anosov representations are linked to specific spacetimes.
problem Understanding representations of groups into Lie groups.
method Holonomy of Anosov representations into O0(2, n) and spatial compactness of spacetimes.
result Anosov representations are the holonomy of CGHM conformally flat spacetimes.
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.
Embeds Riemannian manifolds with Anosov flows, linking classical and new theorems.
problem Embedding Riemannian manifolds with Anosov flows.
method Isometric embedding into a closed Riemannian manifold with Anosov geodesic flow.
result Direct link between classical and new theorems.
Study shows how to detect representation extendability using conformal measures.
problem Detecting extendability of representations using conformal measures.
method Using higher rank conformal measures and self-joinings of groups.
result Affirmative answer to detect extendability of representations.
The paper proves identities for hyperconvex Anosov representations and their applications to Cantor sets.
problem Establishing identities for hyperconvex Anosov representations.
method Analyzing holomorphic families of Cantor non-conformal repellers and studying series identities.
result The series is absolutely summable if and only if the Hausdorff dimension of the Cantor set is less than 1.
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.
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.
The paper proves rigidity and ergodicity of horospherical foliations.
problem Rigidity and ergodicity of horospherical foliations in higher rank.
method Establishes higher rank extensions of rigidity theorems for representations of discrete subgroups of divergence type, using conformal measures and boundary maps.
result Proves conformal measure rigidity and ergodicity of horospherical foliations for hypertransverse subgroups.
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.
The paper studies proper discontinuity of actions on Weyl chamber flow spaces.
problem Properly discontinuous actions on Weyl chamber flow spaces for transverse subgroups.
method Analyzes limit sets and quotient spaces, introduces growth indicators and conformal measures.
result Establishes ergodic dichotomy for Weyl chamber flow and introduces new measures.
In this paper we investigate the Hausdorff dimension of limit sets of Anosov representations. In this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them, analogue to a differentiability property. As an application of this convergence, we prove that t…
This paper studies generic properties of connections on vector bundles, solving cohomological equations and proving opaque connections.
problem Generic properties of unitary connections on vector bundles over Riemannian manifolds.
method Introduction of operators of uniform divergence type and perturbative arguments from spectral theory.
result The existence of twisted Conformal Killing Tensors (CKTs) is generically solved, and connections are generically opaque.
Paper proves non-arithmetic Teichmüller length spectra for subgroup of mapping class groups.
problem Proving non-arithmetic Teichmüller length spectra for subgroups of mapping class groups.
method Introducing cross-ratios on Teichmüller and projectable mapping classes, studying their geometric and dynamical properties.
result Every non-elementary subgroup of the mapping class group has non-arithmetic Teichmüller length spectrum.
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.
Anosov diffeomorphisms with integrable subbundles have coherent dynamics and spectral rigidity.
problem Characterizing Anosov diffeomorphisms with integrable subbundles.
method Joint integrability of strong stable and unstable subbundles leads to coherent dynamics and spectral rigidity.
result Anosov diffeomorphisms with integrable subbundles are dynamically coherent and have spectral rigidity.
The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.
problem Understanding non-differentiability points in limit sets of convex projective structures.
method Introduces hyperplane conicality for θ-Anosov representations and uses it to prove properties of boundary maps. result Hilbert entropy is linked to the Hausdorff dimension of non-differentiability points in flag spaces.
A recent result of M. Kourganoff states that if D is a closed, reducible, non-flat, Weyl connection on a compact conformal manifold M, then the universal covering of M, endowed with the metric whose Levi-Civita covariant derivative is the pull-back of D, is isometric to Rq×N for some irreducib…
The paper studies limit sets on P(R3) using stationary measures.
problem Investigating the Hausdorff dimension of limit sets on P(R3) for SL3(R). method Using stationary measures to generalize the Patterson-Sullivan formula and establish dimension formulas.
result Sharp lower bounds and Hausdorff dimensions for Anosov representations and the Rauzy gasket.
We study pairs of Engel structures on four-manifolds whose intersection has constant rank one and which define the same even contact structure, but induce different orientations on it. We establish a correspondence between such pairs of Engel structures and a class of weakly hyperbolic flows. This correspondence is ana…
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.
The paper introduces Patterson-Sullivan systems and proves their rigidity, with applications to random walks and entropy rigidity.
problem Understanding the rigidity of Patterson-Sullivan systems and their applications.
method Generalization of Tukia's measurable boundary rigidity theorem for Patterson-Sullivan systems.
result Entropy rigidity for Anosov groups with Lipschitz limit sets.
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.
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 …
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.
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…
Characterizes Anosov flows in 3D using symplectic and contact geometry.
problem Understanding Anosov flows in 3D.
method Purely contact and symplectic geometric methods.
result Characterization of Anosov flows based on Reeb flows and underlying (bi)-contact structures.
Study on 3-manifolds admitting pseudo-Anosov maps on subsurfaces.
problem Which 3-manifolds admit pseudo-Anosov maps on incompressible subsurfaces?
method Determine self-homeomorphisms of 3-manifolds that restrict to pseudo-Anosov maps on subsurfaces.
result Self-homeomorphisms of irreducible 3-manifolds are isotopic to partially pseudo-Anosov homeomorphisms.
Study shows partial hyperbolicity leads to Anosov dynamics in 3-manifolds.
problem Understanding dynamics in hyperbolic 3-manifolds and Seifert manifolds.
method Classification of partially hyperbolic diffeomorphisms and pseudo-Anosov dynamics.
result Complete classification of partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds and Seifert manifolds.
Survey on affine Anosov representations and their implications.
problem Generalizing Anosov representations to affine settings.
method Discussion and survey of existing work.
result Implications of affine Anosov representations.
Study Agol cycles for pseudo-Anosov 3-braids.
problem Conditions for equivalent Agol cycles of pseudo-Anosov 3-braids.
method Investigate necessary and sufficient conditions.
result Necessary and sufficient conditions for equivalent Agol cycles of pseudo-Anosov 3-braids.
No contact Anosov diffeomorphisms exist on odd-dimensional manifolds.
problem Existence of contact Anosov diffeomorphisms on odd-dimensional manifolds.
method Analysis of Anosov diffeomorphisms and invariant contact structures.
result No invariant contact structure exists for Anosov diffeomorphisms on odd-dimensional manifolds.
Characterizes groups for specific types of representations into SL(d,R).
problem Understanding groups admitting certain types of representations into SL(d,R).
method Characterization and bounds on cohomological dimension.
result Bounds on cohomological dimension and characterizations of representations.
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.
Generalizes Anosov flows to partially hyperbolic diffeomorphisms.
problem Classifying partially hyperbolic diffeomorphisms.
method Introducing collapsed Anosov flows and self orbit equivalences.
result All examples in Bonatti et al. belong to the collapsed Anosov flow class.
Anosov magnetic flows on surfaces are characterized.
problem Characterizing Anosov magnetic flows on surfaces.
method Using Wojtkowski's quotient bundle, necessary and sufficient conditions are derived.
result Necessary and sufficient conditions for Anosov magnetic flows on surfaces are established.
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.
New representations for surface groups expand known Anosov classes.
problem Understanding new types of representations for surface groups.
method Introducing and studying simple Anosov representations of closed hyperbolic surface groups.
result Simple Anosov representations strictly contain Anosov representations.
Anosov maps on torus curve graphs have positive integer translation lengths.
problem Understanding the translation lengths of Anosov maps on curve graphs of tori.
method Constructive proof and algorithm for calculating exact translation lengths.
result The stable translation length of an Anosov map on the curve graph is always a positive integer.
New findings on cusped Borel Anosov representations and their properties.
problem Characterizing and understanding cusped Borel Anosov representations.
method Analyzing representations of lattices in PGL2(R) to PGLd(R). result Cusped Borel Anosov representations with specific properties are Hitchin representations.