28 results on '"Kariane Calta"'
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2. Infinitely many lattice surfaces with special pseudo-Anosov maps
- Author
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Thomas A. Schmidt and Kariane Calta
- Subjects
Large class ,37D99 ,Pure mathematics ,Mathematics::Dynamical Systems ,Algebra and Number Theory ,Applied Mathematics ,Dynamical Systems (math.DS) ,Mathematics::Geometric Topology ,Translation surface ,FOS: Mathematics ,Affine transformation ,Mathematics - Dynamical Systems ,Invariant (mathematics) ,Analysis ,Mathematics - Abstract
We give explicit pseudo-Anosov homeomorphisms with vanishing Sah-Arnoux-Fathi invariant. Any translation surface whose Veech group is commensurable to any of a large class of triangle groups is shown to have an affine pseudo-Anosov homeomorphism of this type. We also apply a reduction to finite triangle groups and thereby show the existence of non-parabolic elements in the periodic field of certain translation surfaces., Comment: 13 pages, 2 figures
- Published
- 2013
- Full Text
- View/download PDF
3. CONTINUED FRACTIONS FOR A CLASS OF TRIANGLE GROUPS
- Author
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Kariane Calta and Thomas A. Schmidt
- Subjects
Fuchsian group ,Discrete mathematics ,Pure mathematics ,General Mathematics ,Euler's continued fraction formula ,Diophantine approximation ,symbols.namesake ,Conjugacy class ,Modular group ,symbols ,Periodic continued fraction ,Generalized continued fraction ,Continued fraction ,Mathematics - Abstract
We give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature $(3, n, \infty )$, with $n\geq 4$. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results.
- Published
- 2012
- Full Text
- View/download PDF
4. On unipotent flows in ℋ(1,1)
- Author
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Kariane Calta and Kevin Wortman
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Dynamical Systems ,Applied Mathematics ,General Mathematics ,Unipotent ,Moduli space ,Mathematics::Algebraic Geometry ,Horocycle ,Flow (mathematics) ,Ergodic theory ,Abelian group ,Invariant (mathematics) ,Mathematics ,Probability measure - Abstract
We study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures that are invariant and ergodic under the horocycle flow on the stratum ℋ(1,1).
- Published
- 2009
- Full Text
- View/download PDF
5. The 𝐽-invariant, exceptional surfaces and notions of periodicity
- Author
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Kariane Calta and John Smillie
- Published
- 2007
- Full Text
- View/download PDF
6. Veech surfaces and complete periodicity in genus two
- Author
-
Kariane Calta
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Genus (mathematics) ,Translation surface ,Calculus ,Mathematics - Abstract
We present several results pertaining to Veech surfaces and completely periodic translation surfaces in genus two. A translation surface is a pair ( M , ω ) (M, \omega ) where M M is a Riemann surface and ω \omega is an Abelian differential on M M . Equivalently, a translation surface is a two-manifold which has transition functions which are translations and a finite number of conical singularities arising from the zeros of ω \omega . A direction v v on a translation surface is completely periodic if any trajectory in the direction v v is either closed or ends in a singularity, i.e., if the surface decomposes as a union of cylinders in the direction v v . Then, we say that a translation surface is completely periodic if any direction in which there is at least one cylinder of closed trajectories is completely periodic. There is an action of the group S L ( 2 , R ) SL(2, \mathbb {R}) on the space of translation surfaces. A surface which has a lattice stabilizer under this action is said to be Veech. Veech proved that any Veech surface is completely periodic, but the converse is false. In this paper, we use the J J -invariant of Kenyon and Smillie to obtain a classification of all Veech surfaces in the space H ( 2 ) {\mathcal H}(2) of genus two translation surfaces with corresponding Abelian differentials which have a single double zero. Furthermore, we obtain a classification of all completely periodic surfaces in genus two.
- Published
- 2004
- Full Text
- View/download PDF
7. Algebraically periodic translation surfaces
- Author
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John Smillie and Kariane Calta
- Subjects
37D50 ,57M50 ,Algebra and Number Theory ,j-invariant ,Applied Mathematics ,Mathematical analysis ,General Topology (math.GN) ,Periodic sequence ,Dynamical Systems (math.DS) ,Algebraic number field ,Translation (geometry) ,Automorphism ,Translation surface ,FOS: Mathematics ,Algebraic number ,Mathematics - Dynamical Systems ,Slope field ,Analysis ,Mathematics ,Mathematics - General Topology - Abstract
Algebraically periodic directions on translation surfaces were introduced by Calta in her study of genus two translation surfaces. We say that a translation surface with three or more algebraically periodic directions is an algebraically periodic surface. We show that for an algebraically periodic surface the slopes of the algebraically periodic directions are given by a number field which we call the periodic direction field. We show that translation surfaces with pseudo-Anosov automorphisms provide examples. In this case the periodic direction field is the holonomy field. We show that every algebraic field arises as the periodic direction field of a translation surface arising from a right-angled billiard table. The J-invariant of a translation surface was introduced by Kenyon and Smillie. We analyze the $J$ invariants of algebraically periodic surfaces and show that in some cases they are determined by the periodic direction field. We give explicit formulas for $J$ invariants in these cases. The Homological Affine Group was introduced by McMullen in his study of translation surfaces in genus two. We calculate this group for many algebraically periodic surfaces and relate it to the automorphism group of the J-invariant. We show that surfaces which admit certain decompositions into squares have totally real periodic direction field. This is related to a result of Hubert and Lanneau., typos and errors corrected
- Published
- 2007
8. Determinantal conditions for modules of generalized splines.
- Author
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Calta, Kariane and Rose, Lauren L.
- Subjects
COMMUTATIVE algebra ,INTEGRAL domains ,NUMBER theory ,GRAPH theory ,SPLINES - Abstract
Generalized splines on a graph G with edge labels in a commutative ring R are vertex labelings such that if two vertices share an edge in G , the difference between the vertex labels lies in the ideal generated by the edge label. When R is an integral domain, the set of all such splines is a finitely generated R -module R G of rank n , the number of vertices of G. We find determinantal conditions on subsets of R G that determine whether R G is a free module, and if so, whether a so-called "flow-up class basis" exists. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
9. Basis condition for generalized spline modules.
- Author
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Fişekci, Seher and Sarıoğlan, Samet
- Abstract
A generalized spline on an edge-labeled graph (G , α) is defined as a vertex labeling, such that the difference of labels on adjacent vertices lies in the ideal generated by the edge label. We study generalized splines over greatest common divisor domains and present a determinantal basis condition for generalized spline modules on arbitrary graphs. The main result of the paper answers a conjecture that appeared in several papers. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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10. A combinatorial approach to Rauzy-type dynamics II: The labelling method and a second proof of the KZB classification theorem.
- Author
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Mourgues, Quentin De
- Subjects
POINCARE maps (Mathematics) ,ALGEBRAIC geometry ,DYNAMICAL systems ,COMBINATORICS ,CLASSIFICATION - Abstract
Rauzy-type dynamics are group actions on a collection of combinatorial objects. The first and best known example (the Rauzy dynamics) concerns an action on permutations, associated to interval exchange transformations (IET) for the Poincaré map on compact orientable translation surfaces. The equivalence classes on the objects induced by the group action have been classified by Kontsevich and Zorich, and by Boissy through methods involving both combinatorics algebraic geometry, topology and dynamical systems. Our precedent paper [ 5 ] as well as the one of Fickenscher [ 8 ] proposed an ad hoc combinatorial proof of this classification. However, unlike those two previous combinatorial proofs, we develop in this paper a general method, called the labelling method, which allows one to classify Rauzy-type dynamics in a much more systematic way. We apply the method to the Rauzy dynamics and obtain a third combinatorial proof of the classification. The method is versatile and will be used to classify three other Rauzy-type dynamics in follow-up works. Another feature of this paper is to introduce an algorithmic method to work with the sign invariant of the Rauzy dynamics. With this method, we can prove most of the identities appearing in the literature so far ([ 10 ], [ 6 ], [ 2 ], [ 5 ]...) in an automatic way. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
11. Pure rolling motion of hyperquadrics in pseudo-Euclidean spaces.
- Author
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Marques, André and Leite, Fátima Silva
- Subjects
NONHOLONOMIC constraints ,LIE groups ,SUBMANIFOLDS ,CARLEMAN theorem - Abstract
This paper is devoted to rolling motions of one manifold over another of equal dimension, subject to the nonholonomic constraints of no-slip and no-twist, assuming that these motions occur inside a pseudo-Euclidean space. We first introduce a definition of rolling map adjusted to this situation, which generalizes the classical definition of Sharpe [ 26 ] for submanifolds of an Euclidean space. We also prove some important properties of these rolling maps. After presenting the general framework, we analyse the particular rolling of hyperquadrics embedded in pseudo-Euclidean spaces. The central topic is the rolling of a pseudo-hyperbolic space over the affine space associated with its tangent space at a point. We derive the kinematic equations, as well as the corresponding explicit solutions for two specific cases, and prove the existence of a rolling map along any curve in that rolling space. Rolling of a pseudo-hyperbolic space on another and rolling of pseudo-spheres are equally treated. Finally, for the central theme, we write the kinematic equations as a control system evolving on a certain Lie group and prove its controllability. The choice of the controls corresponds to the choice of a rolling curve. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
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12. VEECH SURFACES AND COMPLETE PERIODICITY IN GENUS TWO.
- Published
- 2004
13. A note on the lattice structure for matching markets via linear programming.
- Author
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Neme, Pablo and Oviedo, Jorge
- Subjects
MARKET design & structure (Economics) ,MATCHING theory - Abstract
Given two stable matchings in a many-to-one matching market with q-responsive preferences, by manipulating the objective function of the linear program that characterizes the stable matching set, we compute the least upper bound and greatest lower bound between them. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
14. Thurston's algorithm and rational maps from quadratic polynomial matings.
- Author
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Wilkerson, Mary
- Subjects
POLYNOMIALS ,ALGORITHMS ,SPHERES ,EVIDENCE ,ITERATIVE methods (Mathematics) ,RATIONAL points (Geometry) - Abstract
Topological mating is a combination that takes two same-degree polynomials and produces a new map with dynamics inherited from this initial pair. This process frequently yields a map that is Thurston-equivalent to a rational map F on the Riemann sphere. Given a pair of polynomials of the form z
2 + c that are postcritically finite, there is a fast test on the constant parameters to determine whether this map F exists-but this test does not give a construction of F. We present an iterative method that utilizes finite subdivision rules and Thurston's algorithm to approximate this rational map, F. This manuscript expands upon results given by the Medusa algorithm in [9]. We provide a proof of the algorithm's efficacy, details on its implementation, the settings in which it is most successful, and examples generated with the algorithm. [ABSTRACT FROM AUTHOR]- Published
- 2019
- Full Text
- View/download PDF
15. WEAK MIXING DIRECTIONS IN NON-ARITHMETIC VEECH SURFACES.
- Author
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AVILA, ARTUR and DELECROIX, VINCENT
- Subjects
POLYGONS ,REAL numbers ,ARITHMETIC ,NUMBER systems ,LATTICE theory - Abstract
The article focuses on the weak mixing directions in non-arithmatic regular polygons. It is stated that weak mixing can be interpreted as the complete breakdown of the nice lattice behavior and means that there is no remainder of periodicity or quasiperiodicity from the measurable point of view. It is noted that weak mixing might hold outside a countable set of exceptions and is not only a prevalent property.
- Published
- 2016
- Full Text
- View/download PDF
16. Problems in dynamical systems and related topics.
- Published
- 2007
- Full Text
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17. Fusion: a general framework for hierarchical tilings of $$\mathbb{R }^d$$.
- Author
-
Frank, Natalie and Sadun, Lorenzo
- Abstract
We introduce a formalism for handling general spaces of hierarchical tilings, a category that includes substitution tilings, Bratteli-Vershik systems, S-adic transformations, and multi-dimensional cut-and-stack transformations. We explore ergodic, spectral and topological properties of these spaces. We show that familiar properties of substitution tilings carry over under appropriate assumptions, and give counter-examples where these assumptions are not met. For instance, we exhibit a minimal tiling space that is not uniquely ergodic, with one ergodic measure having pure point spectrum and another ergodic measure having mixed spectrum. We also exhibit a 2-dimensional tiling space that has pure point measure-theoretic spectrum but is topologically weakly mixing. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
18. Transcendence with Rosen continued fractions.
- Author
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Bugeaud, Yann, Hubert, Pascal, and Schmidt, Thomas A.
- Subjects
FRACTIONS ,REAL numbers ,ALGEBRAIC numbers ,LIOUVILLE'S theorem ,HECKE algebras - Abstract
We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers. [ABSTRACT FROM AUTHOR]
- Published
- 2013
- Full Text
- View/download PDF
19. CONTINUED FRACTIONS FOR A CLASS OF TRIANGLE GROUPS.
- Author
-
CALTA, KARIANE and SCHMIDT, THOMAS A.
- Subjects
CONTINUED fractions ,TRIANGLES ,FUCHSIAN groups ,DIOPHANTINE approximation ,ALGORITHMS - Abstract
We give continued fraction algorithms for each conjugacy class of triangle Fuchsian group of signature $(3, n, \infty )$, with $n\geq 4$. In particular, we give an explicit form of the group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study Diophantine properties of approximation in terms of the continued fractions and show that these continued fractions are appropriate to obtain transcendence results. [ABSTRACT FROM AUTHOR]
- Published
- 2012
- Full Text
- View/download PDF
20. On unipotent flows in ℋ(1, 1).
- Author
-
Calta, Kariane and Wortman, Kevin
- Subjects
ABELIAN equations ,MODULI theory ,ERGODIC theory ,POTENTIAL theory (Mathematics) ,DISTRIBUTION (Probability theory) ,PROBABILITY measures - Abstract
We study the action of the horocycle flow on the moduli space of abelian differentials in genus two. In particular, we exhibit a classification of a specific class of probability measures that are invariant and ergodic under the horocycle flow on the stratum ℋ(1; 1). [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
21. Unipotent flows on the space of branched covers of Veech surfaces.
- Author
-
ALEX ESKIN, JENS MARKLOF, and DAVE WITTE MORRIS
- Published
- 2006
22. Twisted Teichmüller curves.
- Author
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SCHMIDT, THOMAS A.
- Subjects
MATHEMATICS ,NONFICTION - Published
- 2016
- Full Text
- View/download PDF
23. Dynamical Systems and Random Processes
- Author
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Jane Hawkins, Rachel L. Rossetti, Jim Wiseman, Jane Hawkins, Rachel L. Rossetti, and Jim Wiseman
- Subjects
- Random dynamical systems--Congresses, Stochastic processes--Congresses, Differentiable dynamical systems--Congresses, Geometry, Differential--Congresses
- Abstract
This volume contains the proceedings of the 16th Carolina Dynamics Symposium, held from April 13–15, 2018, at Agnes Scott College, Decatur, Georgia. The papers cover various topics in dynamics and randomness, including complex dynamics, ergodic theory, topological dynamics, celestial mechanics, symbolic dynamics, computational topology, random processes, and regular languages. The intent is to provide a glimpse of the richness of the field and of the common threads that tie the different specialties together.
- Published
- 2019
24. Surveys on Recent Developments in Algebraic Geometry
- Author
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Izzet Coskun, Tommaso de Fernex, Angela Gibney, Izzet Coskun, Tommaso de Fernex, and Angela Gibney
- Subjects
- Geometry, Algebraic--Congresses, Algebraic geometry--Curves--Families, moduli (, Algebraic geometry--Birational geometry--Minim, Algebraic geometry--Birational geometry--Ratio, Algebraic geometry--Families, fibrations--Vari, Algebraic geometry--Projective and enumerative g, Algebraic geometry--Surfaces and higher-dimensio, Algebraic geometry--Arithmetic problems. Diophan, Commutative algebra--Homological methods--Syzy, $K$-theory--Higher algebraic $K$-theory--$Q$-
- Abstract
The algebraic geometry community has a tradition of running a summer research institute every ten years. During these influential meetings a large number of mathematicians from around the world convene to overview the developments of the past decade and to outline the most fundamental and far-reaching problems for the next. The meeting is preceded by a Bootcamp aimed at graduate students and young researchers. This volume collects ten surveys that grew out of the Bootcamp, held July 6–10, 2015, at University of Utah, Salt Lake City, Utah. These papers give succinct and thorough introductions to some of the most important and exciting developments in algebraic geometry in the last decade. Included are descriptions of the striking advances in the Minimal Model Program, moduli spaces, derived categories, Bridgeland stability, motivic homotopy theory, methods in characteristic $p$ and Hodge theory. Surveys contain many examples, exercises and open problems, which will make this volume an invaluable and enduring resource for researchers looking for new directions.
- Published
- 2017
25. Algebraic and Topological Dynamics
- Author
-
Sergiy Kolyada, Yuri Manin, Thomas Ward, Sergiy Kolyada, Yuri Manin, and Thomas Ward
- Abstract
This volume contains a collection of articles from the special program on algebraic and topological dynamics and a workshop on dynamical systems held at the Max-Planck Institute (Bonn, Germany). It reflects the extraordinary vitality of dynamical systems in its interaction with a broad range of mathematical subjects. Topics covered in the book include asymptotic geometric analysis, transformation groups, arithmetic dynamics, complex dynamics, symbolic dynamics, statistical properties of dynamical systems, and the theory of entropy and chaos. The book is suitable for graduate students and researchers interested in dynamical systems.
- Published
- 2011
26. In the Tradition of Ahlfors-Bers, IV
- Author
-
Dick Canary, Jane Gilman, Juha Heinonen, Howard Masur, Dick Canary, Jane Gilman, Juha Heinonen, and Howard Masur
- Abstract
The Ahlfors–Bers Colloquia commemorate the mathematical legacy of Lars Ahlfors and Lipman Bers. The core of this legacy lies in the fields of geometric function theory, Teichmüller theory, hyperbolic manifolds, and partial differential equations. However, the work of Ahlfors and Bers has impacted and created interactions with many other fields, such as algebraic geometry, mathematical physics, dynamics, geometric group theory, number theory, and topology. The triannual Ahlford–Bers colloquia serve as a venue to disseminate the relevant work to the wider mathematical community and bring the key participants together to ponder future directions in the field. The present volume includes a wide range of articles in the fields central to this legacy. The majority of articles present new results, but there are expository articles as well.
- Published
- 2011
27. Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow
- Author
-
Giovanni Forni, Mikhail Lyubich, Charles Pugh, Michael Shub, Giovanni Forni, Mikhail Lyubich, Charles Pugh, and Michael Shub
- Subjects
- Differentiable dynamical systems, Geometry, Hyperbolic, Hyperbolic spaces, Teichmu¨ller spaces
- Abstract
This volume collects a set of contributions by participants of the Workshop'Partially hyperbolic dynamics, laminations, and Teichmüller flow'held at the Fields Institute in Toronto in January 2006. The Workshop brought together several leading experts in two very active fields of contemporary dynamical systems theory: partially hyperbolic dynamics and Teichmüller dynamics. They are unified by ideas coming from the theory of laminations and foliations, dynamical hyperbolicity, and ergodic theory. These are the main themes of the current volume. The volume contains both surveys and research papers on non-uniform and partial hyperbolicity, on dominated splitting and beyond (in Part I), Teichmüller dynamics with applications to interval exchange transformations and on the topology of moduli spaces of quadratic differentials (in Part II), foliations and laminations and other miscellaneous papers (in Part III). Taken together these papers provide a snapshot of the state of the art in some of the most active topics at the crossroads between dynamical systems, smooth ergodic theory, geometry and topology, suitable for advanced graduate students and researchers. Non-specialists will find the extensive, in-depth surveys especially useful.
- Published
- 2007
28. Dynamics, Ergodic Theory and Geometry
- Author
-
Boris Hasselblatt and Boris Hasselblatt
- Subjects
- Geometry, Ergodic theory, Differentiable dynamical systems
- Abstract
Based on the subjects from the Clay Mathematics Institute/Mathematical Sciences Research Institute Workshop titled'Recent Progress in Dynamics'in September and October 2004, this volume contains surveys and research articles by leading experts in several areas of dynamical systems that have experienced substantial progress. One of the major surveys is on symplectic geometry, which is closely related to classical mechanics and an exciting addition to modern geometry. The survey on local rigidity of group actions gives a broad and up-to-date account of another flourishing subject. Other papers cover hyperbolic, parabolic, and symbolic dynamics as well as ergodic theory. Students and researchers in dynamical systems, geometry, and related areas will find this book fascinating. The book also includes a fifty-page commented problem list that takes the reader beyond the areas covered by the surveys, to inspire and guide further research.
- Published
- 2007
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