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Journal of Modern Dynamics

2007 , Volume 1 , Issue 4

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Fifty years of entropy in dynamics: 1958--2007
Anatole Katok
2007, 1(4): 545-596 doi: 10.3934/jmd.2007.1.545 +[Abstract](1548) +[PDF](824.1KB)
These notes combine an analysis of what the author considers (admittedly subjectively) as the most important trends and developments related to the notion of entropy, with information of more “historical” nature including allusions to certain episodes and discussion of attitudes and contributions of various participants. I directly participated in many of those developments for the last forty three or forty four years of the fifty-year period under discussion and on numerous occasions was fairly close to the center of action. Thus, there is also an element of personal recollections with all attendant peculiarities of this genre.

For the full preface, please click the "Full Text" link above.
A note on Reeb dynamics on the tight 3-sphere
Frédéric Bourgeois, Kai Cieliebak and Tobias Ekholm
2007, 1(4): 597-613 doi: 10.3934/jmd.2007.1.597 +[Abstract](389) +[PDF](197.3KB)
We show that a nondegenerate tight contact form on the 3-sphere has exactly two simple closed Reeb orbits if and only if the differential in linearized contact homology vanishes. Moreover, in this case the Floquet multipliers and Conley-Zehnder indices of the two Reeb orbits agree with those of a suitable irrational ellipsoid in 4-space.
Critical points for surface diffeomorphisms
Enrique R. Pujals and Federico Rodriguez Hertz
2007, 1(4): 615-648 doi: 10.3934/jmd.2007.1.615 +[Abstract](394) +[PDF](373.4KB)
Using the definition of dominated splitting, we introduce the notion of critical set for any dissipative surface diffeomorphism as an intrinsically well-defined object. We obtain a series of results related to this concept.
Lower bounds on growth rates of periodic billiard trajectories in some irrational polygons
W. Patrick Hooper
2007, 1(4): 649-663 doi: 10.3934/jmd.2007.1.649 +[Abstract](314) +[PDF](230.5KB)
In this paper we show that there exist irrational polygons $P$ where the number of periodic billiard paths of length less than $n$, $f_P(n)$, grows superlinearly. In fact, if we fix the number of sides of our polygon, for any $k \in \N$ there is an open set of polygons where $f_P(n)$ grows faster than $n \log^k n$.
Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions
Danijela Damjanović
2007, 1(4): 665-688 doi: 10.3934/jmd.2007.1.665 +[Abstract](409) +[PDF](290.7KB)
We consider partially hyperbolic abelian algebraic higher-rank actions on compact homogeneous spaces obtained from simple split Lie groups of nonsymplectic type. We show that smooth, real-valued cocycles trivialize as well as small cocycles taking values in groups of diffeomorphisms of compact manifolds and some semisimple Lie groups. In the second part of the paper, we show local differentiable rigidity for such actions.
Slow soliton interaction with delta impurities
Justin Holmer and Maciej Zworski
2007, 1(4): 689-718 doi: 10.3934/jmd.2007.1.689 +[Abstract](1549) +[PDF](530.1KB)
We study the Gross--Pitaevskii equation with a delta function potential, $ q \delta_0 $, where $ |q| $ is small and analyze the solutions for which the initial condition is a soliton with initial velocity $ v_0 $. We show that up to time $ (|q| + v_0^2 )^{-1/2} \log$($1$/$|q|$) the bulk of the solution is a soliton evolving according to the classical dynamics of a natural effective Hamiltonian, $ (\xi^2 + q \, \sech^2 ( x ) )$/$2$.

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