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Journal of Modern Dynamics

2007 , Volume 1 , Issue 2

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On configuration spaces of plane polygons, sub-Riemannian geometry and periodic orbits of outer billiards
Daniel Genin and Serge Tabachnikov
2007, 1(2): 155-173 doi: 10.3934/jmd.2007.1.155 +[Abstract](82) +[PDF](192.8KB)
Abstract:
Following a recent paper by Baryshnikov and Zharnitsky, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian geometry. We also prove that the set of 3-periodic outer billiard orbits has empty interior.
Solving the heptic by iteration in two dimensions: Geometry and dynamics under Klein's group of order 168
Scott Crass
2007, 1(2): 175-203 doi: 10.3934/jmd.2007.1.175 +[Abstract](68) +[PDF](370.0KB)
Abstract:
There is a family of seventh-degree polynomials $H$ whose members possess the symmetries of a simple group of order $168$. This group has an elegant action on the complex projective plane. Developing some of the action's rich algebraic and geometric properties rewards us with a special map that also realizes the $168$-fold symmetry. The map's dynamics provides the main tool in an algorithm that solves certain "heptic" equations in $H$.
Construction of ergodic cocycles that are fundamental solutions to linear systems of a special form
Mahesh G. Nerurkar and Héctor J. Sussmann
2007, 1(2): 205-253 doi: 10.3934/jmd.2007.1.205 +[Abstract](84) +[PDF](405.7KB)
Abstract:
If $T=\{T_t\}_{t\in\mathbb R}$ is an aperiodic measure-preserving jointly continuous flow on a compact metric space $\Omega$ endowed with a Borel probability measure $m$, and $G$ is a compact Lie group with Lie algebra $L$, then to each continuous map $A: \Omega \to L$ associate the solution $\Omega\times\mathbb R$ ∋ $(\omega,t)\mapsto X^A(\omega,t)\in G$ of the family of time-dependent initial-value problems $X'(t) = A(T_t\omega)X(t)$, $X(0) =$ identity, $X(t) \in G$ for $\omega\in \Omega$. The corresponding skew-product flow $T^A=\{T_t^A\}_{t\in\mathbb R}$ on $G\times\Omega$ is then defined by letting $T^A_t(g,\omega ) = (X^A(\omega ,t)g,T_t\omega)$ for $(g,\omega)\in G\times\Omega$, $t\in\mathbb R$. The flow $T^A$ is measure-preserving on $(G\times \Omega,\nu_{_G}\otimes m)$ (where $\nu_{_G}$ is normalized Haar measure on $G$) and jointly continuous. For a given closed convex subset $S$ of $L$, we study the set $C_{erg}(\Omega ,S)$ of all continuous maps $A: \Omega\to S$ for which the flow $T^A$ is ergodic. We develop a new technique to determine a necessary and sufficient condition for the set $C_{erg}(\Omega ,S)$ to be residual. Since the dimension of $S$ can be much smaller than that of $L$, our result proves that ergodicity is typical even within very "thin'' classes of cocycles. This covers a number of differential equations arising in mathematical physics, and in particular applies to the widely studied example of the Rabi oscillator.
Prequantum chaos: Resonances of the prequantum cat map
Frédéric Faure
2007, 1(2): 255-285 doi: 10.3934/jmd.2007.1.255 +[Abstract](85) +[PDF](362.7KB)
Abstract:
Prequantum dynamics was introduced in the 70s by Kostant, Souriau and Kirillov as an intermediate between classical and quantum dynamics. In common with the classical dynamics, prequantum dynamics transports functions on phase space, but adds some phases which are important in quantum interference effects. In the case of hyperbolic dynamical systems, it is believed that the study of the prequantum dynamics will give a better understanding of the quantum interference effects for large time, and of their statistical properties. We consider a linear hyperbolic map $M$ in SL $(2,\mathbb{Z})$ which generates a chaotic dynamical system on the torus. The dynamics is lifted to a prequantum fiber bundle. This gives a unitary prequantum (partially hyperbolic) map. We calculate its resonances and show that they are related to the quantum eigenvalues. A remarkable consequence is that quantum dynamics emerges from long-term behavior of prequantum dynamics. We present trace formulas, and discuss perspectives of this approach in the nonlinear case.
Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data
Anatole Katok and Federico Rodriguez Hertz
2007, 1(2): 287-300 doi: 10.3934/jmd.2007.1.287 +[Abstract](76) +[PDF](156.2KB)
Abstract:
Every $C^2$ action $\a$ of $\mathbb{Z}k$, $k\ge 2$, on the $(k+1)$-dimensional torus whose elements are homotopic to the corresponding elements of an action $\ao$ by hyperbolic linear maps has exactly one invariant measure that projects to Lebesgue measure under the semiconjugacy between $\a$ and $\a_0$. This measure is absolutely continuous and the semiconjugacy provides a measure-theoretic isomorphism. The semiconjugacy has certain monotonicity properties and preimages of all points are connected. There are many periodic points for $\a$ for which the eigenvalues for $\a$ and $\a_0$ coincide. We describe some nontrivial examples of actions of this kind.
Smooth Anosov flows: Correlation spectra and stability
Oliver Butterley and Carlangelo Liverani
2007, 1(2): 301-322 doi: 10.3934/jmd.2007.1.301 +[Abstract](1216) +[PDF](251.1KB)
Abstract:
By introducing appropriate Banach spaces one can study the spectral properties of the generator of the semigroup defined by an Anosov flow. Consequently, it is possible to easily obtain sharp results on the Ruelle resonances and the differentiability of the SRB measure.

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