Narrow Results

Let $M$ be a $2n$-dimensional ($n\ge 2$), compact smooth Riemannian manifold endowed with a symplectic form $\omega$. In this paper, we show that, if a symplectic diffeomorphism $f$ is $C^1$-robustly measure expansive, then it is Anosov and a $C^1$ generic measure expansive symplectic diffeomorphism $f$ is mixing Anosov. Moreover, for a Hamiltonian systems, if a Hamiltonian system $(H,e,{{ℰ}}_{H,e})$ is robustly measure expansive, then $(H,e,{{ℰ}}_{H,e})$ is Anosov.

Cosmic microwave background data shows the observable universe to be nearly flat, but leaves open the question of whether it is simply or multiply connected. Several authors have investigated whether the topology of a multiconnected hyperbolic universe would be detectable when 0.9<Ω<1. However, the possibility of detecting a given topology varies depending on the location of the observer within the space. Recent studies have assumed the observer sits at a favorable location. The present paper extends that work to consider observers at all points in the space, and (for given values of Ω_m and Ω_Λ and a given topology) computes the probability that a randomly placed observer could detect the topology. The computations show that when Ω=0.98 a randomly placed observer has a reasonable chance (~50%) of detecting a hyperbolic topology, but when Ω=0.99 the chances are low (<10%) and decrease still further as Ω approaches one.

We investigate the possibility of discrete groups furnishing a kinematic framework for systems whose thermodynamic behavior may be given by nonadditive entropies. Relying on the well-known result of the growth rate of balls of nilpotent groups, we see that maintaining extensivity of the entropy of a nilpotent group requires using a non-Boltzmann/Gibbs/Shannon (BGS) entropic form. We use the Tsallis entropy as an indicative alternative. Using basic results from hyperbolic and random groups, we investigate the genericity and possible range of applicability of the BGS entropy in this context. We propose a sufficient condition for phase transitions, in the context of (multi-) parameter families of nonadditive entropies.

In recent years, significant attention has been attracted to hyperbolic (or indefinite) materials resulting from their extraordinary optical properties, such as all-angle negative refraction. Here, tunable biaxial hyperbolic dispersion and negative refraction were found in graphite crystal. First principles were used to calculate the permittivity of graphite in $x$-, $y$- and $z$-directions, the results indicate that graphite crystals exhibit indefinite dielectric constant tensor in certain frequency range. By adjusting strain, the optical properties of materials changed, and even switched from hyperbolic to elliptical materials in some frequency. This work demonstrates the feasibility of achieving high-performance tunable biaxial natural hyperbolic materials which can be used in various ways.

Shadowing of a dynamical system is often used to justify the validity of computer simulations of the system, and in numerical calculations an inverse form of the shadowing concept is also of some interest. In this paper we characterize the notion of shadowing in terms of stability, and express the notion of hyperbolicity using the concept of inverse shadowing.

In this paper, we analyze chaotic dynamics for two-dimensional nonautonomous maps through the use of a nonautonomous version of the Conley–Moser conditions given previously. With this approach we are able to give a precise definition of what is meant by a chaotic invariant set for nonautonomous maps. We extend the nonautonomous Conley–Moser conditions by deriving a new sufficient condition for the nonautonomous chaotic invariant set to be hyperbolic. We consider the specific example of a nonautonomous Hénon map and give sufficient conditions, in terms of the parameters defining the map, for the nonautonomous Hénon map to have a hyperbolic chaotic invariant set.

A kind of higher-dimensional complex polynomial mappings $F:{ℂ}^n\to {ℂ}^n$ is considered:

Let π : Aut(F_n) → Aut(ℤⁿ) be the epimorphism induced by the isomorphism and define . We prove that the subset of consisting of all non-irreducible with irreducible powers and all non-hyperbolic elements is exponentially small.

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.

The aim of this paper is to study the reflection-transmission of geometrical optic rays described by semi-linear symmetric hyperbolic systems such as the Maxwell–Lorentz equations with the anharmonic model of polarisation. The framework is both that of Donnat and Williams since we consider dispersive media and profiles with hyperbolic (imaginary) phases and elliptic phases (complex with non-null real part). We first give hypothesis close to the Maxwell equation. Then we introduce a decomposition for both profile into boundary (tangential) and normal part and we solve the so-called "microscopic" equation of the small scales for each boundary frequency. Then we show that the non-linearities generate harmonics which interact at the boundary and generate new resonant profiles with harmonic tangential frequency. Lastly we make a WKB expansion at any order and give a precise description of the correctors.

In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S³ for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get "as close as possible" to a counter-example. Specifically, we construct a sequence of hyperbolic knots {K_n} with complements containing closed embedded essential surfaces having principal curvatures converging to zero as n tends to infinity. We also construct a family of two-component links for which the complements contain closed embedded totally geodesic surfaces of arbitrarily large genera. In addition, we prove that a closed embedded surface with sufficiently small principal curvatures is not only quasi-Fuchsian (a result of Thurston's), but it is also either acylindrical or the boundary of a twisted I-bundle.

In the closed, non-Haken, hyperbolic class of examples generated by (2p, q) Dehn fillings of Figure 8 knot space, the geometrically incompressible one-sided surfaces are identified by the filling ratio and determined to be unique in all cases. When applied to one-sided Heegaard splitting, this can be used to classify all geometrically incompressible splittings in this class of closed, hyperbolic examples; no analogous classification exists for two-sided Heegaard splitting.

Agol has conjectured that minimally twisted n-chain links are the smallest volume hyperbolic manifolds with n cusps, for n ≤ 10. In his thesis, Venzke mentions that these cannot be smallest volume for n ≥ 11, but does not provide a proof. In this paper, we give a proof of Venzke's statement for a number of cases. For n ≥ 60 we use a formula from work of Futer, Kalfagianni and Purcell to obtain a lower bound for volume. The proof for n between 12 and 25 inclusive uses a rigorous computer computation that follows methods of Moser and Milley. Finally, we prove that the n-chain link with 2m or 2m + 1 half-twists cannot be the minimal volume hyperbolic manifold with n cusps, provided n ≥ 60 or |m| ≥ 8, and we give computational data indicating this remains true for smaller n and |m|.

We introduce the side parameter map from the space of cone hyperbolic structures on the torus with a single cone point to the hyperbolic plane ℍ², based on the idea in the Jorgensen theory for punctured torus Kleinian groups. Then, we show that the side parameter map is a homeomorphism.

The main result of this paper is the construction of two Hyperbolic manifolds, $M_1$ and $M_2$, with several remarkable properties:

• ⁽¹⁾Every closed orientable $3$-manifold is homeomorphic to the quotient space of the action of a group of order $16$ on some covering space of $M_1$ or $M_2$.
• ⁽²⁾$M_1$ and $M_2$ are tesselated by 16 dodecahedra such that the pentagonal faces of the dodecahedra fit together in a certain way.
• ⁽³⁾There are 12 closed non-orientable hyperbolic surfaces of Euler characteristic $-2$ each of which is tesselated by regular right angled pentagons and embedded in $M_1$ or $M_2$. The union of the pentagonal faces of the tesselating dodecahedra equals the union of the 12 images of the embedded surfaces of Euler characteristic $-2$.

Canonical triangulations provide an algorithm to compute the symmetry groups of hyperbolic links and their complements, along with information on how each symmetry acts on the link components. The latter information determines the chirality and invertibility of the link. Symmetry groups of knots to ten crossings and multicomponent hyperbolic links to nine crossing are listed, with some exceptions. A Macintosh computer program, available free of charge, computes symmetry groups interactively.

We shall show that the stable isometry structure determined by a suitable Seifert hypersurface of a doubly null concordant knot is hyperbolic and we prove a converse for stable knots. This suggests a “universal” source for the known homological invariants of DNC-equivalence. As an application of our main result we shall show that if the homology of the universal cover of the complement of a stable n-knot is torsion, involving only primes >(n+10)/6, and is 0 in the middle dimensions then the knot is doubly null concordant.

Multiscale stochastic homogenization is studied for quasilinear monotone hyperbolic problems with a linear damping term. It is shown by classical G-convergence methods that the sequence of solutions to a class of multi-scale highly oscillatory (possibly random) hyperbolic problems converges in the appropriate Sobolev space to the solution to a homogenized quasilinear hyperbolic problem.

We are interested in nonlinear partial differential equations of hyperbolic type, written as first-order balance laws posed on a curved spacetime, whose unknown is typically a section of the bundle of future-oriented, timelike vectors. We are especially interested in dealing with systems and solutions with low regularity and, to this end, we introduce here the notion of "strictly convex entropy field". We then prove that a system of balance laws endowed with such an entropy admits a symmetrization which, however, may be "degenerate" if the entropy is not uniformly convex or if the constitutive laws defining the balance laws have limited regularity. Next, we establish a weak–strong uniqueness and stability theorem for the initial value problem associated with balance laws endowed with a strictly convex entropy field. We compare two solutions with limited regularity: on the one hand, a continuous solution that need not be Lipschitz continuous and, on the other hand, a weak solution that satisfies an "entropy inequality". Finally, we apply our theory to the Euler system for compressible fluid flows on a curved spacetime, and we exhibit an entropy field, which is strictly convex but fails to be uniformly convex as vacuum is approached. This leads us to a uniqueness theorem for (both the relativistic and non-relativistic versions of) the Euler system, which applies to a continuous solution with vacuum and an entropy weak solution with vacuum.

We study the Ericksen–Leslie hyperbolic system for compressible liquid crystal model in three spatial dimensions. Global regularity and scattering for small and smooth initial data near equilibrium are proved for the case that the system is a nonlinear coupling of compressible Navier–Stokes equations with wave map to ${𝕊}^2$. The main strategy relies on an interplay between the control of high order energies and decay estimates, which is based on the idea inspired by the method of space-time resonances. Unlike the incompressible model, the different behaviors of the decay properties of the density and velocity field for compressible fluids at different frequencies play a key role, which is a particular feature of compressible model.