ISSN:
1930-5311
eISSN:
1930-532X
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Journal of Modern Dynamics
October 2007 , Volume 1 , Issue 4
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2007, 1(4): 545-596
doi: 10.3934/jmd.2007.1.545
+[Abstract](4809)
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Abstract:
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.
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.
2007, 1(4): 597-613
doi: 10.3934/jmd.2007.1.597
+[Abstract](2135)
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Abstract:
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.
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.
2007, 1(4): 615-648
doi: 10.3934/jmd.2007.1.615
+[Abstract](2172)
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Abstract:
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.
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.
2007, 1(4): 649-663
doi: 10.3934/jmd.2007.1.649
+[Abstract](1670)
+[PDF](230.5KB)
Abstract:
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 $n \log^k n$.
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 $n \log^k n$.
2007, 1(4): 665-688
doi: 10.3934/jmd.2007.1.665
+[Abstract](2056)
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Abstract:
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.
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.
2007, 1(4): 689-718
doi: 10.3934/jmd.2007.1.689
+[Abstract](4440)
+[PDF](530.1KB)
Abstract:
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 ) )$/$2$.
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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