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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 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 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 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 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
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 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
On Lagrange polynomials and the rate of approximation of planar sets by polynomial Julia setsSep 19 2017Apr 06 2018We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed by Lindsey and Younsi and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show ... More
A fresh look into monoid rings and formal power series ringsSep 05 2017Feb 22 2018In this article, the ring of polynomials is studied in a systematic way through the theory of monoid rings. As a consequence, this study provides natural and canonical approaches in order to find easy and rigorous proofs and methods for many facts on ... More
Infinitely Many Moduli of Stability at the Dissipative Boundary of ChaosSep 04 2017In the family of area-contracting H\'enon-like maps with zero topological entropy we show that there are maps with infinitely many moduli of stability. Thus one cannot find all the possible topological types for non-chaotic area-contracting H\'enon-like ... More
Existence of an attractor for a geometric tetrahedron transformationAug 28 2017We analyze the dynamical properties of a tetrahedron transformation on the space of non-degenerate tetrahedra which can be identified with the non-compact globally symmetric $8$-dimensional space $\mbox{Sl}(3,\mathbb{R}) / \mbox{So}(3,\mathbb{R})$. We ... More
The strict AKNS hierarchy: its structure and solutionsAug 22 2017In this paper we discuss an integrable hierarchy of compatible Lax equations that is obtained by a wider deformation of a commutative algebra in the loop space of ${\rm sl}_{2}$ than that in the AKNS-case and whose Lax equations are based on a different ... More
The inverse problem of the calculus of variations for discrete systemsAug 14 2017We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also provide a transition ... More
Equidistribution in measure-preserving actions of semisimple groups : case of $SL_2(\mathbb{R})$Aug 13 2017We prove pointwise convergence for the semi-radial averages on $G=SL_2(\mathbb{R})$ given by $\int_t^{t+1} m_K\ast \delta_{a_{s}}ds$ (and similar variants), acting on $K$-finite $L^p$-functions in a probability-measure-preserving action of the group, ... More
The rigidity of pseudo-rotations on the two-torus and a question of Norton-SullivanAug 08 2017We show that under certain boundedness condition, a $C^{r}$ conservative irrational pseudo-rotations on $\mathbb{T}^2$ with a generic rotation vector is $C^{r-1}$-rigid. We also obtain $C^0$-rigidity for H\"older pseudo-rotations with similar properties. ... More
Weil-Petersson geometry on the space of Bridgeland stability conditionsAug 07 2017Dec 22 2017Inspired by mirror symmetry, we investigate some differential geometric aspects of the space of Bridgeland stability conditions on a Calabi-Yau triangulated category. The aim is to develop theory of Weil-Petersson geometry on the stringy K\"ahler moduli ... More
Smooth invariant densities for random switching on the torusAug 04 2017Mar 21 2018We consider a random dynamical system obtained by switching between the flows generated by two smooth vector fields on the 2d-torus, with the random switchings happening according to a Poisson process. Assuming that the driving vector fields are transversal ... More
Small asymptotic translation lengths of pseudo-Anosov maps on the curve complexJul 19 2017Oct 02 2018Let $M$ be a hyperbolic fibered 3-manifold with $b_1(M) \geq 2$ and let $S$ be a fiber with pseudo-Anosov monodromy $\psi$. We show that there exists a sequence $(R_n, \psi_n)$ of fibers and monodromies contained in the fibered cone of $(S,\psi)$ such ... More
Delay-coordinate maps and the spectra of Koopman operatorsJun 26 2017Nov 05 2018The Koopman operator induced by a dynamical system is inherently linear and provides an alternate method of studying many properties of the system, including attractor reconstruction and forecasting. Koopman eigenfunctions represent the non-mixing component ... More
The Bruinier--Funke pairing and the orthogonal complement of unary theta functionsJun 23 2017We describe an algorithm for computing the inner product between a holomorphic modular form and a unary theta function, in order to determine whether the form is orthogonal to unary theta functions without needing a basis of the entire space of modular ... More
On polynomially integrable Birkhoff billiards on surfaces of constant curvatureJun 13 2017Feb 22 2019We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result to billiards ... More
Lozi-like mapsMay 23 2017We define a broad class of piecewise smooth plane homeomorphisms which have properties similar to the properties of Lozi maps, including the existence of a hyperbolic attractor. We call those maps Lozi-like. For those maps one can apply our previous results ... More
Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon MapsMay 14 2017Feb 05 2018This article extends the theorem of the absence of wandering domains from unimodal maps to infinitely period-doubling renormalizable H\'enon-like maps in the strongly dissipative (area contracting) regime. The theorem solves an open problem proposed by ... More
The structure of random homeomorphismsMay 04 2017Aug 16 2018In order to understand the structure of the "typical" element of a homeomorphism group, one has to study how large the conjugacy classes of the group are. When typical means generic in the sense of Baire category, this is well understood, see e.g. the ... More
On Minkowski type question mark functions associated with even or odd continued fractionsMay 03 2017Jun 06 2018We study analogues of Minkowski's question mark function $?(x)$ related to continued fractions with even or odd partial quotients. We prove that these functions are H\"older continuous with precise exponents, and that they linearize the appropriate versions ... More
Approximation of Periodic PDE Solutions with Anisotropic Translation Invariant SpacesMay 02 2017We approximate the quasi-static equation of linear elasticity in translation invariant spaces on the torus. This unifies different FFT-based discretisation methods into a common framework and extends them to anisotropic lattices. We analyse the connection ... More
Natural extensions of unimodal maps: virtual sphere homeomorphisms and prime ends of basin boundariesApr 21 2017Sep 19 2018Let $\{f_t\colon I\to I\}$ be a family of unimodal maps with topological entropies $h(f_t)>\frac12\log 2$, and ${\widehat{f}}_t\colon{\widehat{I}}_t\to{\widehat{I}}_t$ be their natural extensions, where ${\widehat{I}}_t=\varprojlim(I,f_t)$. Subject to ... More
On the Unique Ergodicity of Quadratic Differentials and the Orientation Double CoverApr 20 2017Nov 04 2018We construct an example of a uniquely ergodic measured foliation on a surface such that the associated translation flow on the orientation double cover is minimal but not uniquely ergodic. We then prove a geometric criterion for the horizontal foliation ... More
Rotational deviations and invariant pseudo-foliations for periodic point free torus homeomorphismsApr 16 2017Mar 11 2018This article deals with directional rotational deviations for non-wandering periodic point free homeomorphisms of the 2-torus which are homotopic to the identity. We prove that under mild assumptions, such a homeomorphism exhibits uniformly bounded rotational ... More
Integral representation of binary quadratic forms over rational function fieldsApr 06 2017For diophantine equations of the form ax^2+bxy+cy^2+g=0 over k[t], the ring of integers of rational funtional fields, we show that a condition with respect to the Artin reciprocity map, which we call the Artin condition, is the only obstruction to the ... More
Weight multiplicities and Young tableaux through affine crystalsMar 30 2017The weight multiplicities of finite dimensional simple Lie algebras can be computed individually using various methods. Still, it is hard to derive explicit closed formulas. Similarly, explicit closed formulas for the multiplicities of maximal weights ... More
Quasi-isometry type of the metric space derived from the kernel of the Calabi homomorphismMar 27 2017We prove that the set of symmetrized conjugacy classes of the kernel of the Calabi homomorphism on the group of area-preserving diffeomorphisms of the $2$-disk is not quasi-isometric to the half line.
An algorithm to compute the Teichmueller polynomial from matricesMar 27 2017Mar 12 2018In their precedent work, the authors constructed closed oriented hyperbolic surfaces with pseudo-Anosov homeomorphisms from certain class of integral matrices. In this paper, we present a very simple algorithm to compute the Teichmueller polynomial corresponding ... More
Symmetries and conservation laws of a nonlinear sigma model with gravitinoMar 20 2017We show that the action functional of the nonlinear sigma model with gravitino considered in a previous article [18] is invariant under rescaled conformal transformations, super Weyl transformations and diffeomorphisms. We give a careful geometric explanation ... More