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.
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 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.
Constructs entropy-minimizing pseudo-Anosov diffeomorphisms on K3 surfaces.
problem Finding minimal entropy diffeomorphisms on K3 surfaces.
method Constructs pseudo-Anosov diffeomorphisms minimizing entropy.
result Obtains infinitely many entropy-minimizing diffeomorphisms.
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.
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.
We show that any pseudo-Anosov map that is a lift of pseudo-Anosov homeomorphism of a nonorientable surface has vanishing SAF invariant. We also provide a criterion to certify that a pseudo-Anosov map is not such a lift.
This paper simplifies Anosov geodesic flows on surfaces.
problem Understanding Anosov geodesic flows on surfaces.
method Exposition of Eberlein's work, focusing on surface case.
result Provides a more accessible introduction to Anosov geodesic flows.
New algorithm verifies Anosov condition for surface groups efficiently.
problem Verifying Anosov condition for surface groups in higher dimensions.
method Finite criteria to certify projective Anosov subgroups, practical algorithm.
result Practical algorithm reduces verification from 2 million to 8-length words.
Maximal and Borel Anosov representations in Sp(4,R) are proven to be Hitchin.
problem Characterizing representations of surface groups into Sp(4,R) that are Borel Anosov and maximal. method Proving representations are Hitchin if they have maximal Toledo invariant and are Borel Anosov.
result Maximal and Borel Anosov representations in Sp(4,R) are Hitchin. 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…
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.
Pseudo-Anosov subgroups in surface bundles over tori are convex cocompact.
problem Understanding the structure of pseudo-Anosov subgroups in surface bundles over tori.
method Using the Birman exact sequence to show convex cocompactness.
result Finitely generated, purely pseudo-Anosov subgroups are convex cocompact in surface bundles over tori.
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.
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. Characterizes transverse surfaces for pseudo-Anosov flows in 3-manifolds.
problem Characterizing surfaces transverse to pseudo-Anosov flows.
method Correspondence between surfaces and veering triangulations, Thurston norm minimization.
result Thurston-norm minimizing surfaces are almost transverse to pseudo-Anosov flows.
Anosov representations of surface groups are complex manifolds.
problem Characterizing the geometry of Anosov representations of surface groups.
method Analyzing the character varieties of Anosov representations into $\SL(n , \C)$.
result Character varieties of Anosov representations are complex manifolds of specific dimension.
The paper finds pseudo-Anosov-like maps on an infinite ladder surface.
problem Exploring dynamics on infinite surfaces.
method Lifts Penner-type pseudo-Anosov maps from a closed surface to an infinite ladder surface.
result Existence and properties of pseudo-Anosov-like maps on the infinite ladder surface.
Positive representations of surface groups in PO(p,q) form connected components of character varieties.
problem Characterizing representations of surface groups in special orthogonal groups PO(p,q).
method Using Anosov representations and root versus weight collar lemmas.
result Connected components of character varieties are formed by Θ-positive Anosov representations. 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.
The study restricts Anosov subgroups of Sp(2n,R) based on subset Θ.
problem Characterizing Anosov subgroups of Sp(2n,R) based on subset Θ.
method Analyzing the structure of Anosov subgroups in terms of subset Θ.
result Anosov subgroups of Sp(2n,R) are virtually free or surface groups if Θ contains an odd integer, otherwise they are not.
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…
Continuum-wise hyperbolicity is exactly the pseudo-Anosov dynamics with spine singularities.
problem Classification of continuum-wise hyperbolic surface homeomorphisms
method Proving a complete structural classification
result Every cwF-hyperbolic homeomorphism is pseudo-Anosov with spine singularities Formula estimates pseudo-Anosov maps' fixed points, linking to surface properties.
problem Estimating fixed points of pseudo-Anosov maps.
method Formula using Teichmüller translation length for fixed points of strong irreducible maps.
result Log of fixed points coarsely equals Teichmüller translation length for strong irreducible maps.
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.
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.
Study vortex flows on Riemann surfaces, proving dominated splitting and Anosov properties.
problem Investigate flow properties on Riemann surfaces.
method Associate flow to vortex equations, investigate properties of flow.
result Show that flow always admits a dominated splitting and identify special cases of Anosov flow.
We establish a criterion for certain mapping classes of a surface homeomorphisms to be pseudo-Anosov in terms of the geometry of hyperbolic 3-manifolds and Gromov-hyperbolic surface group extensions. Specifically, any element of the fundamental group of a surface S gives rise to a mapping class on the punctured surface…
This article considers inverse problems on closed Riemannian surfaces whose geodesic flow is Anosov. We prove spectral rigidity for any Anosov surface and injectivity of the geodesic ray transform on solenoidal 2-tensors. We also establish surjectivity results for the adjoint of the geodesic ray transform on solenoidal…
The motivation for this paper is to justify a remark of Thurston that the algebraic degree of stretch factors of pseudo-Anosov maps on a surface S can be as high as the dimension of the Teichmüller space of S. In addition to proving this, we completely determine the set of possible algebraic degrees of pseudo-Anoso…
Minimal stretch factor for non-orientable surfaces is small.
problem Finding the minimal stretch factor for pseudo-Anosov homeomorphisms on non-orientable surfaces.
method Adapting Thurston's theory of fibered faces for non-orientable 3-manifolds.
result The minimal stretch factor is asymptotically on the order of 1/g.
Study bounds topological entropy of maps on surfaces with punctures based on mapping torus homology.
problem Relating topological entropy of pseudo-Anosov maps to homology of mapping tori.
method Analyzing the topological entropy of pseudo-Anosov maps on surfaces with punctures and relating it to the rank of the first homology of their mapping tori.
result Entropy of a pseudo-Anosov map is bounded by a formula involving the genus, number of punctures, and homology rank.
We prove that the dilatation of any pseudo-Anosov homeomorphism on a translation surface that belong to a hyperelliptic component is bounded from below uniformly by sqrt{2}. This is in contrast to Penner's asymptotic. Penner proved that the logarithm of the least dilatation of any pseudo-Anosov homeomorphism on a surfa…
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…
Let F′,F be any two closed orientable surfaces of genus g′>g≥1, and f:F→F be any pseudo-Anosov map. Then we can "extend" f to be a pseudo-Anosov map f′:F′→F′ so that there is a fiber preserving degree one map M(F′,f′)→M(F,f) between the hyperbolic surface bundles. Moreover the extension f′ can…
New representations of surface groups into higher-dimensional PSL generalize pleated surfaces.
problem Generalizing pleated surfaces to higher-dimensional PSL groups.
method λ-Borel Anosov representations of surface groups into PSL_d(C).
result Holomorphic parametrization of space of (λ,d)-pleated surfaces.
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…
We show that the minimum of asymptotic translation lengths of all point-pushing pseudo-Anosov maps on any one punctured Riemann surface is one.
Study on mapping classes of real rational surface automorphisms, focusing on reducible maps and pseudo-Anosov maps.
problem Investigating the mapping classes of real rational surface automorphisms and their restrictions.
method Analysis of reducible maps, determination of pseudo-Anosov mapping classes, and comparison with Penner's construction.
result Realized Lehmer's number as the stretch factor of a pseudo-Anosov map on a specific surface.
Study of pseudo-Anosov actions on SU(2)-character variety for genus 2 surfaces.
problem Characterizing pseudo-Anosov actions on SU(2)-character variety. method Analysis of mapping class group subgroups and their pseudo-Anosov elements.
result Existence of a subgroup containing infinitely many pseudo-Anosov elements with invariant rational functions.
Study on measurable pseudo-Anosov maps on surfaces.
problem Characterize dynamics of pseudo-Anosov maps on surfaces.
method Analyze measurable pseudo-Anosov homeomorphisms with specific properties.
result Prove transitivity, dense periodic points, sensitivity, and ergodicity.
For a pseudo-Anosov homeomorphism f on a closed surface of genus g≥2, for which the entropy is on the order g1 (the lowest possible order), Farb-Leininger-Margalit showed that the volume of the mapping torus is bounded, independent of g. We show that the analogous result fails for a surface of fixe…
This thesis classifies pseudo-Anosov homeomorphisms using geometric Markov partitions.
problem Classifying pseudo-Anosov homeomorphisms up to topological conjugacy.
method Algorithmic approach using geometric Markov partitions.
result Geometric type is a complete invariant of conjugation.
The study of pseudo-Anosovs through mapping torus geometry.
problem Understanding pseudo-Anosov diffeomorphisms via mapping torus geometry.
method Analyzing simplicial complexes and mapping tori to relate dynamics and geometry.
result Relating pseudo-Anosov actions to fixed points in isotopy classes.
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.
Maximal dilatation found on nonorientable surfaces.
problem Finding maximal dilatation on nonorientable surfaces.
method Proving irreducibility of a polynomial to show maximal dilatation.
result Maximal dilatation is achieved by the Liechti-Strenner polynomial.
We prove that Anosov representations from a surface group to SL(3,R) are uniquely determined by their boundary maps if and only if they do not factor over a completely reducible representation. Furthermore we discuss representations not distinguished by their boundary maps, and we give a lower bound for the number of m…
New examples of surface bundles found over surfaces.
problem Finding compact atoroidal surface bundles.
method Type-preserving homomorphism from knot complement to mapping class group.
result Infinitely many commensurability classes of surface subgroups.