Related papers: Linear Triangle Dynamics: The Pedal Map and Beyond
In this paper we study in detail, both analytically and numerically, the dynamical properties of the triangle map, a piecewise parabolic automorphism of the two-dimensional torus, for different values of the two independent parameters…
Polygonal slap maps are piecewise affine expanding maps of the interval obtained by projecting the sides of a polygon along their normals onto the perimeter of the polygon. These maps arise in the study of polygonal billiards with…
The main purpose of this paper is to present a kneading theory for two-dimensional triangular maps. This is done by defining a tensor product between the polynomials and matrices corresponding to the one-dimensional basis map and fiber map.…
As an introduction to the concept of "moduli space" we consider the moduli space of similarity classes of acute and right triangles in the plane. This has a map to the moduli space of elliptic curves which is onto and generically…
In this article we make the concept of a continuous family of triangles precise and prove the moduli functor classifying oriented triangles admits a fine moduli space but the functor classifying non-oriented triangles only admits a coarse…
The space of topological decompositions into triangulations of a surface has a natural graph structure where two triangulations share an edge if they are related by a so-called flip. This space is a sort of combinatorial Teichm\"uller space…
We prove that a Lebesgue measure-preserving linked-twist map defined in the plane is metrically isomorphic to a Bernoulli shift (and thus strongly mixing). This is the first such result for an explicitly defined linked-twist map on a…
We study an area preserving parabolic map which emerges from the Poincar\' e map of a billiard particle inside an elongated triangle. We provide numerical evidence that the motion is ergodic and mixing. Moreover, when considered on the…
We describe the moduli space G^r_d of triples consisting of a curve C, a line bundle L on C of degree d, and a linear system V on L of dimension r. This moduli space extends over a partial compactification {\tilde M_g} of M_g inside {\bar…
The moduli space of regular stable maps with values in a complex manifold admits naturally the structure of a complex orbifold. Our proof uses the methods of differential geometry rather than algebraic geometry. It is based on Hardy…
Let $M$ be the space of triangles, defined up to shifts, rotations and dilations. We define two maps $f:M\to M$ and $g:M\to M$. The map $f$ corresponds to a triangle of perimeter $\pi$ the triangle with angles numerically equal to edges of…
Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry,…
We introduce the notion of a logarithmic stable map from a minimal log prestable curve to a log twisted semi-stable variety of form $xy=0$. We study the compactification of the moduli spaces of such maps and provide a perfect obstruction…
We study triangle decompositions of graphs. We consider constructions of classes of graphs where every edge lies on a triangle and the addition of the minimum number of multiple edges between already adjacent vertices results in a strongly…
A construction is given of Markov partitions for some rational maps, which persist over regions of parameter space, not confined to single hyperbolic components. The set on which the Markov partition exists, and its boundary, are analysed.
The moduli space of triangles is a two-dimensional space that records triangle shapes in the plane, considered up to similarity. We study the subset corresponding to \textit{lattice triangles}, which are triangles whose vertices have…
Ergodic systems, being indecomposable are important part of the study of dynamical systems but if a system is not ergodic, it is natural to ask the following question: Is it possible to split it into ergodic systems in such a way that the…
We consider the time dependent probability distribution of a coarse grained observable Y whose evolution is governed by a discrete time map. If the map is mixing, the time dependent one-step transition probabilities converge in the long…
Iterations of odd piecewise continuous maps with two discontinuities, i.e., symmetric discontinuous bimodal maps, are studied. Symbolic dynamics is introduced. The tools of kneading theory are used to study the homology of the discrete…
For $L \hookrightarrow X$ a Lagrangian embedding associated with a real homogeneous space, we construct the moduli space of stable holomorphic discs mapping to $(X,L)$ as an orbifold with corners equipped with a group action. Some essential…