Related papers: Counting Hyperbolic Components

We study Lefschetz fixed point formulas for constructible sheaves with higher-dimensional fixed point sets. Under fairly weak assumptions, we prove that the local contributions from them are expressed by some constructible functions…

The notion of the characteristic Lie algebra of the discrete hyperbolic type equation is introduced. An effective algorithm to compute the algebra for the equation given is suggested. Examples and further applications are discussed.

For a composite $n$ and an odd $c$ with $c$ not dividing $n$, the number of solutions to the equation $n+a\equiv b\mod c$ with $a,b$ quadratic residues modulus $c$ is calculated. We establish a direct relation with those modular solutions…

In this article, we study the hyperbolic components of McMullen maps. We show that the boundaries of all hyperbolic components are Jordan curves. This settles a problem posed by Devaney. As a consequence, we show that cusps are dense on the…

We show that resonance zones near an elliptic periodic point of a reversible map must, generically, contain asymptotically stable and asymptotically unstable periodic orbits, along with wild hyperbolic sets.

Moduli spaces of quadratic differentials with prescribed singularities are not necessarily connected. We describe here all cases when they have a special hyperelliptic connected component. We announce the general classification theorem: up…

The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a…

We prove that the SL(2, C) character variety of a hyperbolic, freely 2-periodic knot has two canonical components. We also prove that the hyperbolic torsion polynomial of such a knot satisfies a factorization condition which seems to be…

We consider groups defined by cyclic presentations where the defining word has length three and the cyclic presentation satisfies the T(6) small cancellation condition. We classify when these groups are hyperbolic. When combined with known…

In this paper, we study additively indecomposable quadratic forms over real biquadratic and simplest cubic fields. In particular, we show that over these fields, we can always find such a classical form in 2 variables, which differs from…

Symmetric hyperbolic systems of equations are explicitly constructed for a general class of tensor fields by considering their structure as r-fold forms. The hyperbolizations depend on 2r-1 arbitrary timelike vectors. The importance of the…

A triangular field of rational numbers is characterized, with relations to Stirling numbers 2nd, Hyperbolic functions, and centered Binomial distribution. A Generating function is given.

Let (X_i,d_i), i=1,2, be proper geodesic hyperbolic metric spaces. We give a general construction for a ``hyperbolic product'' X_1{times}_h X_2 which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.

We discuss the space of complex exponential maps $\Ek\colon z\mapsto e^{z}+\kappa$. We prove that every hyperbolic component $W$ has connected boundary, and there is a conformal isomorphism $\Phi_W\colon W\to\half^-$ which extends to a…

We study Kostant's partial order on the elements of a semisimple Lie group in relations with the finite dimensional representations. In particular, we prove the converse statement of [3, Theorem 6.1] on hyperbolic elements.

We consider a class of parabolic equations with critical electromagnetic potentials, for which we obtain a classification of local asymptotics, unique continuation results, and an integral representation formula for solutions.

We establish some statistical properties of the hyperbolic times for a class of nonuniformly expanding dynamical systems. The maps arise as factors of area preserving maps of the unit square via a geometric Baker's map type construction,…

Functions of hyperbolic type encode representations on real or complex hyperbolic spaces, usually infinite-dimensional. These notes set up the complex case. As applications, we prove the existence of a non-trivial deformation family of…

We describe a new method of producing equations for the canonical component of representation variety of a knot group into $PSL_2(\mathbb{C})$. Unlike known methods, this one does not involve any polyhedral decomposition or triangulation of…

In the space of cubic polynomials, Milnor defined a notable curve $\mathcal S_p$, consisting of cubic polynomials with a periodic critical point, whose period is exactly $p$. In this paper, we show that for any integer $p\geq 1$, any…