Study on minimal foliations in 3D manifolds with specific conditions.
problem Characterizing minimal foliations in 3D manifolds.
method Analyzing Anosov foliations and their intersections.
result Necessary and sufficient conditions for orbit foliation of Anosov flows.
A foliation is R-covered if the leaf space in the universal cover is homeomorphic to the real numbers. We show that, up to topological conjugacy, there are at most two pseudo-Anosov flows transverse to such a foliation. If there are two, then the foliation is weakly conjugate to the the stable foliation of an R-covered…
The paper studies Liouville structures for taut foliations and Anosov flows, proving their topological invariance.
problem Understanding the topological invariance of Liouville structures for taut foliations and Anosov flows.
method Combining smoothing schemes for topological conjugacies and a refinement of Vogel's uniqueness result.
result Liouville structures are topological invariants of taut foliations and orbit equivalent Anosov flows.
New maps connect universal circles to ideal sphere for hyperbolic manifolds.
problem Understanding universal circles for Anosov foliations with branching.
method Introduced a new type of Cannon--Thurston map for leftmost universal circles.
result Fundamental group acts on leftmost universal circle with pseudo-Anosov dynamics.
New proof associates partitions to isotopic pseudo-Anosov homeomorphisms.
problem Stable and unstable foliations for pseudo-Anosov homeomorphisms.
method Geometric realization of Fathi's result for isotopic homeomorphisms.
result Associated stable and unstable partitions for isotopic pseudo-Anosov homeomorphisms.
We prove the existence of foliations transverse to pseudo-Anosov flows using veering triangulations.
problem Existence of foliations transverse to pseudo-Anosov flows in 3-manifolds.
method Combinatorial approach via veering triangulations and branched surfaces.
result Existence of transverse foliations for certain pseudo-Anosov flows.
Quadratic-time algorithm computes stretch factors and foliations for pseudo-Anosov mapping classes.
problem Computing stretch factors and foliations for pseudo-Anosov mapping classes efficiently.
method Quadratic-time algorithm using input word and length as complexity measure.
result First algorithm to compute stretch factors and foliations in sub-exponential time.
Study shows how certain foliations in unit tangent bundles behave.
problem Characterizing behavior of foliations in unit tangent bundles.
method Analyzing intersections and properties of foliations.
result Certain partially hyperbolic diffeomorphisms are collapsed Anosov flows.
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.
The paper explores universal circles for Anosov foliations and their uniqueness.
problem Exploring the uniqueness of universal circles for Anosov foliations.
method Using the flow space of an Anosov flow to parameterize the circle bundle at infinity of the foliations.
result Several constructions of a universal circle are typically distinct and not conjugate.
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.
New 3D shapes found without certain flows.
problem Finding 3D shapes without specific flows.
method Using foliations and pseudo-Anosov flows, analyzing cusped hyperbolic 3-manifolds.
result First examples of 3D shapes without veering triangulations.
Anosov flow found in specific partially hyperbolic systems.
problem Characterizing partially hyperbolic diffeomorphisms with center foliation.
method Analyzing transitive dynamically coherent systems with one-dimensional center foliation.
result Discretized Anosov flow found in systems satisfying f(W)=W for center leaves. Proof of contact structure from taut foliation for certain knots.
problem Contact structures from taut foliations for specific knots.
method Alternative proof using pseudo-Anosov monodromy and fractional Dehn twist coefficient.
result Fibered knots with certain properties support contact structures that are taut foliation perturbations.
Completes preliminary structures in 3D flows to foliations.
problem Characterizing completability of lamination pairs in 3-manifolds.
method General approach for various types of foliations and flows.
result Characterizes when lamination pairs can be completed to foliations.
New proof shows left orderability of 3-manifold groups with specific foliations.
problem Left orderability of 3-manifold groups with co-orientable taut foliations.
method Analyzes co-orientable taut foliations with orderable cataclysm to prove left orderability of 3-manifold fundamental groups.
result Proves left orderability of 3-manifold groups with specific foliations.
Let F be a foliation in a closed 3-manifold with negatively curved fundamental group and suppose that F is almost transverse to a quasigeodesic pseudo-Anosov flow. We show that the leaves of the foliation in the universal cover extend continuously to the sphere at infinity, hence the limit sets are continuous images of…
We construct infinite sequences of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of abelian differentials. The existence of such sequences show that all pseudo-Anosov homeomorphisms f…
Paper translates train track concepts to cluster algebras for pseudo-Anosov mapping classes.
problem Understanding pseudo-Anosov mapping classes on surfaces.
method Using Goncharov--Shen's potential function, the paper translates train track concepts into cluster algebra language.
result Proves sign stability of general pseudo-Anosov mapping classes.
Anosov flows in hyperbolic 3-manifolds are quasigeodesic if not R-covered.
problem Characterizing Anosov flows in hyperbolic 3-manifolds.
method Analyzing the properties of Anosov flows and foliations.
result Anosov flows in hyperbolic 3-manifolds are quasigeodesic if not R-covered.
The invariant measured foliations of a pseudo-Anosov homeomorphism induce a natural (singular) Sol structure on mapping tori of surfaces with pseudo-Anosov monodromy. We show that when the pseudo-Anosov φ:S→S has orientable foliations and does not have 1 as an eigenvalue of the induced cohomology action on…
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.
The paper characterizes when a surface can be completed to a depth one lamination transverse to a pseudo-Anosov flow.
problem Characterizing when a surface can be completed to a depth one lamination transverse to a pseudo-Anosov flow.
method Homological characterization and foliation cone analysis.
result The cone of classes in H1(M\\S) that are positive on the closed orbits of φ is an entire foliation cone of M\\S. Paper shows how to transform certain flows into R-covered ones.
problem Transforming Anosov flows into R-covered ones.
method Goodman-Fried surgery along periodic points.
result Any topologically transitive Anosov flow can be transformed into an R-covered one.
An infinite family of generalized pseudo-Anosov homeomorphisms of the sphere S is constructed, and their invariant foliations and singular orbits are described explicitly by means of generalized train tracks. The complex strucure induced by the invariant foliations is described, and is shown to make S into a complex sp…
It is proved, that a foliation on a modular curve given by the vertical trajectories of holomorphic differential corresponding to the Hecke eigenform is either the Strebel foliation or the pseudo-Anosov foliation.
Study Veech groups in fibered 3-manifolds, proving no parabolics for fibers.
problem Characterizing Veech groups in fibered 3-manifolds.
method Analyzing pseudo-Anosov monodromies and foliations, proving properties of Veech groups.
result Veech groups in fibers typically contain no parabolic elements.
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 investigates certain foliations of three-manifolds that are hybrids of fibrations over the circle with foliated circle bundles over surfaces: a 3-manifold slithers around the circle when its universal cover fibers over the circle so that deck transformations are bundle automorphisms. Examples include hyperbo…
We determine the smallest stretch factor among pseudo-Anosov maps with an orientable invariant foliation on the closed nonorientable surfaces of genus 4, 5, 6, 7, 8, 10, 12, 14, 16, 18 and 20. We also determine the smallest stretch factor of an orientation-reversing pseudo-Anosov map with orientable invariant foliation…
The paper proves left-orderability for certain Dehn fillings of pseudo-Anosov mapping tori.
problem Left-orderability of fundamental groups in Dehn fillings of pseudo-Anosov mapping tori.
method Two approaches: one using R-covered foliations and the other using one-sided branching. result All such Dehn fillings have left-orderable fundamental groups.
Study shows universal circles for taut foliations in 3-manifolds.
problem Understanding foliations in 3-manifolds.
method Analyzes pseudo-Anosov flows and taut foliations, defines universal circles.
result Action of pi_1(M) on flow space defines universal circles for taut foliations.
Classifies Anosov flows on figure-eight knot surgeries.
problem Classifying Anosov flows on Dehn surgeries on the figure-eight knot.
method Combining Plante's classification of M(0) with Schwider's branched surfaces.
result M(r) carries a unique Anosov flow if r is an integer, and no Anosov flow if r is not an integer.
The paper studies stability of discretized Anosov flows.
problem Global stability of discretized Anosov flows.
method Defined and proved equivalence with previous definitions, showed properties through C1 openness and closedness, and established integrability and uniqueness of invariant foliations. result Discretized Anosov flows are globally stable.
An alternate proof shows how foliation extensions work in 3D spaces.
problem Continuous extension of foliations in 3D spaces.
method Uses the universal circle and properties of pseudo-Anosov flows.
result Shows how all continuous extensions organize in the boundary.
Classifies 3D partially hyperbolic systems, proving ergodicity.
problem Ergodicity of partially hyperbolic diffeomorphisms in 3-manifolds.
method Topological classification, Anosov flows, foliations, Gromov hyperbolicity.
result Complete answer to Hertz-Hertz-Ures conjecture for 3D systems.
This note gives a brief survey of the minimum dilatation problem for pseudo-Anosov mapping classes, and the first explicit train track description of an infinite family of pseudo-Anosov mapping classes with orientable stable foliations and the conjectural minimum dilatation for closed surfaces of even genus g≥2.
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.
Study partially hyperbolic diffeomorphisms in 3D, focusing on foliations and dynamics.
problem Classify 3D partially hyperbolic diffeomorphisms homotopic to the identity.
method Analyze Burago and Ivanov's branching foliations in Seifert fibered and hyperbolic manifolds.
result Complete classification of diffeomorphisms in Seifert fibered manifolds, and new potential class in hyperbolic manifolds.
In this article we study the topological structure of the lifts to the universal of the stable and unstable foliations of 3-dimensional Anosov flows. In particular we consider the case when these foliations do not have Hausdorff leaf space. We completely determine the structure of the set of non separated leaves from…
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
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.
Shows Anosov flows with genus one sections, supporting a conjecture.
problem Finding genus one Birkhoff sections for Anosov flows.
method Utilizes horizontal Goodman surgery operation and correspondence with veering triangulations.
result Provides evidence for Fried and Ghys conjecture.
Given a general pseudo-Anosov flow in a three manifold, the orbit space of the lifted flow to the universal cover is homeomorphic to an open disk. We compactify this orbit space with an ideal circle boundary. If there are no perfect fits between stable and unstable leaves and the flow is not topologically conjugate to …
Paper finds optimal pseudo-Anosov homeomorphisms for surfaces with specific properties.
problem Finding optimal pseudo-Anosov homeomorphisms on surfaces.
method Analyzing Liechti-Strenner's examples and proving minimality conditions.
result Optimal pseudo-Anosov homeomorphisms are identified for both orientable and nonorientable surfaces.
This paper concerns the set M^ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3-manifold N by Dehn filling three cusps with a mild restriction. We prove that for each g (resp. g≡0(mod6)), the minimum among dilatations of elements (res…
Let Γ be a surface group of higher genus. Let ρ_0:Γ→PGL(V) be a discrete faithful representation with image contained in the natural embedding of SL(2,R) in PGL(3,R) as a group preserving a point and a disjoint projective line in the projective plane. We prove that such a repres…
We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham's proof of the least dilatation of pseudo-Anosov homeomorphisms on a genus two surf…