Search

It is known that all uniformly expanding dynamics have no phase transition with respect to Hölder continuous potentials. In this paper we show that given a local diffeomorphism f on the circle, that is neither a uniformly expanding dynamics nor invertible, the topological pressure function ℝ ∋ t ↦ P t o p (f , − t log | D f |) is not analytical. In other words, f has a thermodynamic phase transition with respect to geometric potential. Assuming that f is transitive and that Df is Hölder continuous, we show that there exists t 0 ∈ (0 , 1 ] such that the transfer operator L f , − t log | D f | , acting on the space of Hölder continuous functions, has the spectral gap property for all t < t 0 and has not the spectral gap property for all t ⩾ t 0 . Similar results are also obtained when the transfer operator acts on the space of bounded variations functions and smooth functions. In particular, we show that in the transitive case f has a unique thermodynamic phase transition and it occurs in t 0. In addition, if the loss of expansion of the dynamics occurs because of an indifferent fixed point or the dynamics admits an absolutely continuous invariant probability with positive Lyapunov exponent then t 0 = 1. [ABSTRACT FROM AUTHOR]

Loop quantum gravity (LQG) is a quantization program for gravity based on the principles of QFT and general covariance of general relativity. Quantum states of LQG describe gravitational excitations based on graphs embedded in a spatial slice of spacetime. We show that, under certain assumptions on the class of diffeomorphisms, the space of diffeomorphism invariant states carries a Fock space structure. The role of one-particle excitations for this structure is played by the diffeomorphism invariant states based on graphs with a single (linked) component. This means, however, that a lot of the structure of the diffeomorphism invariant Hilbert space remains unresolved by this structure. We show how the Fock structure allows to write at least some condensate states of group field theory as diffeomorphism invariant coherent states of LQG in a precise sense. We also show how to construct other interesting states using this Fock structure. We finally explore the quantum geometry of single- and multi-particle states and tentatively observe some resemblance to geometries with a single or multiple components, respectively. [ABSTRACT FROM AUTHOR]

We present diffeomorphisms of wild blender-horseshoes which belong to C r (1 ⩽ r < ∞) closures of two types of diffeomorphisms, one of which has a historic contracting wandering domain, and the other has a non-trivial Dirac physical measure supported by saddle periodic orbit. It is a non-trivial extension of Colli–Vargas' model (Colli and Vargas 2001 Ergod. Theory Dyn. Syst. 21 1657–81) to the higher dimensional dynamics with the use of wild blender-horseshoes. [ABSTRACT FROM AUTHOR]

We study bifurcation mechanisms for the appearance of hyperchaotic attractors in three-dimensional diffeomorphisms, i.e. such attractors whose orbits have two positive Lyapunov exponents in numerical experiments. In particular, periodic orbits belonging to the attractor should have two-dimensional unstable invariant manifolds. We discuss several bifurcation scenarios which create such periodic orbits inside the attractor. This includes cascades of supercritical period-doubling bifurcations of saddle periodic orbits and supercritical Neimark–Sacker bifurcations of stable periodic orbits, as well as various combinations of these cascades. These scenarios are illustrated by an example of the three-dimensional Mirá map. [ABSTRACT FROM AUTHOR]

In (2017 Nonlinearity 30 2667–86) Slipantschuk, Bandtlow and Just gave concrete examples of Anosov diffeomorphisms of T 2 for which their resonances could be completely described. Their approach was based on composition operators acting on analytic anisotropic Hilbert spaces. In this note we present a construction of alternative anisotropic Hilbert spaces which helps to simplify parts of their analysis and gives scope for constructing further examples. [ABSTRACT FROM AUTHOR]

In this paper, we study non-wandering homeomorphisms of the two-dimen-sional torus homotopic to the identity, whose rotation sets are non-trivial segments from (0, 0) to some totally irrational point (α, β). We show that for any r â©ľ 1, this rotation set only appears for C r diffeomorphisms satisfying some degenerate conditions. And when such a rotation set does appear, assuming several natural conditions that are generically satisfied in the area-preserving world, we give a clearer description of its rotational behaviour. More precisely, the dynamics admits bounded deviation along the direction âˆ'(α, β) in the lift, and the rotation set is locked inside an arbitrarily small cone with respect to small C 0-perturbations of the dynamics. On the other hand, for any non-wandering homeomorphism f with this kind of rotation set, we also present a perturbation scheme in order for the rotation set to be eaten by the rotation set of some nearby dynamics, in the sense that the later set has non-empty interior and contains the former one. These two flavours interplay and share the common goal of understanding the stability/instability properties of this kind of rotation set. [ABSTRACT FROM AUTHOR]

Nontrivial diffeomorphisms act on the horizon of a generic 4D black holes and create distinguishing features referred to as soft hair. Amongst these are a leftâ€"right pair of Virasoro algebras with associated charges that reproduce the Bekensteinâ€"Hawking entropy for Kerr black holes. In this paper we show that if one adds a negative cosmological constant, there is a similar set of infinitesimal diffeomorphisms that act non-trivially on the horizon. The algebra of these diffeomorphisms gives rise to a central charge. Adding a boundary counterterm, justified to achieve integrability, leads to well-defined central charges with c L = c R. The macroscopic area law for Kerr-AdS black holes follows from the assumption of a Cardy formula governing the black hole microstates. [ABSTRACT FROM AUTHOR]

The authors previously introduced a diffeomorphism-invariant definition of a homogeneous and isotropic sector of loop quantum gravity (LQG), along with a program to embed loop quantum cosmology (LQC) into it. The present paper works out that program in detail for the simpler, but still physically non-trivial, case where the target of the embedding is the homogeneous, but not isotropic, Bianchi I model. The diffeomorphism-invariant conditions imposing homogeneity and isotropy in the full theory reduce to conditions imposing isotropy on an already homogeneous Bianchi I spacetime. The reduced conditions are invariant under the residual diffeomorphisms still allowed after gauge fixing the Bianchi I model. We show that there is a unique embedding of the quantum isotropic model into the homogeneous quantum Bianchi I model that (a) is covariant with respect to the actions of such residual diffeomorphisms, and (b) intertwines both the (signed) volume operator and at least one directional Hubble rate. That embedding also intertwines all other operators of interest in the respective loop quantum cosmological models, including their Hamiltonian constraints. It thus establishes a precise equivalence between dynamics in the isotropic sector of the Bianchi I model and the quantized isotropic model, and not just their kinematics. We also discuss the adjoint relationship between the embedding map defined here and a projection map previously defined by Ashtekar and Wilson-Ewing. Finally, we highlight certain features that simplify this reduced embedding problem, but which may not have direct analogues in the embedding of homogeneous and isotropic LQC into full LQG. [ABSTRACT FROM AUTHOR]

Motivated by the image reconstruction in spatiotemporal imaging, we introduce a concept named diffeomorphic optimal transportation (DOT), which combines the Wasserstein distance with Benamouâ€"Brenier formula in optimal transportation and the flow of diffeomorphisms in large deformation diffeomorphic metric mapping. Using DOT, we propose a new variational model for joint image reconstruction and motion estimation, which is suitable for spatiotemporal imaging involving mass-preserving large diffeomorphic deformations. We further get its equivalent PDE-constrained optimal control formulation. The proposed model is easy-to-implement and solved by an alternating gradient descent algorithm, which is compared against existing alternatives theoretically and numerically. The performance is validated by several numerical experiments in spatiotemporal tomography, where the projection data is time-dependent sparse and/or high-noise. Moreover, we present several extensions based on DOT. Under appropriate conditions, the proposed algorithm can be adapted as a new algorithm to solve the models using quadratic Wasserstein distance. [ABSTRACT FROM AUTHOR]

The standard way of deriving Euler–Lagrange (EL) equations given a point particle action is to vary the trajectory and set the first variation of the action to zero. However, if the action is (i) reparameterization invariant, and (ii) generally covariant, I show that one may derive the EL equations by suitably nullifying the variation through a judicious coordinate transformation. The net result of this is that the curve remains fixed, while all other geometrical objects in the action undergo a change, given precisely by the Lie derivatives along the variation vector field. This, then, is the most direct and transparent way to elucidate the connection between general covariance, diffeomorphism invariance, and Lie derivatives, without referring to covariant derivative. I highlight the geometric underpinnings and generality of above ideas by applying them to simplest of field theories, keeping the discussion at a level easily accessible to advanced undergraduates. As non-trivial applications of these ideas, I (i) derive the geodesic deviation equation using first order diffeomorphisms, and (ii) demonstrate how they can highlight the connection between canonical and metric stress–energy tensors in field theories. [ABSTRACT FROM AUTHOR]

Null shells are a useful geometric construction to study the propagation of infinitesimally thin concentrations of massless particles or impulsive waves. In this paper, we determine and study the necessary and sufficient conditions for the matching of two spacetimes with respective null embedded hypersurfaces as boundaries. Whenever the matching is possible, it is shown to depend on a diffeomorphism between the set of null generators in each boundary and a scalar function, called step function, that determines a shift of points along the null generators. Generically there exists at most one possible matching but in some circumstances this is not so. When the null boundaries are totally geodesic, the point-to-point identification between them introduces a freedom whose nature and consequences are analysed in detail. The expression for the energy–momentum tensor of a general null shell is also derived. Finally, we find the most general shell (allowing non-zero energy, energy flux and pressure) that can be generated by matching two Minkowski regions across a null hyperplane. This allows us to show how the original Penrose's cut-and-paste construction fits and connects with the standard matching formalism. [ABSTRACT FROM AUTHOR]

We construct a general effective dynamics for diffeomorphisms of spacetime, in a fixed external metric. Though related to familiar models of scalar fields as coordinates, our models have subtly different properties, both at kinematical and dynamical level. The energy–momentum (EM) tensor consists of two independently conserved parts. The background solution is the identity diffeomorphism and the EM tensor of this solution gives rise to an effective cosmological constant. [ABSTRACT FROM AUTHOR]

In this work we address the problem of existence and uniqueness (finiteness) of ergodic equilibrium states for partially hyperbolic diffeomorphisms isotopic to Anosov on , with two-dimensional center foliation. To do so we propose to study the disintegration of measures along one-dimensional subfoliations of the center bundle. Moreover, we obtain a more general result characterizing the disintegration of ergodic measures in our context. [ABSTRACT FROM AUTHOR]

We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with controlled geometry) and such that their induced action on is hyperbolic. In absence of the simplicity condition we construct a robustly transitive counter-example. [ABSTRACT FROM AUTHOR]

We prove that in the isotopy class of any volume preserving partially hyperbolic diffeomorphism in a three-dimensional manifold, there is a non-partially hyperbolic stably ergodic diffeomorphism. In particular, we provide new examples of stably ergodic diffeomorphisms in three-dimensional manifolds with respect to a smooth volume measure. [ABSTRACT FROM AUTHOR]

In this paper we consider the semi-continuity of the physical-like measures for diffeomorphisms with dominated splittings. We prove that any weak-* limit of physical-like measures along a sequence of C1 diffeomorphisms {fn} must be a Gibbs F-state for the limiting map f. As a consequence, we establish the statistical stability for the C1 perturbation of the time-one map of three-dimensional Lorenz attractors, and the continuity of the physical measure for the diffeomorphisms constructed by Bonatti and Viana. [ABSTRACT FROM AUTHOR]

In this paper, we define and study unstable measure theoretic pressure for C1-smooth partially hyperbolic diffeomorphisms with sub-additive potentials. We show that this measure theoretic pressure for any ergodic measure equals the corresponding unstable measure theoretic entropy plus the Lyapunov exponent of the potentials with respect to the ergodic measure. On the other hand, we also give other definitions of unstable metric pressure, in terms of Bowen's picture and the capacity picture. We show that all definitions of unstable metric pressure, including the one defined at the beginning, actually coincide for any ergodic measure. [ABSTRACT FROM AUTHOR]

We consider the class G(Sn) of orientation preserving Morse–Smale diffeomorphisms of the sphere Sn of dimension n > 3, assuming that invariant manifolds of different saddle periodic points have no intersection. For any diffeomorphism f ∈ G(Sn), we define a coloured graph Γf that describes a mutual arrangement of invariant manifolds of saddle periodic points of the diffeomorphism f. We enrich the graph Γf by an automorphism Pf induced by dynamics of f and define the isomorphism notion between two coloured graphs. The aim of the paper is to show that two diffeomorphisms f, f′ ∈ G(Sn) are topologically conjugated if and only if the graphs Γf, Γf′ are isomorphic. Moreover, we establish the existence of a linear-time algorithm to distinguish coloured graphs of diffeomorphisms from the class G(Sn). [ABSTRACT FROM AUTHOR]

In the last few years, there has been significant interest in understanding the stationary comparison version of the first law of black hole mechanics in the vielbein formulation of gravity. Several authors have pointed out that to discuss the first law in the vielbein formulation one must extend the Iyer–Wald Noether charge formalism appropriately. Jacobson and Mohd (2015 Phys. Rev. D 92 124010) and Prabhu (2017 Class. Quantum Grav. 34 035011) formulated such a generalisation for symmetry under combined spacetime diffeomorphisms and local Lorentz transformations. In this paper, we apply and appropriately adapt their formalism to four-dimensional gravity coupled to a Majorana field and to a Rarita–Schwinger field. We explore the first law of black hole mechanics and the construction of the Lorentz-diffeomorphism Noether charges in the presence of fermionic fields, relevant for simple supergravity. [ABSTRACT FROM AUTHOR]

Here we start from a dual version of Vasiliev's first order action for massless spin-2 particles (linearized first order Einstein–Hilbert) and derive, via Kaluza–Klein dimensional reduction from D + 1 to D dimensions, a set of dual massive spin-2 models. This set includes the massive 'BR' model, a spin-2 analogue of the spin-1 Cremmer–Scherk model. In our approach the linearized Riemann curvature emerges from a solution of a functional constraint. In D = 2 + 1 the BR model can be written as a linearized version of a new bimetric model for massive gravitons. We also have a new massive spin-2 model, in arbitrary dimensions, invariant under linearized diffeomorphisms. It is given in terms of a non symmetric rank-2 tensor and a mixed symmetry tensor. [ABSTRACT FROM AUTHOR]

A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension-2 submanifold of an energy level that can be spanned by two compact codimension-1 surfaces of unidirectional flux whose union, called a dividing surface, locally separates the energy level into two components and has no local recrossings. For this to happen robustly to all smooth perturbations, the transition state must be normally hyperbolic. The dividing surface then has locally minimal geometric flux through it, giving an upper bound on the rate of transport in either direction.Transition states diffeomorphic to are known to exist for energies just above any index-1 critical point of a Hamiltonian of m degrees of freedom, with dividing surfaces . The question addressed here is what qualitative changes in the transition state, and consequently the dividing surface, may occur as the energy or other parameters are varied? We find that there is a class of systems for which the transition state becomes singular and then regains normal hyperbolicity with a change in diffeomorphism class. These are Morse bifurcations.Various examples are considered. Firstly, some simple examples in which transition states connect or disconnect, and the dividing surface may become a torus or other. Then, we show how sequences of Morse bifurcations producing various interesting forms of transition state and dividing surface are present in reacting systems, by considering a hypothetical class of bimolecular reactions in gas phase. [ABSTRACT FROM AUTHOR]

The discrete spatial geometry underlying loop quantum gravity (LQG) is degenerate almost everywhere. This is at apparent odds with the non-degeneracy of asymptotically flat metrics near spatial infinity. Koslowski generalized the LQG representation so as to describe states labeled by smooth non-degenerate triad fields. His representation was further studied by Sahlmann with a view to imposing gauge and spatial diffeomorphism invariance through group averaging methods. Motivated by the desire to model asymptotically flat quantum geometry by states with triad labels which are non-degenerate at infinity but not necessarily so in the interior, we initiate a generalization of Sahlmann’s considerations to triads of varying degeneracy. In doing so, we include delicate phase contributions to the averaging procedure which are crucial for the correct implementation of the gauge and diffeomorphism constraints, and whose existence can be traced to the background exponential functions recently constructed by one of us. Our treatment emphasizes the role of symmetries of quantum states in the averaging procedure. Semianalyticity, influential in the proofs of the beautiful uniqueness results for LQG, plays a key role in our considerations. As a by product, we re-derive the group averaging map for standard LQG, highlighting the role of state symmetries and explicitly exhibiting the essential uniqueness of its specification. [ABSTRACT FROM AUTHOR]

A reconstruction theorem in terms of the topology and geometrical structures on the spaces of light rays and skies of a given spacetime is discussed. This result can be seen as a part of Penrose and Low’s programme intending to describe the causal structure of a spacetime M in terms of the topological and geometrical properties of the space of light rays, i.e., unparametrized time-oriented null geodesics, . In the analysis of the reconstruction problem, the structure of the space of skies, i.e., of congruences of light rays, becomes instrumental. It will be shown that the space of skies Σ of a strongly causal non-refocusing spacetime M carries a canonical differentiable structure diffeomorphic to the original manifold M. Celestial curves, this is, curves in which are everywhere tangent to skies, play a fundamental role in the analysis of the geometry of the space of light rays. It will be shown that a celestial curve is induced by a past causal curve of events iff the Legendrian isotopy defined by it is non-negative. This result extends in a nontrivial way some recent results by Chernov et al on Low’s Legendrian conjecture. Finally, it will be shown that a celestial causal map between the space of light rays of two strongly causal spaces (provided that the target space is null non-conjugate) is necessarily induced from a conformal immersion and conversely. These results make explicit the fundamental role played by the collection of skies, a collection of Legendrian spheres with respect to the canonical contact structure on , in characterizing the causal structure of spacetime. [ABSTRACT FROM AUTHOR]

An outstanding issue in the treatment of boundaries in general relativity is the lack of a local geometric interpretation of the necessary boundary data. For the Cauchy problem, the initial data is supplied by the 3-metric and extrinsic curvature of the initial Cauchy hypersurface, subject to constraints. This Cauchy data determine a solution to Einstein’s equations which is unique up to a diffeomorphism. Here, we show how three pieces of unconstrained boundary data, which are associated locally with the geometry of the boundary, likewise determine a solution of the initial-boundary value problem which is unique, up to a diffeomorphism. Two pieces of this data constitute a conformal class of rank-2 metrics, which represent the two gravitational degrees of freedom. The third piece, constructed from the extrinsic curvature of the boundary, determines the dynamical evolution of the boundary. [ABSTRACT FROM AUTHOR]

The canonical formalism of the (2+2) formulation of general relativity of four spacetime dimensions is studied under no symmetry assumptions, where the spacetime is viewed as a local product of a two dimensional base manifold of Lorentzian signature with the vertical space as its complement. The affine null parameter is chosen as the time coordinate whose level surfaces are three dimensional spacelike hypersurfaces. From the first-order action principle, Hamilton's equations of motion and the constraints are obtained, which are found to be equivalent to the Einstein's equations. The constraint algebra is also presented, which has interesting subalgebras such as the infinite dimensional Lie algebra of the diffeomorphisms of the two dimensional vertical space, infinite dimensional Virasoro algebra associated with the two dimensional base manifold, and an analogue of supertranslation. The symmetry algebra may be viewed as a generalization of the BMS or Spi group to a finite distance. [ABSTRACT FROM AUTHOR]

The action integral for a matter system composed of 0- and 2-forms, C and Bμν, topologically coupled to 3D spin-3 gravity is considered first in the frame-like formalism. The field C satisfies an equation of motion, , where Aμ and are the Chern–Simons gauge fields. With a suitable gauge fixing of a new local symmetry and diffeomorphism, only one component of Bμν, say Bϕr, remains non-vanishing and satisfies . These equations are the same as those for 3D (free) Vasiliev scalars, C and . The spin connection is eliminated by solving the equation of motion for the total action, and it is shown that in the resulting metric-like formalism, (BC)2 interaction terms are induced because of the torsion. The world-volume components of the matter field, C0, Cμ and C(μν), are introduced by contracting the local-frame index of C with those of the inverse vielbeins, and , which were defined by the present authors in (2013 Class. Quantum Grav.30 035003). 3D higher spin gravity theory contains various metric-like fields. These metric-like fields, as well as the new connections and the generalized curvature tensors, introduced in the above mentioned paper, are explicitly expressed in terms of the metric gμν and the spin-3 field ϕμνλ by means of the ϕ-expansion. The action integral for the pure spin-3 gravity in the metric-like formalism up to , obtained before in the literature, is re-derived. Then the matter action is re-expressed in terms of gμν, ϕμνρ and the covariant derivatives for spin-3 geometry. Spin-3 gauge transformation is extended to the matter fields. It is also found that the action of the matter-coupled theory in the metric-like formalism has larger symmetry than that of the pure spin-3 gravity. The matter-coupled theory in the metric-like formalism is invariant under the ordinary diffeomorphisms, because the vielbein and the spin connection are covariant vectors of diffeomorphisms. They are also gauge fields for the local translation. In the pure spin-3 gravity this symmetry provides the ordinary diffeomorphism and the spin-3 transformation in the metric-like formalism. When the matter is coupled, the local translation yields a new symmetry in the metric-like formalism, which does not contain diffeomorphism. [ABSTRACT FROM AUTHOR]

For non-convex Hamiltonians, the viscosity solution and the more geometric minimax solution of the Hamilton–Jacobi equation do not coincide in general. They are nevertheless related: we show that iterating the minimax procedure during shorter and shorter time intervals one recovers the viscosity solution. [ABSTRACT FROM AUTHOR]

A generalization of the representation, underlying the discrete spatial geometry of loop quantum gravity, to accomodate states labelled by smooth spatial geometries, was discovered by Koslowski and further studied by Sahlmann. We show how to construct the diffeomorphism constraint operator in this Koslowski-Sahlmann (KS) representation from suitable connection and triad dependent operators. We show that the KS representation supports the action of hitherto unnoticed 'background exponential' operators which, in contrast to the holonomy-flux operators, change the smooth spatial geometry label of the states. These operators are shown to be quantizations of certain connection dependent functions and play a key role in the construction of the diffeomorphism constraint operator. [ABSTRACT FROM AUTHOR]

Double field theory (DFT) is a proposal to incorporate T-duality, a distinctive symmetry of string theory, as a symmetry of a field theory defined on a double configuration space. The aim of this review is to provide a pedagogical presentation of DFT and its applications. We first introduce some basic ideas on T-duality and supergravity in order to proceed to the construction of generalized diffeomorphisms and an invariant action on the double space. Steps towards the construction of a geometry on the double space are discussed. We then address generalized Scherk-Schwarz compactifications of DFT and their connection to gauged supergravity and flux compactifications. We also discuss U-duality extensions and present a brief parcours on worldsheet approaches to DFT. Finally, we provide a summary of other developments and applications that are not discussed in detail in the review. [ABSTRACT FROM AUTHOR]

A hypersurface N formed of two null sheets, or 'light fronts', swept out by the future null normal geodesics emerging from a common spacelike 2-disc can serve as a Cauchy surface for a region of spacetime. Already in the 1960s, free (unconstrained) initial data for general relativity were found for such hypersurfaces. Here, an expression is obtained for the symplectic 2-form of vacuum general relativity in terms of such free data. This can be done, even though variations of the geometry do not in general preserve the nullness of the initial hypersurface, because of the diffeomorphism gauge invariance of general relativity. The present expression for the symplectic 2-form has been used previously (Reisenberger 2008 Phys. Rev. Lett. 101 211101) to calculate the Poisson brackets of the free data. [ABSTRACT FROM AUTHOR]

It has been shown that the canonical structure of the first-order Einstein-Hilbert (1EH) action involves three generations of constraints and that these can be used to find the generator of a gauge transformation which leaves the action invariant; this transformation is a diffeomorphism with field-dependent gauge function while on shell. In this paper we examine the relationship between the canonical structure of this action and that of the first-order spin-2 (1S2) action, which is the weak field limit of the Einstein-Hilbert action. We find that the weak field limit of the Possion brackets algebra of first class constraints associated with the 1EH action is not that of the 1S2 action. [ABSTRACT FROM AUTHOR]

We obtain sufficient conditions for an invariant splitting over a compact invariant subset of a C1 flow Xt to be dominated. In particular, we reduce the requirements to obtain sectional hyperbolicity and hyperbolicity. [ABSTRACT FROM AUTHOR]

Unimodular gravity is classically equivalent to standard Einstein gravity, but differs when it comes to the quantum theory: the conformal factor is non-dynamical, and the gauge symmetry consists of transverse diffeomorphisms only. Furthermore, the cosmological constant is not renormalized. Thus the quantum theory is distinct from a quantization of standard Einstein gravity. Here we show that within a truncation of the full renormalization group flow of unimodular quantum gravity, there is a non-trivial ultraviolet (uv)-attractive fixed point, yielding a UV completion for unimodular gravity. We discuss important differences to the standard asymptotic-safety scenario for gravity, and provide further evidence for this scenario by investigating a new form of the gauge-fixing and ghost sector. [ABSTRACT FROM AUTHOR]

A discrete model of Lorentzian quantum gravity is proposed. The theory is completely background free, containing no reference to absolute space, time, or simultaneity. The states at one slice of time are networks in which each vertex is labelled with two arrows, which point along an adjacent edge, or to the vertex itself. The dynamics is specified by a set of unitary replacement rules, which causally propagate the local degrees of freedom. The inner product between any two states is given by a sum over histories. Assuming it converges (or can be Abel resummed), this inner product is proven to be Hermitian and fully gauge-degenerate under spacetime diffeomorphisms. At least for states with a finite past, the inner product is also positive. This allows a Hilbert space of physical states to be constructed. [ABSTRACT FROM AUTHOR]

We prove that a C3 unimodal interval map with a non-flat critical point and a wild attractor satisfies the slow recurrence condition and is stochastically stable. [ABSTRACT FROM AUTHOR]

We present a non-perturbative canonical analysis of the D = 3 quadraticcurvature, yet ghost-free, model to exemplify a novel, 'constraint bifurcation', effect. Consequences include a jump in excitation count: a linearized level gauge variable is promoted to a dynamical one in the full theory. We illustrate these results with their concrete perturbative counterparts. They are of course mutually consistent, as are perturbative findings in related models. A geometrical interpretation in terms of propagating torsion reveals the model's relation to an (improved) version of Einstein-Weyl gravity at the linearized level. Finally, we list some necessary conditions for triggering the bifurcation phenomenon in general interacting gauge systems. [ABSTRACT FROM AUTHOR]

An important goal is to understand better the relation between full loop quantum gravity (LQG) and the simplified, reduced theory known as loop quantum cosmology (LQC), directly at the quantum level. Such a firmer understanding would increase confidence in the reduced theory as a tool for formulating predictions of the full theory, as well as permitting lessons from the reduced theory to guide further development in the full theory. This paper constructs an embedding of the usual state space of LQC into that of standard LQG, that is, LQG based on piecewise analytic paths. The embedding is well defined even prior to solving the diffeomorphism constraint, at no point is a graph fixed and at no point is the piecewise linear category used. This motivates for the first time a definition of operators in LQC corresponding to holonomies along non-piecewise linear paths, without changing the usual kinematics of LQC in any way. The new embedding intertwines all operators corresponding to such holonomies, and all elements in its image satisfy an operator equation which classically implies homogeneity and isotropy. The construction is made possible by a recent result proven by Fleischhack. [ABSTRACT FROM AUTHOR]

Viewing gravitational energy-momentum pµG as equal by observation, but different in essence from inertial energy-momentum pµ1 naturally leads to the gauge theory of volume-preserving diffeomorphisms of an inner Minkowski space M . The generalized asymptotic free scalar, Dirac and gauge fields in that theory are canonically quantized, the Fock spaces of stationary states are constructed and the gravitational limit--mapping the gravitational energy--momentum onto the inertial energy--momentum to account for their observed equality--is introduced. Next the S-matrix in quantum gravity is defined as the gravitational limit of the transition amplitudes of asymptotic into out-states in the gauge theory of volume-preserving diffeomorphisms. The so-defined S-matrix relates in- and out-states of observable particles carrying gravitational equal to inertial energy--momentum. Finally, generalized Lehmann-Symanzik-Zimmermann reduction formulae for scalar, Dirac and gauge fields are established which allow us to express S-matrix elements as the gravitational limit of truncated Fourier-transformed vacuum expectation values of time-ordered products of field operators of the interacting theory. Together with the generating functional of the latter established in Wiesendanger (2011 arXiv:1103.1012) any transition amplitude can in principle be computed consistently to any order in perturbative quantum gravity. [ABSTRACT FROM AUTHOR]

We study a type of geometric theory with a non-dynamical 1-form field. Its dynamical variables are an su(2) gauge field and a triad of su(2) valued 1-forms. Hamiltonian decomposition reveals that the theory has a true Hamiltonian, together with spatial diffeomorphism and Gauss lawconstraints, which generate the only local symmetries. Although perturbatively non-renormalizable, the model provides a test bed for the non-perturbative quantization techniques of loop quantum gravity. [ABSTRACT FROM AUTHOR]

A second-order formalism for the theory of 3D spin-3 gravity is considered. Such a formalism is obtained by solving the torsion-free condition for the spin connection ωaμ, and substituting the result into the action integral. In the first-order formalism of the spin-3 gravity defined in terms of SL(3, R) × SL(3, R) Chern-Simons (CS) theory, however, the generalized torsion-free condition cannot be easily solved for the spin connection, because the vielbein eaμ itself is not invertible. To circumvent this problem, extra vielbein-like fields ea (μv) are introduced as a functional of ea μ. New sets of affine-like connections ΓNμM are defined in terms of the metric-like fields, and a generalization of the Riemann curvature tensor is also presented. In terms of this generalized Riemann tensor the action integral in the second-order formalism is expressed. The transformation rules of the metric and the spin-3 gauge field under the generalized diffeomorphims are obtained explicitly. As in Einstein gravity, the new affine-like connections are related to the spin connection by a certain gauge transformation and a gravitational CS term expressed in terms of the new connections is also presented. [ABSTRACT FROM AUTHOR]

The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold M, to each spin structure σ and Riemannian metric g there is associated a space Sσ,g of spinor fields on M and a Hilbert space Hσ,g = L2(Sσ,g, volg(M)) of L2-spinors of Sσ,g. The group Diff+(M) of orientation-preserving diffeomorphisms of M acts both on g (by pullback) and on [σ] (by a suitably defined pullback f*s). Any f ∊ Diff+(M) lifts in exactly twoways to a unitary operator U from Hs,g to Hf*σ, f*g. The canonically defined Dirac operator is shown to be equivariant with respect to the action of U, so in particular its spectrum is invariant under the diffeomorphisms. [ABSTRACT FROM AUTHOR]

The holonomy-flux algebra μ of loop quantum gravity is known to admit a natural representation that is uniquely singled out by the requirement of covariance under spatial diffeomorphisms. In the cosmological context, the requirement of spatial homogeneity naturally reduces μ to a much smaller algebra, μRed, used in loop quantum cosmology. In Bianchi Imodels, it is shown that the requirement of covariance under residual diffeomorphism symmetries again uniquely selects the representation of μRed that has been commonly used. We discuss the close parallel between the two uniqueness results and also point out a difference. [ABSTRACT FROM AUTHOR]

We analyse in detail the three-moment maximum entropy radiation hydrodynamics which describes the motion of a gas of massless particles obeying Maxwell-Boltzmann, Bose-Einstein or Fermi-Dirac statistics. The primitive fields of this hydrodynamics are the energy density, the heat flux and the flux of the heat flux. We focus on the study of one-dimensional flows. Thus, the independent or non-vanishing variables defining the state of the gas are 3 in number. Two formulations are used: the formulation in terms of the Lagrange multipliers of the constrained optimization problem and the formulation in terms of the primitive fields. We prove that there exists a diffeomorphism between the open convex domain of Lagrange multipliers and the whole natural domain of primitive fields. [ABSTRACT FROM AUTHOR]

We reviewthe application of the spectral zeta function to the one-loop properties of quantum field theories on manifolds with boundary, with emphasis on Euclidean quantum gravity and quantum cosmology. As was shown in the literature some time ago, the only boundary conditions that are completely invariant under infinitesimal diffeomorphisms on metric perturbations suffer from a drawback, i.e. lack of strong ellipticity of the resulting boundaryvalue problem. Nevertheless, at least on the Euclidean 4-ball background, it remains possible to evaluate the ζ (0) value, which describes in this case a universe which, in the limit of small 3-geometry, has vanishing probability of approaching the cosmological singularity. An assessment of this result is performed here, discussing its physical and mathematical implications. This article is part of a special issue of Journal of Physics A:Mathematical and Theoretical in honour of StuartDowker's 75th birthday devoted to 'Applications of zeta functions and other spectral functions in mathematics and physics'. [ABSTRACT FROM AUTHOR]

Diffusion-weighted magnetic resonance imaging is a key investigation technique in modern neuroscience. In clinical settings, diffusion-weighted imaging and its extension to diffusion tensor imaging (DTI) are usually performed applying the technique of echo-planar imaging (EPI). EPI is the commonly available ultrafast acquisition technique for single-shot acquisition with spatial encoding in a Cartesian system. A drawback of these sequences is their high sensitivity against small perturbations of the magnetic field, caused, e.g., by differences in magnetic susceptibility of soft tissue, bone and air. The resulting magnetic field inhomogeneities thus cause geometrical distortions and intensity modulations in diffusion-weighted images. This complicates the fusion with anatomical T1- or T2-weighted MR images obtained with conventional spin- or gradient-echo images and negligible distortion. In order to limit the degradation of diffusion-weighted MR data, we present here a variational approach based on a reference scan pair with reversed polarity of the phase- and frequency-encoding gradients and hence reversed distortion. The key novelty is a tailored nonlinear regularization functional to obtain smooth and diffeomorphic transformations.We incorporate the physical distortion model into a variational image registration framework and derive an accurate and fast correction algorithm. We evaluate the applicability of our approach to distorted DTI brain scans of six healthy volunteers. For all datasets, the automatic correction algorithm considerably reduced the image degradation. We show that, after correction, fusion with T1- or T2- weighted images can be obtained by a simple rigid registration. Furthermore, we demonstrate the improvement due to the novel regularization scheme. Most importantly, we show that it provides meaningful, i.e. diffeomorphic, geometric transformations, independent of the actual choice of the regularization parameters. [ABSTRACT FROM AUTHOR]

In 2006, Gorodetski proved that central fibres of perturbed skew products are Hölder continuous with respect to the base point. In this paper, we give an explicit estimate of this Hölder exponent. Moreover, we extend Gorodetski's result from the case when the fibre maps are close to the identity to a much wider class of maps that satisfy the so-called modified dominated splitting condition. In many cases (for example, in the case of skew products over the solenoid or over linear Anosov diffeomorphisms of the torus), the Hölder exponent is close to 1. This allows one to overcome the so-called Fubini nightmare, in some sense. Namely, we prove that the union of central fibres that are strongly atypical from the point of view of ergodic theory, has Lebesgue measure zero despite the lack of absolute continuity of the holonomy map for the central foliation. This result is based on a new kind of ergodic theorem, which we call special. To prove our main result, we revisit the theory of Hirsch, Pugh and Shub, and estimate the contraction constant of the graph transform map. [ABSTRACT FROM AUTHOR]

The Kastor-Traschen metric is a time-dependent solution of the Einstein- Maxwell equations with positive cosmological constant Λ which can be used to describe an arbitrary number of charged dynamical black holes. In this paper, we consider the null geodesic structure of this solution, in particular, focusing on the projection to the space of orbits of the timelike conformal retraction. It is found that these projected light rays arise as integral curves of a system of third-order ordinary differential equations. This system is not uniquely defined, however, and we use the inherent freedom to construct a new system whose integral curves coincide with the projection of distinguished null curves of Kastor-Traschen arising from a magnetic flow. We discuss our results in the one-centre case and demonstrate a link to conformal circles in the limit Λ 0. We also show how to construct analytic expressions for the projected null geodesics of this metric by exploiting a well-known diffeomorphism between the K-T metric and extremal Reissner-Nordstrom-de Sitter. We make some remarks about the two-centre solution and demonstrate a link with the onecentre case. [ABSTRACT FROM AUTHOR]

We investigate the construction of coherent states for quantum theories of connections based on graphs embedded in a spatial manifold, as in loop quantum gravity. We discuss the many subtleties of the construction, mainly related to the diffeomorphism invariance of the theory. Aiming at approximating a continuum geometry in terms of discrete, graph-based data, we focus on coherent states for collective observables characterizing both the intrinsic and extrinsic geometry of the hypersurface, and we argue that one needs to revise accordingly the more local definitions of coherent states considered in the literature so far. In order to clarify the concepts introduced, we work through a concrete example that we hope will be useful in applying coherent state techniques to cosmology. [ABSTRACT FROM AUTHOR]

For a fixed initial reference measure, we study the dependence of the escape rate on the hole for a smooth or piecewise smooth hyperbolic map. First, we prove the existence and Hölder continuity of the escape rate for systems with small holes admitting Young towers. Then we consider general holes for Anosov diffeomorphisms, without size or Markovian restrictions. We prove bounds on the upper and lower escape rates using the notion of pressure on the survivor set and show that a variational principle holds under generic conditions. However, we also show that the escape rate function forms a devil's staircase with jumps along sequences of regular holes and present examples to elucidate some of the difficulties involved in formulating a general theory. [ABSTRACT FROM AUTHOR]

In this paper, we show two related results on circle diffeomorphisms. The first result is on quasi-reducibility: for a Baire-dense set of α, for any diffeomorphism f of rotation number α, it is possible to accumulate Rα with a sequence hnfh-1n, hn being a diffeomorphism. The second result is: for a Baire-dense set of α, given two commuting diffeomorphisms f and g, such that f has α for rotation number, it is possible to approach each of them by commuting diffeomorphisms fn and gn that are differentiably conjugated to rotations. In particular, it implies that if α is in this Baire-dense set, and if β is an irrational number such that (a, β) are not simultaneously Diophantine, then the set of commuting diffeomorphisms (f, g) with singular conjugacy, and with rotation numbers (α, β), respectively, is C∞-dense in the set of commuting diffeomorphisms with rotation numbers (α, β). [ABSTRACT FROM AUTHOR]