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Panos E. Livadas - One of the best experts on this subject based on the ideXlab platform.

  • The Kobayashi and Carathe´odory pseudodistances for complex Analytic Manifolds
    Information Sciences, 1994
    Co-Authors: Panos E. Livadas
    Abstract:

    Pseudodistances defined on Analytic Banach Manifolds permit one to obtain a number of results on that space by purely topological methods. In addition, they enable one to give geometric insight into function theoretic results. In this paper, we extend the notion of a complex Analytic Manifold to a complex Analytic Banach Manifold over a complex Banach space. Then we show that if M is a complex Analytic Banach Manifold, then the Kobayashi pseudodistance is the largest for which every holomorphic mapping from the unit disk of the complex plane into a complex Analytic Banach Manifold is distance decreasing, whereas the Caratheodory pseudodistance is the smallest pseudodistance from which every holomorphic mapping from the complex Analytic Manifold to the unit disk of the complex plane is distance decreasing.

  • the kobayashi and carathe odory pseudodistances for complex Analytic Manifolds
    Information Sciences, 1994
    Co-Authors: Panos E. Livadas
    Abstract:

    Pseudodistances defined on Analytic Banach Manifolds permit one to obtain a number of results on that space by purely topological methods. In addition, they enable one to give geometric insight into function theoretic results. In this paper, we extend the notion of a complex Analytic Manifold to a complex Analytic Banach Manifold over a complex Banach space. Then we show that if M is a complex Analytic Banach Manifold, then the Kobayashi pseudodistance is the largest for which every holomorphic mapping from the unit disk of the complex plane into a complex Analytic Banach Manifold is distance decreasing, whereas the Caratheodory pseudodistance is the smallest pseudodistance from which every holomorphic mapping from the complex Analytic Manifold to the unit disk of the complex plane is distance decreasing.

Hou-yi Chen - One of the best experts on this subject based on the ideXlab platform.

Daniel Panazzolo - One of the best experts on this subject based on the ideXlab platform.

Helge Glöckner - One of the best experts on this subject based on the ideXlab platform.

  • Invariant Manifolds for finite-dimensional non-archimedean dynamical systems
    arXiv: Dynamical Systems, 2014
    Co-Authors: Helge Glöckner
    Abstract:

    Let M be an Analytic Manifold modelled on an ultrametric Banach space over a complete ultrametric field. Let f be an Analytic diffeomorphism from M onto itself and p be a fixed point of f. We discuss invariant Manifolds around p, like stable Manifolds, centre-stable Manifolds and centre Manifolds, with an emphasis on results specific to the case that M has finite dimension. The results have applications in the theory of Lie groups over totally disconnected local fields.

  • Complexifications of infinite-dimensional Manifolds and new constructions of infinite-dimensional Lie groups
    arXiv: Differential Geometry, 2014
    Co-Authors: Rafael Dahmen, Helge Glöckner, Alexander Schmeding
    Abstract:

    Let M be a real Analytic Manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of the latter (as a complex Manifold) is uniquely determined. If M is regular and the complexified modeling space of M is normal, then a regular complexification exists for some neighborhood of K. For each (real or complex) Analytic regular Manifold M modeled on a metrizable locally convex space and Banach-Lie group H, this allows the group Germ(K,H) of germs of H-valued Analytic maps around K in M to be turned into an Analytic Lie group which is regular in Milnor's sense. A special case is a regular real Analytic Lie group structure on the group of real Analytic H-valued maps on a compact real Analytic Manifold M (which, previously, had only been treated in the convenient setting of analysis). Combining our results concerning groups of germs with an idea by Neeb and Wagemann, one can also obtain a regular Lie group structure on the group of all real Analytic H-valued mappings on the real line.

  • Invariant Manifolds for Analytic dynamical systems over ultrametric fields
    Expositiones Mathematicae, 2013
    Co-Authors: Helge Glöckner
    Abstract:

    Abstract We give an exposition of the theory of invariant Manifolds around a fixed point, in the case of time-discrete, Analytic dynamical systems over a complete ultrametric field K . Typically, we consider an Analytic Manifold M modelled on an ultrametric Banach space over K , an Analytic diffeomorphism f : M → M , and a fixed point p of f . Under suitable assumptions on the tangent map T p ( f ) , we construct a centre–stable Manifold, a centre Manifold, respectively, an a -stable Manifold around  p , for a given real number a ∈ ] 0 , 1 ] .

  • Invariant Manifolds for Analytic dynamical systems over ultrametric fields
    arXiv: Dynamical Systems, 2008
    Co-Authors: Helge Glöckner
    Abstract:

    We give an exposition of the theory of invariant Manifolds around a fixed point, in the case of time-discrete, Analytic dynamical systems over a complete ultrametric field K. Typically, we consider an Analytic Manifold M modelled on an ultrametric Banach space over K, an Analytic self-map f of M, and a fixed point p of f. Under suitable conditions on the tangent map of f at p, we construct a centre-stable Manifold, a centre Manifold, respectively, an r-stable Manifold around p, for a given positive real number r not exceeding 1. The invariant Manifolds are useful in the theory of Lie groups over local fields, where they allow results to be extended to the case of positive characteristic which previously were only available in characteristic zero (i.e., for p-adic Lie groups).

Alexander Schmeding - One of the best experts on this subject based on the ideXlab platform.

  • The Lie group of real Analytic diffeomorphisms is not real Analytic
    Studia Mathematica, 2015
    Co-Authors: Rafael Dahmen, Alexander Schmeding
    Abstract:

    We construct an infinite dimensional real Analytic Manifold structure for the space of real Analytic mappings from a compact Manifold to a locally convex Manifold. Here a map is real Analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the construction turns the group of real Analytic diffeomorphisms into a smooth locally convex Lie group. We prove then that the diffeomorphism group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real Analytic diffeomorphisms even form a real Analytic Lie group. However, we prove that the Lie group structure on the group of real Analytic diffeomorphisms is in general not real Analytic in our sense.

  • The Lie group of real Analytic diffeomorphisms is not real Analytic
    Studia Mathematica, 2015
    Co-Authors: Rafael Dahmen, Alexander Schmeding
    Abstract:

    We construct an infinite dimensional real Analytic Manifold structure for the space of real Analytic mappings from a compact Manifold to a locally convex Manifold. Here a map is real Analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the construction turns the group of real Analytic diffeomorphisms into a smooth locally convex Lie group. We prove then that the diffeomorphism group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real Analytic diffeomorphisms even form a real Analytic Lie group. However, we prove that the Lie group structure on the group of real Analytic diffeomorphisms is in general not real Analytic in our sense.Comment: 33 pages, LaTex, v2: now includes a proof for the regularity of the real Analytic diffeomorphism grou

  • Complexifications of infinite-dimensional Manifolds and new constructions of infinite-dimensional Lie groups
    arXiv: Differential Geometry, 2014
    Co-Authors: Rafael Dahmen, Helge Glöckner, Alexander Schmeding
    Abstract:

    Let M be a real Analytic Manifold modeled on a locally convex space and K be a non-empty compact subset of M. We show that if an open neighborhood of K in M admits a complexification which is a regular topological space, then the germ of the latter (as a complex Manifold) is uniquely determined. If M is regular and the complexified modeling space of M is normal, then a regular complexification exists for some neighborhood of K. For each (real or complex) Analytic regular Manifold M modeled on a metrizable locally convex space and Banach-Lie group H, this allows the group Germ(K,H) of germs of H-valued Analytic maps around K in M to be turned into an Analytic Lie group which is regular in Milnor's sense. A special case is a regular real Analytic Lie group structure on the group of real Analytic H-valued maps on a compact real Analytic Manifold M (which, previously, had only been treated in the convenient setting of analysis). Combining our results concerning groups of germs with an idea by Neeb and Wagemann, one can also obtain a regular Lie group structure on the group of all real Analytic H-valued mappings on the real line.