Persistence of laminations

We present a modern proof of some extensions of the well known Hirsch-Pugh-Shub theorem on persistence of normally hyperbolic compact laminations. Our extensions consist of allowing the dynamics to be an endomorphism, the complex analytic case and of allowing the laminations to be non compact. To study the analytic case, we use the formalism of deformations of complex structures. We present various persistent complex laminations which appear in dynamics of several complex variables: Hénon maps, fibered holomorphic maps. In order to prove the persistence theorems,weconstructalaminarstructureonthestableandunstablesetofthenormally hyperbolic laminations.     

Hyperbolic Structures on the Configuration Space of Six Points in the Projective Line

The oriented configuration space X⁺₆ of six points on the real projective line is a noncompact three-dimensional manifold which admits a unique complete hyperbolic structure of finite volume with ten cusps. On the other hand, it decomposes naturally into 120 cells each of which can be interpreted as the set of equiangular hexagons with unit area. Similar hyperbolic structures can be obtained by considering nonequiangular hexagons so that the standard hyperbolic structure on X⁺₆ is at the center of a five parameter family of hyperbolic structures of finite volume. This paper contributes to investigations of the properties of this family. In particular, we exhibit two real analytic maps from the set of prescribed angles of hexagons into R¹⁰ whose components are the traces of the monodromies at the ten cusps. We show that this map has maximal rank 5 at the center.     

Curved flats,pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudo-Riemannian space forms. As a corollary, we obtain a non-immersibility theorem for spheres into certain pseudo-Riemannian spheres and hyperbolic spaces. The second aspect pursued is to clarify the relationship between the loop group formulation of isometric immersions of space forms and that of pluriharmonic maps into symmetric spaces. We show that the objects in the first class are, in the real analytic case, extended pluriharmonic maps into certain symmetric spaces which satisfy an extra reality condition along a totally real submanifold. We show how to construct such pluriharmonic maps for general symmetric spaces from curved flats, using a generalised DPW method.      

On differentiable compactifications of the hyperbolic plane and algebraic actions of $${\rm SL}_2(\mathbb{R})$$ on surfaces

Real-analytic actions of SL(2;R) on surfaces have been classified, up to analytic change of coordinates. In particular it is known that there exists countably many analytic equivariant compactification of the isometric action on the hyperbolic plane. In this paper we study the algebraicity of these actions. We get a classification of the algebraic actions of SL(2;R) on surfaces. In particular, we classify the algebraic equivariant compactifications of the hyperbolic plane. The online version of the original article can be found under doi: .     

Geometric bounds on the linearization domain and analytic dependence on parameters for families of analytic vector fields in a neighborhood of a singular point

We study families of holomorphic vector fields, holomorphically depending on parameters, in a neighborhood of an isolated singular point. When the singular point is in the Poincaré domain for every vector field of the family we prove, through a modification of classical Sternberg's linearization argument, cf. Nelson (1969) [7] too, analytic dependence on parameters of the linearizing maps and geometric bounds on the linearization domain: each vector field of the family is linearizable inside the smallest Euclidean sphere which is not transverse to the vector field, cf. Brushlinskaya (1971) [2], Ilyashenko and Yakovenko (2008) [5] for related results. We also prove, developing ideas in Martinet (1980) [6], a version of Brjuno's Theorem in the case of linearization of families of vector fields near a singular point of Siegel type, and apply it to study some 1-parameter families of vector fields in two dimensions.     

How to work with one-dimensional quadratic maps

We analyse some identifiers which can univocally identify hyperbolic components and Misiurewicz points of one-dimensional quadratic maps. After seeing the equivalence among the different identifiers and how to go from one to another, we show which are the best for some specific tasks. Likewise, we present the analytic expressions, some of them shown for the first time in this paper, to calculate these identifiers. Some experimental considerations are taken into account.     

On Differentiable Compactifications of the Hyperbolic Plane and Algebraic Actions of SL2(\mathbb R) on Surfaces

Real-analytic actions of SL(2;R) on surfaces have been classified, up to analytic change of coordinates. In particular it is known that there exists countably many analytic equivariant compactification of the isometric action on the hyperbolic plane. In this paper we study the algebraicity of these actions. We get a classification of the algebraic actions of SL(2,R) on surfaces. In particular, we classify the algebraic equivariant compactifications of the hyperbolic plane. An erratum to this article can be found at     

Analytic regularity for a class of semi-linear weakly hyperbolic equations of second order

A class of semi-linear weakly hyperbolic equations of second order in 1 space dimension is considered. Using the properties of the analytic functions we give energy estimates for the solutions and then we prove the propagation of the analytic regularity.     

Analytic torsion of complete hyperbolic manifolds of finite volume

In this paper we define the analytic torsion for a complete oriented hyperbolic manifold of finite volume. It depends on a representation of the fundamental group. For manifolds of odd dimension, we study the asymptotic behavior of the analytic torsion with respect to certain sequences of representations obtained by restriction of irreducible representations of the group of isometries of the hyperbolic space to the fundamental group.     

Dynamical systems on lattices with decaying interaction II: Hyperbolic sets and their invariant manifolds

This is the second part of the work devoted to the study of maps with decay in lattices. Here we apply the general theory developed in Fontich et al. (2011) [3] to the study of hyperbolic sets. In particular, we establish that any close enough perturbation with decay of an uncoupled lattice map with a hyperbolic set has also a hyperbolic set, with dynamics on the hyperbolic set conjugated to the corresponding of the uncoupled map. We also describe how the decay properties of the maps are inherited by the corresponding invariant manifolds.     

Unique continuation along curves and hypersurfaces for second order anisotropic hyperbolic systems with real analytic coefficients

In this paper we prove the following kind of unique continuation property. That is, the zero on each geodesic of the solution in a real analytic hypersurface for second order anisotropic hyperbolic systems with real analytic coefficients can be continued along this curve.

     

Dynamical convergence of polynomials to the exponential

In this paper we investigate the relationship between the dynamics of the polynomials maps p_d,λ(z)=(1+z/d)^d and the exponential family E_λ(z)=λc^z. We show that the hyperbolic components of the parameter planes for the polynomials converge to those for the exponential family as the degree d tends to infinity. We also show that certain "hairs"in the parameter plane for the exponential are limits of correspondings external rays for the polynomial families. For parameter values on the hairs, the Julia sets for the corresponding exponentials are the entire plane whereas, for polynomial parameters on the external rays, the Julia sets are Cantor sets     

Unstable directions and fractal dimension for skew products with overlaps in fibers

A unique feature of smooth hyperbolic non-invertible maps is that of having different unstable directions corresponding to different prehistories of the same point. In this paper we construct a new class of examples of non-invertible hyperbolic skew products with thick fibers for which we prove that there exist uncountably many points in the locally maximal invariant set ?? (actually a Cantor set in each fiber), having different unstable directions corresponding to different prehistories; also we estimate the angle between such unstable directions. We discuss then the Hausdorff dimension of the fibers of ?? for these maps by employing the thickness of Cantor sets, the inverse pressure, and also by use of continuous bounds for the preimage counting function. We prove that in certain examples, there are uncountably many points in ?? with two preimages belonging to ??, as well as uncountably many points having only one preimage in ??. In the end we give examples which, also from the point of view of Hausdorff dimension, are far from being homeomorphisms on ??, as well as far from being constant-to-1 maps on ??.     

Dualities and envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-spaces

We consider envelopes of one-parameter families of frontals in hyperbolic and de Sitter 2-space from the viewpoint of duality, respectively. Since the classical notions of envelopes for singular curves do not work, we have to find a new method to define the envelope for singular curves in hyperbolic space or de Sitter space. To do that, we first introduce notions of one-parameter families of Legendrian curves by using the Legendrian dualities. Afterwards, we give definitions of envelopes for the one-parameter families of frontals in hyperbolic and de Sitter 2-space, respectively. We investigate properties of the envelopes. At last, we give relationships among those envelopes.     

Hypercyclic composition operators on Hv0‐spaces

We consider analytic self‐maps φ on $\mathbf {D}$ and prove that the composition operator C_φ acting on $H_{v}^0$ is hypercyclic if φ is an automorphism or a hyperbolic non‐automorphic symbol with no fixed point. We give examples of weights v and parabolic non‐automorphisms φ on $\mathbf {D}$ which yield non‐hypercyclic composition operators C_φ on $H_{v}^0$.     

Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of \(\mathrm{SL }_2(\mathbb {C})\) to the corresponding Reidemeister torsion. While the analytic torsion is a spectral invariant of the manifold, the Reidemeister torsion is of combinatorial nature. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti.     

The rational maps have no Herman rings

We prove that for every , the rational maps in the family have no Herman rings. From this we conclude a dynamical characterization for the parameters in the Mandelbrot set of these families. Further, we show that hyperbolic maps are dense in this family if and only if the set of parameters for which the Julia set is the whole sphere has no interior.

     

A Compactification of the Space of Expanding Maps on the Circle

We show the space of expanding Blaschke products on S¹ is compactified by a sphere of invariant measures, reminiscent of the sphere of geodesic currents for a hyperbolic surface. More generally, we develop a dynamical compactification for the Teichmüller space of all measure preserving topological covering maps of S¹. Research supported in part by the NSF.