scholarly journals Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group

2016 ◽  
Vol 8 (1) ◽  
pp. 1-12 ◽  
Author(s):  
James Benn
Keyword(s):  
Exponential Map ◽  
Symplectomorphism Group

New Parameterizations of \(\mathrm{SL}(2,\mathbb{R})\) and Some Explicit Formulas for Its Logarithm

2021 ◽  
Vol 60 ◽  
pp. 65-81
Author(s):  
Tihomir Valchev ◽  
◽  
Clementina Mladenova ◽  
Ivaïlo Mladenov
Keyword(s):  
Lie Groups ◽  
Analytic Description ◽  
Exponential Map ◽  
Explicit Formulas ◽  
Field Of Values

Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $\mathrm{Sp}(2,\bbr)\cong\mathrm{SL}(2,\bbr)$. Relying on the properties of the exponential map $\mathfrak{sl}(2,\bbr)\to\mathrm{SL}(2,\bbr)$, we have found a few explicit formulas for the logarithm of the matrices in these groups.\\ Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.

STABILIZATION OF CHAOTIC DYNAMICS OF ONE-DIMENSIONAL MAPS BY A CYCLIC PARAMETRIC TRANSFORMATION

1996 ◽  
Vol 06 (04) ◽  
pp. 725-735 ◽  
Author(s):  
ALEXANDER Yu. LOSKUTOV ◽  
VALERY M. TERESHKO ◽  
KONSTANTIN A. VASILIEV
Keyword(s):  
Periodic Orbits ◽  
Chaotic Dynamics ◽  
Chaotic Maps ◽  
Logistic Map ◽  
Exponential Map ◽  
One Dimensional ◽  
Stable Periodic ◽  
One Dimensional Maps ◽  
Parameter Values ◽  
Parametric Transformation

We consider one-dimensional maps, the logistic map and an exponential map, in those sets of parameter values which correspond to their chaotic dynamics. It is proven that such dynamics may be stabilized by a certain cyclic parametric transformation operating strictly within the chaotic set. The stabilization is a result of the creation of stable periodic orbits in the initially chaotic maps. The period of these stable orbits is a multiple of the period of the cyclic transformation. It is shown that stabilized behavior cannot be destroyed by a weak noise smearing of the required parameter values. The regions where the behavior stabilization takes place are numerically estimated. Periods of the created stabile periodic orbits are calculated.

scholarly journals On the normal exponential map in singular conformal metrics

2015 ◽  
Vol 127 ◽  
pp. 35-44 ◽  
Author(s):  
Roberto Giambò ◽  
Fabio Giannoni ◽  
Paolo Piccione
Keyword(s):  
Exponential Map ◽  
Conformal Metrics
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