Related papers: Homoclinic and heteroclinic intersections for lemo…
By a classical theorem transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In this paper we study the fate of homoclinic tangles in parameterized systems from the…
Given a random map (T_1, T_2, T_3, T_4, p_1, p_2, p_3, p_4), we define a random billiard map on a surface of constant curvature (Euclidean plane, hyperbolic plane, or the sphere). The Liouville measure is invariant for this billiard map.…
The complexity of the billiard language of regular polygons in the hyperbolic plane with $p$ sides and $2\pi/q$ internal angles is known to grow exponentially and the exponential growth rate is known to equal the topological entropy of the…
We show that wave functions in planar rational polygonal billiards (all angles rationally related to Pi) can be expanded in a basis of quasi-stationary and spatially regular states. Unlike the energy eigenstates, these states are directly…
In this paper we constructed a special family of semidispersing billiards bounded on a rectangle with a few dispersing scatters. We assume there exists a pair of flat points (with zero curvature) on the boundary of these scatters, whose…
We study holomorphic maps between C$^*$-algebras $A$ and $B$. When $f:B_A (0,\varrho) \longrightarrow B$ is a holomorphic mapping whose Taylor series at zero is uniformly converging in some open unit ball $U=B_{A}(0,\delta)$ and we assume…
We obtain geometric upper bounds on the topological entropy of generalized polygon exchange transformations. As an application of our results, we show that billiards in polygons and rational polytopes have zero topological entropy.
For every quadrilateral sufficiently close to a rectangle, we shall show that it possess a periodic billiard path. This is an REU work done at ICERM in Summer 2012.
We study finite two dimensional spin lattices with definite geometry (spin billiards) demonstrating the display of collective integrable or chaotic dynamics depending on their shape. We show that such systems can be quantum simulated by…
We determine with unprecedented accuracy the lowest 900 eigenvalues of two quantum constant-width billiards from resonance spectra measured with flat, superconducting microwave resonators. While the classical dynamics of the constant-width…
Let $k,a,b$ be positive integers with $a+b=k$. A $k$-uniform hypergraph is called an $(a,b)$-cycle if there is a partition $(A_0,B_0,A_1,B_1,\ldots,A_{t-1},B_{t-1})$ of the vertex set with $|A_i|=a$, $|B_i|=b$ such that $A_i\cup B_i$ and…
We show that a generic analytic strongly convex billiard is "maximally chaotic" in the sense that, for every rational number $\frac{p}{q} \in \mathbb{Q} \cap (0,1)$, all intersections between the stable and unstable manifolds of maximizing…
We investigate the metric structure of the intersection lattice L(B(n,k)) of the discriminantal arrange ment using circuit supports. We show that the cover graph associated with L(B(n,k)) is isometrically embedded into a hypercube, making…
If the homology of the free loop space of a closed manifold B is infinite dimensional then generically there exist infinitely many leaf-wise intersection points for fiber-wise star-shaped hypersurfaces in T*B.
We show that two-dimensional billiard systems are Turing complete, in the sense that the halting of any Turing machine with a given input is equivalent to a certain bounded trajectory in this system entering a specified open set. Billiards…
In this paper we analyze the heteroclinic cycle and the Hopf bifurcation of a generic dynamical system with the symmetry of the group $\mathbf{Q}_8,$ constructed via a Cayley graph. While the Hopf bifurcation is similar to that of a…
The aim of the present paper is to propose and study a dissipative variant of symplectic billiards within planar strictly convex domains. The associated billiard map is dissipative, thus it admits a compact invariant set, the so-called…
We study locally flat disks in $(\mathbb{C} P^2)^\circ:=(\mathbb{C} P^2)\setminus \mathring{B^4}$ with boundary a fixed knot $K$ and whose complement has fundamental group $\mathbb{Z}$. We show that up to topological isotopy rel. boundary,…
We establish an equivalence between infinitely many asymptotically stable periodic solutions and subsumed homoclinic connections for $N$-dimensional piecewise-linear continuous maps. These features arise as a codimension-three phenomenon.…
The aim of the present paper is to establish a Bialy-Mironov type rigidity for centrally symmetric symplectic billiards. For a centrally symmetric $C^2$ strongly-convex domain $D$ with boundary $\partial D$, assume that the symplectic…