Maps minimize periodic points for high periods, but not for low periods.
problem Understanding periodic points of maps on infinite type surfaces.
method Analysis of isotopic maps, spun pseudo-Anosov maps, and Handel-Miller maps.
result Spun pseudo-Anosov maps minimize periodic points for sufficiently high periods.
Paper disproves Wright's periodic map conjecture.
problem Existence of periodic maps on closed hyperbolic surfaces.
method Analyzes automorphisms and curve behavior on surfaces.
result No periodic map can fill every curve with its image.
In this paper, we give a classification of orientation reversing periodic maps on closed surfaces which generalizes the theory of Nielsen for the orientation preserving periodic maps. On one hand, we give a group of data for each orientation reversing periodic map such that two periodic maps with the same data must be …
Embeddings of mapping tori for end-periodic graph maps are proven.
problem Embedding mapping tori of end-periodic graph maps into finite complexes.
method Flowline-preserving homotopy equivalence and π1-injective map. result Every mapping class of Γ arising from an end-periodic homotopy equivalence contains a representative whose mapping torus realizes such an embedding.
Call a periodic map h on the closed orientable surface Σg extendable if h extends to a periodic map over the pair (S3,Σg) for possible embeddings e:Σg→S3. We determine the extendabilities for all periodical maps on Σ2. The results involve various orientation preserving/reversing behalves of the p…
The paper explores methods to decompose periodic maps into Dehn twists.
problem Factoring periodic maps into Dehn twists.
method Various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy.
result Existence of conjugates of periodic maps whose product is pseudo-Anosov.
New bounds on specific torsion lengths for periodic mapping classes.
problem Understanding specific torsion lengths in periodic mapping classes.
method Introduced stable specific torsion length and used it to bound Dehn twists.
result Bounded the stable specific torsion length of Dehn twists by six.
We study the asymptotic behavior of the sequence of the Nielsen numbers {N(fk)}, the essential periodic orbits of f and the homotopy minimal periods of f by using the Nielsen theory of maps f on infra-solvmanifolds of type R. We give a linear lower bound for the number of essential periodic orbits of such …
Algorithm for Teichmüller isometries induced by periodic mapping classes
problem Determining isometries in Teichmüller space induced by periodic mapping classes
method Algorithm using Fenchel-Nielsen coordinates
result Description of isometry induced by periodic mapping class of order 4g+2 We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix repres…
The period map for 4-manifolds is dense and surjective under certain conditions.
problem Understanding the space of metrics on 4-manifolds and their harmonic forms.
method Constructing families of metrics by stretching along hypersurfaces.
result The period map is dense and surjective for 4-manifolds with b+=1. Simplified proof of K3 surface period map surjectivity.
problem Surjectivity of period map on K3 surfaces.
method Utilizes hyperkähler geometry and collapsing techniques.
result Simple proof of Todorov's result on K3 surfaces.
Period maps surjective for certain gravitational instantons.
problem Understanding the uniformization of gravitational instantons.
method Uniformization theorems by Chen--Chen, Chen--Viaclovsky, Collins--Jacob--Lin, and Hein--Sun--Viaclovsky--Zhang.
result Period maps surjective for ALH*, ALG, and ALG* gravitational instantons.
Maps on surfaces can be embedded into spheres with minimal dimensions.
problem Embedding periodic maps of surfaces into spheres with the smallest possible dimensions.
method Determining the minimal dimensions m for embeddings of periodic maps of order n on surfaces of genus g into spheres Sm. result For each integer k>1, there exist infinitely many periodic maps such that the smallest possible m is equal to k. The paper characterizes roots of pseudo-periodic mapping classes in surface groups.
problem Characterizing roots of pseudo-periodic mapping classes in surface groups.
method Using pseudo-periodic mapping classes, canonical decomposition, and algorithms, the paper derives necessary and sufficient conditions for roots and provides an algorithm to determine conjugacy classes of roots.
result Realizable bounds on the degrees of roots of pseudo-periodic mapping classes and the highest possible degree is 3q(F)(g+1)(g+2).
Kulkarni showed that, if g is greater than 3, a periodic map on an oriented surface S_g of genus g with order more than or equal to 4g is uniquely determined by its order, up to conjugation and power. In this paper, we show that, if g is greater than 30, the same phenomenon happens for periodic maps on the surfaces wit…
Paper explains dynamics of homeomorphisms to mapping tori geometry.
problem Understanding dynamics of end-periodic homeomorphisms.
method Illustration-driven overview of recent results.
result Analogue of Brock's theorem for infinite-type surfaces.
Lower bound on stretch factor for periodic maps.
problem Finding a lower bound on stretch factors for periodic maps.
method Using core characteristic of end-periodic homeomorphisms, we derive a lower bound on the Handel-Miller stretch factor.
result The derived bound is sharp and measures topological complexity.
Researchers extend period maps for Calabi-Yau types using modified Kato-Nakayama-Usui construction.
problem Existence of weak fans for non-classical period maps of weight 3 Calabi-Yau type.
method Modified Kato-Nakayama-Usui construction for period maps of weight 3 Calabi-Yau type.
result Existence of weak fans for a large class of period maps of weight 3 Calabi-Yau type.
We study the periods mapping from the moduli space of real hyperelliptic curves with marked point on an oriented oval to the euclidean space. The mapping arises in the analysis of Chebyshev construction used in the constrained optimization of the uniform norm of polynomials and rational functions. The decomposition of …
Periodic Floer homology (PFH) is a Gromov-Floer type invariant for fibered three-manifolds with Hamiltonian structures. The cobordism maps on periodic Floer homology induced by symplectic cobordisms are currently only defined indirectly by using Seiberg-Witten theory. In this paper, we investigate the cobordism maps in…
Lower bound on volumes of special mapping tori.
problem Calculating the minimum volume of compactified mapping tori.
method Using strongly irreducible end-periodic homeomorphisms and properties of pants graphs.
result Volume of compactified mapping tori is comparable to the translation length of the homeomorphism on pants graphs.
Study shows periodic cohomology of non-orientable surface mapping class groups for odd primes.
problem Investigating periodic cohomology of non-orientable surface mapping class groups for odd primes.
method Using Yagita invariant, cohomology classes, Nielsen realization theorem, and properties of cyclic subgroups of order p.
result The p-period of Ngk is bounded below by 4 when Ngk has p-periodic cohomology, g⩾3 and k⩾0. Classifies periodic diffeomorphisms and hyperelliptic involutions on surfaces.
problem Classifying periodic diffeomorphisms and involutions on surfaces.
method Refinement of Ishizaka's result using Dehn twist presentations.
result Dehn twist presentations of hyperelliptic periodic mapping classes.
Global rigidity theorem for curve to abelian variety maps.
problem Characterizing maps between moduli spaces of curves and abelian varieties.
method Analyzing nonconstant holomorphic maps between moduli spaces.
result Holomorphic maps between specific moduli spaces are rigid, with only one possible form.
We study the homotopical minimal periods for maps on infra-solvmanifolds of type (R) using the density of the homotopical minimal period set in the natural numbers. This extends the result of [10] from flat manifolds to infra-solvmanifolds of type (R). Applying our main result we will list all possible maps on infra-so…
We analyze the signature type of a cascade of periodic orbits associated to period doubling renormalizable maps of the two dimensional disk. The signature is a sequence of rational numbers which describes how periodic orbits turn each other and is invariant by topological conjugacies that preserve orientation. We prove…
Scattering theory on Riemann surfaces with explicit matrix and generalized period mappings.
problem Scattering theory for harmonic one-forms on Riemann surfaces.
method Construction of scattering theory from boundary value problems involving systems of curves and jump problems. Explicit expression for scattering matrix using Schiffer operators.
result Unitary scattering matrix and general association of polarizing Lagrangian spaces.
We study the space of linear difference equations with periodic coefficients and (anti)periodic solutions. We show that this space is isomorphic to the space of tame frieze patterns and closely related to the moduli space of configurations of points in the projective space. We define the notion of combinatorial Gale tr…
Positive factorization found for a specific map on surfaces.
problem Balanced superelliptic rotation on surfaces.
method Positive factorization approach.
result Positive factorization for balanced superelliptic rotation.
The paper studies liftable mapping class groups of cyclic covers of spheres.
problem Understanding liftable mapping class groups of cyclic covers of spheres.
method Derived finite generating sets, provided algorithms, determined isomorphism classes, derived presentations, and calculated normalizers and centralizers.
result Presentations and isomorphism classes of liftable mapping class groups for various covers.
The paper introduces invariants to describe period-doubling routes to chaos in dynamical systems.
problem Understanding the dynamics of period-doubling routes to chaos in complex systems.
method Introducing three topological invariants to describe the topology of period-doubling routes to chaos.
result Ascribed symbolic dynamics to perturbations of the Shilnikov homoclinic scenario and dynamics of the Henon map.
The paper explores infinite metacyclic subgroups in mapping class groups of surfaces.
problem Existence and properties of infinite metacyclic subgroups in mapping class groups.
method Provided necessary and sufficient conditions for the existence of infinite metacyclic subgroups.
result Established the existence of specific infinite metacyclic subgroups and derived bounds on their periodic generators.
The periodic Floer homology of a surface symplectomorphism, defined by the first author and M. Thaddeus, is the homology of a chain complex which is generated by certain unions of periodic orbits, and whose differential counts certain embedded pseudoholomorphic curves in R cross the mapping torus. It is conjectured to …
The Bergman kernel and period map for curves are studied.
problem Characterize the Torelli map's second fundamental form on the moduli space of curves.
method Use the Bergman kernel form associated to the curve and its harmonic representative.
result The Bergman kernel form is the harmonic representative of the multiplication by a certain meromorphic form.
Some years ago, V. Markovic proved that there is no section of the Mapping Class Group for a closed surface of genus g larger than 5 (in the case of homeomorphims) and more recently generalized this result with D. Saric to the case where g is larger than 1. We will state a periodicity criterion and will use it to simpl…
We define a generalization of Coxeter graphs and an associated Coxeter system and Coxeter mapping class. These can be used to construct periodic Coxeter mapping classes on surfaces with arbitrarily large genus, preserving lots of symmetries. The periodic mapping classes can in turn be used to construct sequences of pse…
The study proves rigidity for mixed Hodge structures and applies to curve families.
problem Rigidity of period maps for mixed Hodge structures.
method Holomorphic bisectional curvature approach.
result Establishes rigidity in various cases, including curve families.
This research studies end-periodic mapping tori and their hyperbolic structures.
problem Understanding the geometry and dynamics of end-periodic mapping tori.
method Analyzes invariant laminations and hyperbolic structures of mapping tori.
result Establishes a relationship between the geodesic length of boundary components and the infimum of geodesic lengths in hyperbolic structures.
Classifies theories with eight supercharges using pseudo-periodic maps and Riemann surfaces.
problem Classifying theories with eight supercharges using mathematical tools.
method Assumes theories are given by genus g fibrations of Riemann surfaces, uses pseudo-periodic maps of negative type in mapping class group.
result Identifies dual graphs and 3d mirror quivers, unifies various SCFTs in combinatorial framework.
Li-York theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the se…
Upper bound on 3-manifold volumes from surface homeomorphisms.
problem Bounding volumes of 3-manifolds from surface homeomorphisms.
method Using end-periodic homeomorphisms and pants graphs.
result Upper bound on infimal hyperbolic volume is asymptotically sharp.
Study lifts periodic mapping classes under alternating group actions on surfaces.
problem Classifying subgroups of mapping class groups isomorphic to alternating groups.
method Derived conditions for periodic mapping classes to lift under alternating group actions.
result For n≥7, no subgroup of Mod(Sg) can have an irreducible periodic mapping class. The paper constructs moduli spaces for genus one fibered K3 surfaces.
problem Understanding the moduli spaces and period mappings of genus one fibered K3 surfaces.
method Constructing various moduli spaces and period mappings related to locally symmetric spaces.
result Computed fundamental groups of moduli spaces and applied results to mapping class groups.
Study on moduli spaces of sextic curves with simple singularities and their compactifications.
problem Understanding moduli spaces of sextic curves with simple singularities.
method Using period maps of K3 surfaces with ADE singularities, algebraic open embeddings into arithmetic quotients of type IV domains, and GIT and Looijenga compactifications.
result Identifications of GIT and Looijenga compactifications for all cases.
Several new mutation-periodic quivers of period higher than 1 are introduced as well as the associated discrete dynamical systems. The reduction of these systems is developed using either a presymplectic or a Poisson approach. The presymplectic approach leads to a reduced system whose iteration map is symplectic with r…
In this article, we investigate the cobordism maps on periodic Floer homology (PFH). In the first part of the paper, we define the cobordism maps on PFH via Seiberg Witten theory as well as the isomorphism between PFH and Seiberg Witten cohomology. Furthermore, we show that the maps satisfy the holomorphic curve axiom.…
The period mapping assigns to each rank n, marked metric graph Gamma a positive definite quadratic form on H_1(Gamma). This defines maps Phi* and Phi on Culler--Vogtmann's outer space CV_n, and its Torelli space quotient T_n, respectively. The map Phi is a free group analog of the classical period mapping that sends a …