Volume 23, Number 1
Volume 23, Number 1, 2018
de la Llave R.
Uniform Boundedness of Iterates of Analytic Mappings Implies Linearization: a Simple Proof and Extensions
Abstract
A wellknown result in complex dynamics shows that if the iterates of an analytic
map are uniformly bounded in a complex domain, then the map is analytically conjugate to
a linear map. We present a simple proof of this result in any dimension. We also present several
generalizations and relations to other results in the literature.

Karaliolios N.
Local Rigidity of Diophantine Translations in Higherdimensional Tori
Abstract
We prove a theorem asserting that, given a Diophantine
rotation $\alpha $ in a torus $\mathbb{T} ^{d} \equiv \mathbb{R} ^{d} / \mathbb{Z} ^{d}$,
any perturbation, small enough in the $C^{\infty}$ topology,
that does not
destroy all orbits with rotation vector $\alpha$ is actually
smoothly conjugate to the rigid rotation. The proof relies
on a KAM scheme (named after Kolmogorov–Arnol'd–Moser),
where at each step the existence of an invariant measure with rotation
vector $\alpha$ assures that we can linearize the equations
around the same rotation $\alpha$. The proof of the convergence of
the scheme is carried out in the $C^{\infty}$ category.

Kozlov V. V.
Linear Hamiltonian Systems: Quadratic Integrals, Singular Subspaces and Stability
Abstract
A chain of quadratic first integrals of general linear Hamiltonian systems that have
not been represented in canonical form is found. Their involutiveness is established and the
problem of their functional independence is studied. The key role in the study of a Hamiltonian
system is played by an integral cone which is obtained by setting known quadratic first integrals
equal to zero. A singular invariant isotropic subspace is shown to pass through each point
of the integral cone, and its dimension is found. The maximal dimension of such subspaces
estimates from above the degree of instability of the Hamiltonian system. The stability of
typical Hamiltonian systems is shown to be equivalent to the degeneracy of the cone to an
equilibrium point. General results are applied to the investigation of linear mechanical systems
with gyroscopic forces and finitedimensional quantum systems.

MartínezTorres D., Miranda E.
Zeroth Poisson Homology, Foliated Cohomology and Perfect Poisson Manifolds
Abstract
We prove that, for compact regular Poisson manifolds, the zeroth homology group
is isomorphic to the top foliated cohomology group, and we give some applications. In particular,
we show that, for regular unimodular Poisson manifolds, top Poisson and foliated cohomology
groups are isomorphic. Inspired by the symplectic setting, we define what a perfect Poisson
manifold is. We use these Poisson homology computations to provide families of perfect Poisson
manifolds.

Buhovsky L., Kaloshin V.
Nonisometric Domains with the Same Marvizi–Melrose Invariants
Abstract
For any strictly convex planar domain
$\Omega \subset \mathbb R^2$ with a $C^\infty$ boundary
one can associate an infinite sequence of spectral
invariants introduced by Marvizi–Merlose~\cite{MM}.
These invariants can generically be determined using
the spectrum of the Dirichlet problem of the Laplace operator.
A natural question asks if this collection is sufficient to determine
$\Omega$ up to isometry. In this paper we give
a counterexample, namely, we present two nonisometric
domains $\Omega$ and $\bar \Omega$ with the same collection
of Marvizi–Melrose invariants. Moreover, each domain
has countably many periodic orbits $\{S^n\}_{n \geqslant 1}$ (resp.
$\{ \bar S^n\}_{n \geqslant 1}$) of period going to infinity such that
$ S^n $ and $ \bar S^n $ have the same period and perimeter for each $ n $.

Carpenter B. K., Ezra G. S., Farantos S. C., Kramer Z. C., Wiggins S.
Dynamics on the Double Morse Potential: A Paradigm for Roaming Reactions with no Saddle Points
Abstract
In this paper we analyze a twodegreeoffreedom Hamiltonian system constructed
from two planar Morse potentials. The resulting potential energy surface has two potential wells
surrounded by an unbounded flat region containing no critical points. In addition, the model
has an index one saddle between the potential wells. We study the dynamical mechanisms
underlying transport between the two potential wells, with emphasis on the role of the flat
region surrounding the wells. The model allows us to probe many of the features of the “roaming
mechanism” whose reaction dynamics are of current interest in the chemistry community.

Andrade J., Vidal C.
Stability of the Polar Equilibria in a Restricted Threebody Problem on the Sphere
Abstract
In this paper we consider a symmetric restricted circular threebody problem on the surface $\mathbb{S}^2$ of constant
Gaussian curvature $\kappa=1$. This problem consists in the description of the dynamics of an infinitesimal mass particle attracted
by two primaries with identical masses, rotating with constant angular velocity in a fixed parallel of radius $a\in (0,1)$.
It is verified that both poles of $\mathbb{S}^2$ are equilibrium points for any value of the parameter $a$. This problem is
modeled through a Hamiltonian system of two degrees of freedom depending on the parameter $a$. Using results concerning nonlinear
stability, the type of Lyapunov stability (nonlinear) is provided for the polar equilibria, according to the resonances.
It is verified that for the north pole there are two values of bifurcation (on the stability) $a=\dfrac{\sqrt{4\sqrt{2}}}{2}$ and $a=\sqrt{\dfrac{2}{3}}$,
while the south pole has one value of bifurcation $a=\dfrac{\sqrt{3}}{2}$.

Cruz I., MenaMatos H., SousaDias M.
Multiple Reductions, Foliations and the Dynamics of Cluster Maps
Abstract
Reduction of cluster maps via presymplectic and Poisson structures is described in terms of the canonical foliations defined by these structures. In the case where multiple reductions coexist, we establish conditions on the underlying presymplectic and Poisson structures that allow for an interplay between the respective foliations. It is also shown how this interplay may be explored to simplify the analysis and obtain an effective geometric description of the dynamics of the original (not reduced) map. Consequences of the approach we developed to the description of several features of some cluster maps dynamics are illustrated by two examples which include the Somos5 map and an instance of a Somos7 map.

Stankevich N. V., Dvorak A., Astakhov V. V., Jaros P., Kapitaniak M., Perlikowski P., Kapitaniak T.
Chaos and Hyperchaos in Coupled Antiphase Driven Toda Oscillators
Abstract
The dynamics of two coupled antiphase driven Toda oscillators is studied. We demonstrate three different routes of transition to chaotic dynamics associated with different bifurcations of periodic and quasiperiodic regimes. As a result of these, two types of chaotic dynamics with one and two positive Lyapunov exponents are observed. We argue that the results obtained are robust as they can exist in a wide range of the system parameters.
