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Prime Ends Dynamics in Parametrised Families of Rotational AttractorsJun 11 2019We provide several new examples in dynamics on the $2$-sphere, with the emphasis on better understanding the induced boundary dynamics of invariant domains in parametrized families. First, motivated by a topological version of the Poincar\'e-Bendixson ... More

Explicit methods for the Hasse norm principle and applications to $A_n$ and $S_n$ extensionsJun 09 2019Let $K/k$ be an extension of number fields. We describe theoretical results and computational methods for calculating the obstruction to the Hasse norm principle for $K/k$ and the defect of weak approximation for the norm one torus $R^1_{K/k} \mathbb{G}_m$. ... More

Generic invariant measures for iterated systems of interval homeomorphismsMay 31 2019It is known that Iterated Function Systems generated by orientation preserving homeomorphisms of the unit interval admit a unique invariant measure on $(0,1)$. The setup for this result is the positivity of Lyapunov exponents at both fixed points and ... More

Algebras defined by Lyndon words and Artin-Schelter regularityMay 27 2019Let $X= \{x_1, x_2, \cdots, x_n\}$ be a finite alphabet, and let $K$ be a field. We study classes $\mathfrak{C}(X, W)$ of graded $K$-algebras $A = K\langle X\rangle / I$, generated by $X$ and with a fixed set of obstructions $W$. Initially we do not impose ... More

Dynamic mode decomposition for analytic mapsMay 22 2019Extended dynamic mode decomposition (EDMD) provides a class of algorithms to identify patterns and effective degrees of freedom in complex dynamical systems. We show that the modes identified by EDMD correspond to those of compact Perron-Frobenius and ... More

Defect 2 spin blocks of symmetric groups and canonical basis coefficientsMay 10 2019This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect $2$, analogous to Richards's formula ... More

Centers of subgroups of big mapping class groups and the Tits alternativeApr 22 2019In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping ... More

A combinatorial proof of the pointwise ergodic theorem for actions of amenable groups along Tempelman Følner SequencesApr 22 2019We generalize A. Tserunyan's proof of the pointwise ergodic theorem for $\mathbb{Z}$ to give a short and combinatorial proof of the pointwise ergodic theorem for actions of amenable groups along Tempelman F{\o}lner sequences. We do this by finding Vitali ... More

Straightening Billiard Trajectories in Flat Disks and Katok-Zemlyakov ConstructionApr 09 2019Apr 11 2019We provide a method to straighten each billiard trajectory in a flat disk. As an application, we show that for each point on the disk and for almost all directions, the closure of the corresponding billiard trajectory contains a vertex. We generalize ... More

Straightening Billiard Trajectories in Flat Disks and Katok-Zemlyakov ConstructionApr 09 2019We provide a method to straighten each billiard trajectory in a flat disk. As an application, we show that for each point on the disk and for almost all directions, the closure of the corresponding billiard trajectory contains a vertex. We generalize ... More

Intrinsic nature of the Stein-Weiss $H^1$-inequalityApr 08 2019This paper explores the intrinsic nature of the celebrated Stein-Weiss $H^1$-inequality $$ \|I_s u\|_{L^\frac{n}{n-s}}\lesssim \|u\|_{L^1}+\|\vec{R}u\|_{L^{1}}=\|u\|_{H^1} $$ through the tracing and duality laws based on Riesz's singular integral operator ... More

Intrinsic nature of the Stein-Weiss $H^1$-inequalityApr 08 2019May 14 2019This paper explores the intrinsic nature of the celebrated Stein-Weiss $H^1$-inequality $$ \|I_s u\|_{L^\frac{n}{n-s}}\lesssim \|u\|_{L^1}+\|\vec{R}u\|_{L^{1}}=\|u\|_{H^1} $$ through the tracing and duality laws based on Riesz's singular integral operator ... More

Quantitative Oppenheim conjecture for $S$-arithmetic quadratic forms of rank $3$ and $4$Apr 04 2019The celebrated result of Eskin, Margulis and Mozes (1998) and Dani and Margulis (1993) on quantitative Oppenheim conjecture says that for irrational quadratic forms $q$ of rank at least 5, the number of integral vectors $\mathbf v$ such that $q(\mathbf ... More

Quantitative Oppenheim conjecture for $S$-arithmetic quadratic forms of rank $3$ and $4$Apr 04 2019Apr 08 2019The celebrated result of Eskin, Margulis and Mozes (1998) and Dani and Margulis (1993) on quantitative Oppenheim conjecture says that for irrational quadratic forms $q$ of rank at least 5, the number of integral vectors $\mathbf v$ such that $q(\mathbf ... More

Elements of a metric spectral theoryApr 02 2019This paper discusses a general method for spectral type theorems using metric spaces instead of vector spaces. Advantages of this approach are that it applies to genuinely non-linear situations and also to random versions. Metric analogs of operator norm, ... More

SRB measures and Young towers for surface diffeomorphismsMar 29 2019We give geometric conditions that are necessary and sufficient for the existence of Sinai-Ruelle-Bowen (SRB) measures for $C^{1+\alpha}$ surface diffeomorphisms, thus proving a version of the Viana conjecture. As part of our argument we give an original ... More

Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphereMar 22 2019Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M,\mathbb{R})$ be a Morse function, and $\Gamma_f$ be its Kronrod-Reeb graph. Denote by $\mathcal{O}_{f}=\{f \circ h \mid h \in \mathcal{D}\}$ the orbit of $f$ with respect to the natural ... More

Dirichlet-to-Neumann maps on TreesMar 22 2019In this paper we study the Dirichlet-to-Neumann map for solutions to mean value formulas on trees. We give two alternative definition of the Dirichlet-to-Neumann map. For the first definition (that involves the product of a "gradient" with a "normal vector") ... More

Transfer operators, atomic decomposition and the BestiaryMar 16 2019Arbieto and S. recently used atomic decomposition to study transfer operators. We give a long list of old and new expanding dynamical systems for which those results can be applied, obtaining the quasi-compactness of transfer operator acting on Besov ... More

Transfer operators and atomic decompositionMar 16 2019Apr 16 2019We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, ... More

Transfer operators and atomic decompositionMar 16 2019We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, ... More

On the correspondence of external rays under renormalizationMar 03 2019Let $P$ be a monic polynomial of degree $D \geq 3$ whose filled Julia set $K_P$ has a non-degenerate periodic component $K$ of period $k \geq 1$ and renormalization degree $2 \leq d<D$. Let $I=I_K$ denote the set of angles $\theta$ on the circle ${\mathbb ... More

Tropical Dynamics of Area-Preserving MapsMar 02 2019We consider a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.

Fatou-Type Theorems and Boundary Value Problems for Elliptic Systems in the Upper Half-SpaceFeb 21 2019We survey recent progress in a program aimed at proving general Fatou-type results and establishing the well-posedness of a variety of boundary value problems in the upper half-space ${\mathbb{R}}^n_{+}$ for second-order, homogeneous, constant complex ... More

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varietiesFeb 16 2019Feb 23 2019We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group ... More

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varietiesFeb 16 2019Apr 14 2019We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group ... More

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varietiesFeb 16 2019We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group ... More

Limits of sequences of pseudo-Anosov maps and of hyperbolic 3-manifoldsFeb 14 2019There are two objects naturally associated with a braid $\beta\in B_n$ of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism $\varphi_\beta\colon S^2\to S^2$; and the finite volume complete hyperbolic structure on the 3-manifold $M_\beta$ obtained ... More

On automorphisms of moduli spaces of parabolic vector bundlesFeb 11 2019Fix $n\geq 5$ general points $p_1, \dots, p_n\in\mathbb{P}^1$, and a weight vector $\mathcal{A} = (a_{1}, \dots, a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{\mathcal{A}}$ parametrizing rank two parabolic vector ... More

Near actionsJan 14 2019A near permutation of a set is a bijection between two cofinite subsets, modulo coincidence on smaller cofinite subsets. Near permutations of a set form its near symmetric group. In this monograph, we define near actions as homomorphisms into this group, ... More

Monotonicity of entropy for real quadratic rational mapsJan 11 2019The monotonicity of entropy is investigated for real quadratic rational maps on the real circle $\mathbb{R}\cup\{\infty\}$ based on the natural partition of the corresponding moduli space $\mathcal{M}_2(\mathbb{R})$ into its monotonic, covering, unimodal ... More

On closed finite gap curves in spaceforms IIJan 10 2019We prove that the set of closed finite gap curves in hyperbolic 3-space $\mathbb{H}^3$ is $W^{2,2}$-dense in the Sobolev space of all closed $W^{2,2}$-curves in $\mathbb{H}^3$. We also show that the set of closed finite gap curves in any 2-dimensional ... More

Polynomial Dynamical Systems, Reaction Networks, and Toric Differential InclusionsJan 08 2019Some of the most common mathematical models in biology, chemistry, physics, and engineering, are polynomial dynamical systems, i.e., systems of differential equations with polynomial right-hand sides. Inspired by notions and results that have been developed ... More

On the rotation sets of generic homeomorphisms on the torus $\mathbb T^d$Jan 02 2019We study the rotation sets for homeomorphisms homotopic to the identity on the torus $\mathbb T^d$, $d\ge 2$. In the conservative setting, we prove that there exists a Baire residual subset of the set $\text{Homeo}_{0, \lambda}(\mathbb T^2)$ of conservative ... More

Clebsch-Lagrange variational principle and geometric constraint analysis of relativistic field theoriesDec 11 2018Inspired by the Clebsch optimal control problem, we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry. Its special feature is that the Lagrange ... More

Foliations and Conjugacy II: The Mendes Conjecture for Time-One Maps of FlowsDec 11 2018A diffeomorphism $f:\mathbb{R}^2\to\mathbb{R}^2$ in the plane is Anosov if it has a hyperbolic splitting at every point of the plane. The two known topological conjugacy classes of such diffeomorphisms are linear hyperbolic automorphisms and translations ... More

Fukaya categories of two-tori revisitedNov 30 2018We construct an A-infinity structure of the Fukaya category explicitly for any flat symplectic two-torus. The structure constants of the non-transversal A-infinity products are obtained as derivatives of those of transversal A-infinity products.

Homeomorphisms of surfaces of finite typeNov 27 2018We give a proof of the Neilsen-Thurston classification theorem of a homeomorphism f of a standard surface of finite type as either periodic, pseudo-Anosov, or reducible. In the periodic case, we show that there exists an integer n>0 such that f is isotopic ... More

Difference and $(Δ)$ properties for some new classesOct 31 2018\begin{abstract} {In this paper we study difference and $(\Delta)$ properties for the classes of the form $C_0(J,X)$, $\frak {g} \U$, $\U+\frak {g} \V$, where $\U, \V\in \{BUC(J,X), UC(J,X)\}$ and $\frak{g} (t)=e^{it^2}$, $t\in \mathbb{R}$. For functions ... More

Geodesics in the mapping class groupOct 30 2018We construct explicit examples of geodesics in the mapping class group and show that the shadow of a geodesic in mapping class group to the curve graph does not have to be a quasi-geodesic. We also show that the quasi-axis of a pseudo-Anosov element of ... More

Analytic solutions of convolution equations on convex sets with a mixed structure. IIOct 05 2018Oct 19 2018We prove conditions for the existence of a continuous linear right inverse for a surjective convolution operator in spaces of germs of analytic functions on convex subsets of the complex plane. Considered convex sets have a countable neighborhood basis ... More

Holomorphic Legendrian curves in projectivised cotangent bundlesSep 25 2018Dec 01 2018We study holomorphic Legendrian curves in the standard complex contact structure on the projectivised cotangent bundle $X=\mathbb P(T^*Z)$ of a complex manifold $Z$ of dimension at least $2$. We provide a detailed analysis of Legendrian curves degenerating ... More

A question of Norton-Sullivan in the analytic caseSep 19 2018In 1996, A. Norton and D. Sullivan asked the following question: If $f:\mathbb{T}^2\rightarrow\mathbb{T}^2$ is a diffeomorphism, $h:\mathbb{T}^2\rightarrow\mathbb{T}^2$ is a continuous map homotopic to the identity, and $h f=T_{\rho} h$ where $\rho\in\mathbb{R}^2$ ... More

On the transversal dependence of weak K.A.M. solutions for symplectic twist mapsSep 07 2018For a symplectic twist map, we prove that there is a choice of weak K.A.M. solutions that depend in a continuous way on the cohomology class. We thus obtain a continuous function $u(\theta, c)$ in two variables: the angle $\theta$ and the cohomology class ... More

Dolbeault cohomology of complex nilmanifolds foliated in toroidal groupsAug 24 2018It is conjectured that the Dolbeault cohomology of a complex nilmanifold $X$ is computed by left-invariant forms. We prove this under the assumption that $X$ is suitably foliated in toroidal groups and deduce that the conjecture holds in real dimension ... More

Convenient partial Poisson manifoldsAug 08 2018Nov 04 2018We introduce the concept of partial Poisson structure on a manifold $M$ modelled on a convenient space. This is done by specifying a (weak) subbundle $T^{\prime}M$ of $T^{\ast}M$ and an antisymmetric morphism $P:T^{\prime}M\rightarrow TM$ such that the ... More

Minimal Penner dilatations on nonorientable surfacesJul 24 2018For any nonorientable closed surface, we determine the minimal dilatation among pseudo-Anosov mapping classes arising from Penner's construction. We deduce that the sequence of minimal Penner dilatations has exactly two accumulation points, in contrast ... More

Invariant multi-graphs in step skew-productsJul 23 2018We study step skew-products over a finite-state shift (base) space whose fiber maps are $C^1$ injective maps on the unit interval. We show that certain invariant sets have a multi-graph structure and can be written graphs of one, two or more functions ... More

Bowen factors, their degree, and codings of surface diffeomorphismsJul 11 2018We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce H\"older-continuous conjugacies on large sets, sometimes preserving transitivity. We deduce this from their Bowen property. This notion, ... More

On the Measure of Maximal Entropy for Finite Horizon Sinai Billiard MapsJul 06 2018The Sinai billiard map $T$ on the two-torus, i.e., the periodic Lorentz gas, is a discontinuous map. Assuming finite horizon, we propose a definition $h_*$ for the topological entropy of $T$. We prove that $h_*$ is not smaller than the value given by ... More

Lagrangian Shadows and Triangulated CategoriesJun 18 2018Under certain assumptions (such as weak exacteness or monotonicity) we show that splitting Lagrangians through cobordism has an energy cost and, from this cost being smaller than certain explicit bounds, we deduce some strong forms of rigidity of Lagrangian ... More

Ilyashenko algebras based on transserial asymptotic expansionsJun 05 2018Jan 08 2019We construct a Hardy field that contains Ilyashenko's class of germs at infinity of almost regular functions as well as all log-exp-analytic germs. In addition, each germ in this Hardy field is uniquely characterized by an asymptotic expansion that is ... More

Limiting Measure of Lee--Yang Zeros for the Cayley TreeJun 01 2018Jan 07 2019This paper is devoted to an in-depth study of the limiting measure of Lee--Yang zeroes for the Ising Model on the Cayley Tree. We build on previous works of M\"uller-Hartmann-Zittartz (1974 and 1977), Barata--Marchetti (1997), and Barata--Goldbaum (2001), ... More

Mathematical Analysis of Chemical Reaction SystemsMay 25 2018The use of mathematical methods for the analysis of chemical reaction systems has a very long history, and involves many types of models: deterministic versus stochastic, continuous versus discrete, and homogeneous versus spatially distributed. Here we ... More

Quasisymmetric rigidity in one-dimensional dynamicsMay 23 2018In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conjugate are quasisymmetrically conjugate (provided some obvious necessary assumptions are ... More

A descriptive set theorist's proof of the pointwise ergodic theoremMay 18 2018Jun 18 2018We give a short combinatorial proof of the classical pointwise ergodic theorem for probability measure preserving $\mathbb{Z}$-actions. Our approach reduces the theorem to a tiling problem: tightly tile each orbit by intervals with desired averages. This ... More

Normal generators for mapping class groups are abundantMay 09 2018Jul 09 2018We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution ... More

Pointwise ergodic theorem for locally countable quasi-pmp graphsMay 07 2018Jun 21 2018We prove a pointwise ergodic theorem for quasi-probability-measure-preserving (quasi-pmp) locally countable measurable graphs, analogous to pointwise ergodic theorems for group actions, replacing the group with a Schreier graph of the action. For any ... More

On homeomorphisms and $C^{1}$ mapsApr 27 2018Our purpose in this article is first, following [8], to prove that if $\alpha $, $\beta $ are any points of the open unit disc $D(0;1)$ in the complex plane ${\bf C}$ and $r$, $s$ are any positive real numbers such that ${\overline{D}}( \alpha ;r) \subseteq ... More

Explicit solutions to utility maximization problems in a regime-switching market model via Laplace transformsApr 23 2018We study the problem of utility maximization from terminal wealth in which an agent optimally builds her portfolio by investing in a bond and a risky asset. The asset price dynamics follow a diffusion process with regime-switching coefficients modeled ... More

Moduli of surface diffeomorphisms with cubic tangenciesApr 23 2018In this paper, we study conjugacy invariants for 2-dimensional diffeomorphisms with homoclinic cubic tangencies (two-sided tangencies of the lowest order) under certain open conditions. Ordinary arguments used in past studies of conjugacy invariants associated ... More

Duality and approximation of Bergman spacesApr 08 2018Nov 15 2018Expected duality and approximation properties are shown to fail on Bergman spaces of domains in $\mathbb{C}^n$, via examples. When the domain admits an operator satisfying certain mapping properties, positive duality and approximation results are proved. ... More

A fixed point theorem for plane homeomorphisms with the topological shadowing propertyApr 06 2018Apr 24 2019In this paper we show that every homeomorphism of the plane with the topological shadowing property has a fixed point. Also, we show that a linear isomorphism of an Euclidean space has the topological shadowing property if and only if the origin is an ... More

A fixed point theorem for Topologically Anosov Plane homeomorphismsApr 06 2018Apr 09 2018Certain notions of expansivity and shadowing were defined on topological spaces which are dynamical properties and generalize the usual definitions. A Topologically Anosov homeomorphism is a homeomorphism with such properties. We exhibit explicit examples ... More

The $β$-transformation with a hole at 0Mar 20 2018For $\beta\in(1,2]$ the $\beta$-transformation $T_\beta: [0,1) \to [0,1)$ is defined by $T_\beta ( x) = \beta x \pmod 1$. For $t\in[0, 1)$ let $K_\beta(t)$ be the survivor set of $T_\beta$ with hole $(0,t)$ given by \[K_\beta(t):=\{x\in[0, 1): T_\beta^n(x)\not ... More

The compression body graph has infinite diameterMar 16 2018We show that the compression body graph has infinite diameter, and that every subgroup in the Johnson filtration of the mapping class group contains elements which act loxodromically on the compression body graph. Our methods give an alternate proof of ... More

Lyapunov exponents and partial hyperbolicity of chain control sets on flag manifoldsMar 14 2018Nov 03 2018For a right-invariant control system on a flag manifold $\mathbb{F}_{\Theta}$ of a real semisimple Lie group, we relate the $\mathfrak{a}$-Lyapunov exponents to the Lyapunov exponents of the system over regular points. Moreover, we adapt the concept of ... More

Topological horseshoes for surface homeomorphismsMar 12 2018In this work we develop a new criterion for the existence of topological horseshoes for surface homeomorphisms in the isotopy class of the identity. Based on our previous work on forcing theory, this new criterion is purely topological and can be expressed ... More

Real entropy rigidity under quasi-conformal deformationsMar 12 2018May 02 2019We set up a real entropy function $h_\Bbb{R}$ on the space $\mathcal{M}'_d$ of M\"obius conjugacy classes of real rational maps of degree $d$ by assigning to each class the real entropy of a representative $f\in\Bbb{R}(z)$; namely, the topological entropy ... More

Real entropy rigidity under quasi-conformal deformationsMar 12 2018We set up a real entropy function $h_\Bbb{R}$ on the space $\mathcal{M}'_d$ of M\"obius conjugacy classes of real rational maps of degree $d$ by assigning to each class the real entropy of a representative $f\in\Bbb{R}(z)$; namely, the topological entropy ... More

A user-friendly condition for exponential ergodicity in randomly switched environmentsMar 09 2018Jun 30 2018We consider random switching between finitely many vector fields leaving positively invariant a compact set. Recently, Li, Liu and Cui showed that if one the vector fields has a globally asymptotically stable (G.A.S.) equilibrium from which one can reach ... More

An infinite surface with the lattice property II: Dynamics of pseudo-AnosovsMar 05 2018Jul 19 2018We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are ... More

Modified Gauss-Newton method in low-rank signal estimationMar 04 2018The paper is devoted to the solution of a weighted non-linear least-squares problem for low-rank signal estimation, which is related Hankel structured low-rank approximation problems. The solution is constructed by a modified weighted Gauss-Newton method. ... More

Almost uniform and strong convergences in ergodic theorems for symmetric spacesFeb 20 2018Let $(\Omega,\mu)$ be a $\sigma$-finite measure space, and let $X\subset L^1(\Omega)+L^\infty(\Omega)$ be a fully symmetric space of measurable functions on $(\Omega,\mu)$. If $\mu(\Omega)=\infty$, necessary and sufficient conditions are given for almost ... More

Point island dynamics under fixed rate depositionFeb 15 2018Dec 12 2018In this paper we consider the dynamics of point islands during submonolayer deposition, in which the fragmentation of subcritical size islands is allowed. To understand asymptotics of solutions, we use methods of centre manifold theory, and for globalisation, ... More

On a generalized theorem of de Bruijn and Erdös in d-dimensional Fuzzy Linear SpacesFeb 13 2018In this study we follow a new framework for the theory that offers us, other than traditional, a new angle to observe and investigate some relations between finite sets, F-lattice L and their elements. The theory is based on the Fuzzy Linear Spaces (FLS) ... More

Minimal asymptotic translation lengths of Torelli groups and pure braid groups on the curve graphFeb 13 2018Nov 01 2018In this paper, we show that the minimal asymptotic translation length of the Torelli group $\mathcal{I}_g$ of the surface $S_g$ of genus $g$ on the curve graph asymptotically behaves like $1/g$, contrary to the mapping class group $Mod(S_g)$, which behaves ... More

Ball Prolate Spheroidal Wave Functions In Arbitrary DimensionsFeb 11 2018In this paper, we introduce the prolate spheroidal wave functions (PSWFs) of real order $\alpha>-1$ on the unit ball in arbitrary dimension, termed as ball PSWFs. They are eigenfunctions of both a weighted concentration integral operator, and a Sturm-Liouville ... More

On closed finite gap curves in spaceforms IJan 22 2018Jun 12 2019We show that the spaces of closed finite gap curves in $R^3$ and $S^3$ are dense with respect to the Sobolev $W^{1,2}$-norm in the spaces of closed curves in $R^3$ respectively $S^3$.

On closed finite gap curves in spaceforms IJan 22 2018Jan 10 2019We show that the spaces of closed finite gap curves in $R^3$ and $S^3$ are dense with respect to the Sobolev $W^{1,2}$-norm in the spaces of closed curves in $R^3$ respectively $S^3$.

An upper bound on the asymptotic translation lengths on the curve graph and fibered facesJan 20 2018We study the asymptotic behavior of the asymptotic translation lengths on the curve complexes of pseudo-Anosov monodromies in a fibered cone of a fibered hyperbolic 3-manifold $M$ with $b_1(M) \geq 2$. For a sequence $(\Sigma_n, \psi_n)$ of fibers and ... More

A forcing relation of braids from Nielsen fixed point theoryJan 14 2018In this paper, we focus our attention on the connections between the braid group and the Nielsen fixed point theory. A new forcing relation between braids is introduced, and shown that it can be fulfilled by using Nielsen fixed point theory.

Pseudo-Anosov maps with small stretch factorsJan 05 2018Jan 11 2018We consider the problem of estimating the minimum entropy of pseudo-Anosov maps on a surface of genus g with n punctures. We determine the behaviour of this minimum number on a two-dimensional subset of the (g,n) plane, up to a multiplicative constant. ... More

Pseudo-Anosov maps with small stretch factors on punctured surfacesJan 05 2018Apr 30 2019Consider the problem of estimating the minimum entropy of pseudo-Anosov maps on a surface of genus $g$ with $n$ punctures. We determine the behaviour of this minimum number for a certain large subset of the $(g,n)$ plane, up to a multiplicative constant. ... More

Non-isometric domains with the same Marvizi-Melrose invariantsJan 03 2018For any strictly convex planar domain $\Omega \subset \mathbb{R}^2$ with a $C^\infty$ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi-Merlose. These invariants can generically be determined using the spectrum ... More

Boundedness of the Bergman projection on generalized Fock-Sobolev spaces on ${\mathbb C}^n$Dec 14 2017In this paper we solve a problem posed by H. Bommier-Hato, M. Engli\v{s} and E.H. Youssfi in [3] on the boundedness of the Bergman-type projections in generalized Fock spaces. It will be a consequence of two facts: a full description of the embeddings ... More

On periodic groups of homeomorphisms of the 2-dimensional sphereDec 06 2017We prove that every finitely-generated group of homeomorphisms of the 2-dimensional sphere all of whose elements have a finite order which is a power of 2 and so that there exists a uniform bound for the order of group elements is finite. We prove a similar ... More

Ergodic optimization of Birkhoff averages and Lyapunov exponentsDec 05 2017Apr 23 2018Ergodic optimization is the study of extremal values of asymptotic dynamical quantities such as Birkhoff averages or Lyapunov exponents, and of the orbits or invariant measures that attain them. We discuss some results and problems.

Lower bound for the Perron-Frobenius degrees of Perron numbersNov 18 2017Jan 26 2019Using an idea of Doug Lind, we give a lower bound for the Perron-Frobenius degree of a Perron number that is not totally-real. As an application, we prove that there are cubic Perron numbers whose Perron-Frobenius degrees are arbitrary large; a result ... More

On 2d-4d motivic wall-crossing formulasNov 10 2017In this paper we propose definitions and examples of categorical enhancements of the data involved in the $2d$-$4d$ wall-crossing formulas which generalize both Cecotti-Vafa and Kontsevich-Soibelman motivic wall-crossing formulas.

Strict versions of various matrix hierarchies related to SL(n)-loops and their combinationsNov 06 2017Let $\mathfrak{t}$ be a commutative Lie subalgebra of ${\rm sl}_{n}(\mathbb{C})$ of maximal dimension. We consider in this paper three spaces of $\mathfrak{t}$-loops that each get deformed in a different way. We require that the deformed generators of ... More

A triple boundary lemma for surface homeomorphismsNov 02 2017Jun 02 2018Given an orientation-preserving and area-preserving homeomorphism $f$ of the sphere, we prove that every point which is in the common boundary of three pairwise disjoint invariant open topological disks must be a fixed point. As an application, if $K$ ... More

Algebraic integrability of foliations with numerically trivial canonical bundleOct 17 2017Apr 23 2018Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical bundle such ... More

Linking combinatorial and classical dynamics: Conley index and Morse decompositionsOct 16 2017We prove that every combinatorial dynamical system in the sense of Forman, defined on a family of simplices of a simplicial complex, gives rise to a multivalued dynamical system F on the geometric realization of the simplicial complex. Moreover, F may ... More

Quantum-classical correspondence on associated vector bundles over locally symmetric spacesOct 12 2017Jul 10 2018For a compact Riemannian locally symmetric space $\mathcal M$ of rank one and an associated vector bundle $\mathbf V_\tau$ over the unit cosphere bundle $S^\ast\mathcal M$, we give a precise description of those classical (Pollicott-Ruelle) resonant states ... More

Quantum-classical correspondence on associated vector bundles over locally symmetric spacesOct 12 2017Mar 12 2019For a compact Riemannian locally symmetric space $\mathcal M$ of rank one and an associated vector bundle $\mathbf V_\tau$ over the unit cosphere bundle $S^\ast\mathcal M$, we give a precise description of those classical (Pollicott-Ruelle) resonant states ... More

The Fatou coordinate for parabolic Dulac germsOct 02 2017Jul 13 2018We study the class of parabolic Dulac germs of hyperbolic polycycles. For such germs we give a constructive proof of the existence of a unique Fatou coordinate, admitting an asymptotic expansion in the power-iterated log scale.

Geodesic intersections and isoxial Fuchsian groupsSep 26 2017The set of axes of hyperbolic elements in a Fuchsian group depends on the commensurability class of the group. In fact, it has been conjectured that it determines the commensurability class and this has been verified in for groups of the second kind by ... More

Geodesics Currents and Counting ProblemsSep 20 2017Mar 23 2019For every positive, continuous and homogeneous function $f$ on the space of currents on a compact surface $\overline{\Sigma}$, and for every compactly supported filling current $\alpha$, we compute as $L \to \infty$, the number of mapping classes $\phi$ ... More

Geodesics Currents and Counting ProblemsSep 20 2017For every positive, continuous and homogeneous function $f$ on the space of currents on a closed surface $\Sigma$, and for every filling current $\alpha$, we compute as $L \to \infty$, the number of mapping classes $\phi$ so that $f(\phi(\alpha))\leq ... More