scholarly journals On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

2016 ◽  
Vol 182 (2) ◽  
pp. 463-487
Author(s):  
Fernando José Sánchez-Salas
Keyword(s):  
Hyperbolic Behavior ◽  
Nonuniformly Hyperbolic

scholarly journals A note on statistical properties for nonuniformly hyperbolic systems with slow contraction and expansion

2016 ◽  
Vol 16 (03) ◽  
pp. 1660012 ◽  
Author(s):  
Ian Melbourne ◽  
Paulo Varandas
Keyword(s):  
Invariance Principle ◽  
Hyperbolic Systems ◽  
Return Time ◽  
Limit Laws ◽  
Weak Invariance Principle ◽  
Weak Invariance ◽  
The Central Limit Theorem ◽  
Statistical Limit ◽  
Contraction And Expansion ◽  
Nonuniformly Hyperbolic

We provide a systematic approach for deducing statistical limit laws via martingale-coboundary decomposition, for nonuniformly hyperbolic systems with slowly contracting and expanding directions. In particular, if the associated return time function is square-integrable, then we obtain the central limit theorem, the weak invariance principle, and an iterated version of the weak invariance principle.

Some open sets of nonuniformly hyperbolic cocycles

1993 ◽  
Vol 13 (2) ◽  
pp. 409-415 ◽  
Author(s):  
L.-S. Young
Keyword(s):  
Lyapunov Exponents ◽  
Open Set ◽  
Nonuniformly Hyperbolic

AbstractWe consider some very simple examples ofSL(2, ℝ)-cocycles and prove that they have positive Lyapunov exponents. These cocycles form an open set in theC1topology.

AN INTRODUCTION TO QUANTITATIVE POINCARÉ RECURRENCE IN DYNAMICAL SYSTEMS

2009 ◽  
Vol 21 (08) ◽  
pp. 949-979 ◽  
Author(s):  
BENOIT SAUSSOL
Keyword(s):  
Dynamical Systems ◽  
Thermodynamic Formalism ◽  
Dimension Theory ◽  
Recurrence Rates ◽  
One Dimensional ◽  
Expanding Map ◽  
System A ◽  
Higher Dimensional ◽  
Technical Details ◽  
Hyperbolic Behavior

We present some recurrence results in the context of ergodic theory and dynamical systems. The main focus will be on smooth dynamical systems, in particular, those with some chaotic/hyperbolic behavior. The aim is to compute recurrence rates, limiting distributions of return times, and short returns. We choose to give the full proofs of the results directly related to recurrence, avoiding as much as possible to hide the ideas behind technical details. This drove us to consider as our basic dynamical system a one-dimensional expanding map of the interval. We note, however, that most of the arguments still apply to higher dimensional or less uniform situations, so that most of the statements continue to hold. Some basic notions from the thermodynamic formalism and the dimension theory of dynamical systems will be recalled.

scholarly journals Hyperbolic Metamaterials via Hierarchical Block Copolymer Nanostructures

2020 ◽  
Author(s):  
Irdi Murataj ◽  
Marwan Channab ◽  
Eleonora Cara ◽  
Candido Fabrizio Pirri ◽  
Luca Boarino ◽  
...  
Keyword(s):  
Optical Axis ◽  
Visible Spectrum ◽  
Pattern Transfer ◽  
Strong Reduction ◽  
Purcell Factor ◽  
Hierarchical Nanostructures ◽  
Hyperbolic Metamaterials ◽  
Propagation Of Light ◽  
Novel Applications ◽  
Hyperbolic Behavior

Abstract Hyperbolic metamaterials (HMMs) offer unconventional properties in the field of optics, enabling the opportunity for confinement and propagation of light at the nanoscale. In-plane orientation of the optical axis, in the direction coinciding with the anisotropy of the HMMs, is desirable for a variety of novel applications in nanophotonics and imaging. Here, we introduce a method for creating localized HMMs with in-plane optical axis based on block copolymers (BCPs) blend instability. The dewetting of BCP thin film over topographically defined substrates generates droplets composed of highly ordered lamellar nanostructures in hierarchical configuration. The hierarchical nanostructures represent a valuable platform for the subsequent pattern transfer into a Au/air HMM, exhibiting hyperbolic behavior in a broad wavelength range in the visible spectrum. A computed Purcell factor as high as 32 at 580 nm supports the strong reduction in the fluorescence lifetime of defects in nanodiamonds placed on top of the HMM.

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