scholarly journals A non-squeezing theorem for convex symplectic images of the Hilbert ball

2015 ◽  
Vol 54 (2) ◽  
pp. 1469-1506 ◽  
Author(s):  
Alberto Abbondandolo ◽  
Pietro Majer
Keyword(s):  
2014 ◽  
Vol 201 (5) ◽  
pp. 595-613
Author(s):  
M. Elin ◽  
M. Levenshtein ◽  
S. Reich ◽  
D. Shoikhet

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
Yasunori Kimura

We propose a new concept of set convergence in a Hadamard space and obtain its equivalent condition by using the notion of metric projections. Applying this result, we also prove a convergence theorem for an iterative scheme by the shrinking projection method in a real Hilbert ball.


2006 ◽  
Vol 2006 ◽  
pp. 1-16 ◽  
Author(s):  
Eva Kopecká ◽  
Simeon Reich
Keyword(s):  

2015 ◽  
Vol 117 (2) ◽  
pp. 203 ◽  
Author(s):  
P. Mellon

Let $g$ be a fixed-point free biholomorphic self-map of a bounded symmetric domain $B$. It is known that the sequence of iterates $(g^n)$ may not always converge locally uniformly on $B$ even, for example, if $B$ is an infinite dimensional Hilbert ball. However, $g=g_a\circ T$, for a linear isometry $T$, $a=g(0)$ and a transvection $g_a$, and we show that it is possible to determine the dynamics of $g_a$. We prove that the sequence of iterates $(g_a^n)$ converges locally uniformly on $B$ if, and only if, $a$ is regular, in which case, the limit is a holomorphic map of $B$ onto a boundary component (surprisingly though, generally not the boundary component of $\frac{a}{\|a\|}$). We prove $(g_a^n)$ converges to a constant for all non-zero $a$ if, and only if, $B$ is a complex Hilbert ball. The results are new even in finite dimensions where every element is regular.


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