arXiv research

Maps minimize periodic points for high periods, but not for low periods.

Paper disproves Wright's periodic map conjecture.

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.

Call a periodic map $h$ on the closed orientable surface $Σ_g$ extendable if $h$ extends to a periodic map over the pair $(S^3, Σ_g)$ for possible embeddings $e: Σ_g\to S^3$. We determine the extendabilities for all periodical maps on $Σ_2$. The results involve various orientation preserving/reversing behalves of the p…

The paper explores methods to decompose periodic maps into Dehn twists.

New bounds on specific torsion lengths for periodic mapping classes.

We study the asymptotic behavior of the sequence of the Nielsen numbers $\{N(f^k)\}$, 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

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.

Simplified proof of K3 surface period map surjectivity.

Period maps surjective for certain gravitational instantons.

Maps on surfaces can be embedded into spheres with minimal dimensions.

The paper characterizes roots of pseudo-periodic mapping classes in surface groups.

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.

Lower bound on stretch factor for periodic maps.

Researchers extend period maps for Calabi-Yau types using modified Kato-Nakayama-Usui construction.

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.

Study shows periodic cohomology of non-orientable surface mapping class groups for odd primes.

Global rigidity theorem for curve to abelian variety maps.

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.

We classify up to conjugacy the group generated by a commuting pair of a periodic diffeomorphism and a hyperelliptic involution on an oriented closed surface. This result can be viewed as a refinement of Ishizaka's result on classification of the mapping classes of hyperelliptic periodic diffeomorphisms. As an applicat…

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.

The paper studies liftable mapping class groups of cyclic covers of spheres.

The paper introduces invariants to describe period-doubling routes to chaos in dynamical systems.

The paper explores infinite metacyclic subgroups in mapping class groups of surfaces.

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.

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.

This research studies end-periodic mapping tori and their hyperbolic structures.

Classifies theories with eight supercharges using pseudo-periodic maps and Riemann surfaces.

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.

Study lifts periodic mapping classes under alternating group actions on surfaces.

The paper constructs moduli spaces for genus one fibered K3 surfaces.

Study on moduli spaces of sextic curves with simple singularities and their compactifications.

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 …