Invariant rigid geometric structures and expanding maps

2011 ◽  
Vol 32 (3) ◽  
pp. 941-959 ◽  
Author(s):  
YONG FANG
Keyword(s):  
Dynamical Systems ◽  
Geometric Structure ◽  
Closed Manifold ◽  
Expanding Maps ◽  
Geometric Structures ◽  
Global Rigidity ◽  
Chaotic Dynamical Systems ◽  
Expanding Map ◽  
Locally Homogeneous ◽  
Rigid Geometric Structures

AbstractIn the first part of this paper, we consider several natural problems about locally homogeneous rigid geometric structures. In particular, we formulate a notion of topological completeness which is adapted to the study of global rigidity of chaotic dynamical systems. In the second part of the paper, we prove the following result: let φ be a C∞ expanding map of a closed manifold. If φ preserves a topologically complete C∞ rigid geometric structure, then φ is C∞ conjugate to an expanding infra-nilendomorphism.

Fujiki class 𝒞 and holomorphic geometric structures

2020 ◽  
Vol 31 (05) ◽  
pp. 2050039
Author(s):  
Indranil Biswas ◽  
Sorin Dumitrescu
Keyword(s):  
Chern Class ◽  
Geometric Structure ◽  
Geometric Structures ◽  
Cartan Geometry ◽  
Compact Torus ◽  
First Chern Class ◽  
Locally Homogeneous ◽  
Simply Connected ◽  
Affine Type ◽  
Compact Complex Manifolds

For compact complex manifolds with vanishing first Chern class that are compact torus principal bundles over Kähler manifolds, we prove that all holomorphic geometric structures on them, of affine type, are locally homogeneous. For a compact simply connected complex manifold in Fujiki class [Formula: see text], whose dimension is strictly larger than the algebraic dimension, we prove that it does not admit any holomorphic rigid geometric structure, and also it does not admit any holomorphic Cartan geometry of algebraic type. We prove that compact complex simply connected manifolds in Fujiki class [Formula: see text] and with vanishing first Chern class do not admit any holomorphic Cartan geometry of algebraic type.

scholarly journals Symmetries of holomorphic geometric structures on tori

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Sorin Dumitrescu ◽  
Benjamin McKay
Keyword(s):  
Homogeneous Space ◽  
Geometric Structure ◽  
Connected Components ◽  
Deformation Space ◽  
Higher Dimension ◽  
Geometric Structures ◽  
Translation Invariant ◽  
Homogeneous Surface ◽  
Complex Torus ◽  
Locally Homogeneous

AbstractWe prove that any holomorphic locally homogeneous geometric structure on a complex torus of dimension two, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true in any dimension. In higher dimension, we prove it for G nilpotent. We also prove that for any given complex algebraic homogeneous space (X, G), the translation invariant (X, G)-structures on tori form a union of connected components in the deformation space of (X, G)-structures.

scholarly journals Holomorphic affine connections on non-Kähler manifolds

2016 ◽  
Vol 27 (11) ◽  
pp. 1650094 ◽  
Author(s):  
Indranil Biswas ◽  
Sorin Dumitrescu
Keyword(s):  
Fundamental Group ◽  
Geometric Structure ◽  
Riemannian Metric ◽  
Complex Manifolds ◽  
Geometric Structures ◽  
Holomorphic Affine ◽  
Locally Homogeneous ◽  
Dimension One ◽  
Affine Type ◽  
Compact Complex Manifolds

Our aim here is to investigate the holomorphic geometric structures on compact complex manifolds which may not be Kähler. We prove that holomorphic geometric structures of affine type on compact Calabi–Yau manifolds with polystable tangent bundle (with respect to some Gauduchon metric on it) are locally homogeneous. In particular, if the geometric structure is rigid in Gromov’s sense, then the fundamental group of the manifold must be infinite. We also prove that compact complex manifolds of algebraic dimension one bearing a holomorphic Riemannian metric must have infinite fundamental group.

Pseudorandom Number Generators Based on Chaotic Dynamical Systems

2001 ◽  
Vol 08 (02) ◽  
pp. 137-146 ◽  
Author(s):  
Janusz Szczepański ◽  
Zbigniew Kotulski
Keyword(s):  
Dynamical Systems ◽  
Communication Systems ◽  
Initial Conditions ◽  
Pseudorandom Number ◽  
New Approach ◽  
Chaotic Dynamical Systems ◽  
Pseudorandom Number Generators ◽  
Transmission Of Information ◽  
Extreme Sensitivity ◽  
Contemporary Technology

Pseudorandom number generators are used in many areas of contemporary technology such as modern communication systems and engineering applications. In recent years a new approach to secure transmission of information based on the application of the theory of chaotic dynamical systems has been developed. In this paper we present a method of generating pseudorandom numbers applying discrete chaotic dynamical systems. The idea of construction of chaotic pseudorandom number generators (CPRNG) intrinsically exploits the property of extreme sensitivity of trajectories to small changes of initial conditions, since the generated bits are associated with trajectories in an appropriate way. To ensure good statistical properties of the CPRBG (which determine its quality) we assume that the dynamical systems used are also ergodic or preferably mixing. Finally, since chaotic systems often appear in realistic physical situations, we suggest a physical model of CPRNG.

CHARACTERIZING LOSS OF MEMORY IN CHAOTIC DYNAMICAL SYSTEMS

1991 ◽  
Vol 05 (14) ◽  
pp. 2323-2345 ◽  
Author(s):  
R.E. AMRITKAR ◽  
P.M. GADE
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Chaotic Dynamical Systems

We discuss different methods of characterizing the loss of memory of initial conditions in chaotic dynamical systems.

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