This article explains why a paper by Heinz G. Helfenstein entitled "Ovals
with equichordal points", published in J.London Math.Soc.31, 54-57, 1956, is
incorrect. We point out a computational error which renders his conclusions
invalid. More importantly, we explain that the method cannot be used to solve
the equichordal point problem with the method presented there. Today, there is
a solution to the problem: Marek R. Rychlik, "A complete solution to the
equichordal point problem of Fujiwara, Blaschke, Rothe and Weizenb\"ock",
Inventiones Mathematicae 129 (1), 141-212, 1997.

We give a complete description of the qualitative behavior of the
second-order rational difference equation #166. We also establish the
boundedness character for the rational system in the plane #(8,30).

This paper is devoted to study thermodynamic formalism for suspension flows
defined over countable alphabets. We are mostly interested in the regularity
properties of the pressure function. We establish conditions for the pressure
function to be real analytic or to exhibit a phase transition. We also
construct an example of a potential for which the pressure has countably many
phase transitions.

In this work we introduce the notion of weak quasi groups, that are,
quasi-group operations defined almost everywhere on some set. Then we present
sufficient conditions for an expansive ergodic map $T:X\to X$ to be an
automorphism for some topological weak quasi group. Therefore, we find out an
Abelian topological weak group operation and a standard decomposition of the
dynamics of $T$ in terms of $T$-invariant weak sub-groups.

The FitzHugh-Nagumo equation has been investigated with a wide array of
different methods in the last three decades. Recently a version of the
equations with an applied current was analyzed by Champneys, Kirk, Knobloch,
Oldeman and Sneyd using numerical continuation methods. They obtained a
complicated bifurcation diagram in parameter space featuring a C-shaped curve
of homoclinic bifurcations and a U-shaped curve of Hopf bifurcations.

We translate Akin's notion of {\it good} (and related concepts) from measures
on Cantor sets to traces on dimension groups, and particularly for invariant
measures of minimal homeomorphisms (and their corresponding simple dimension
groups), this yields characterizations and examples, which translate back to
the original context. Good traces on a simple dimension group are characterized
by their kernel having dense image in their annihilating set of affine
functions on the trace space; this makes it possible to construct many examples
with seemingly paradoxical properties.

These are expanded lecture notes for the summer school on Berkovich spaces
that took place at the Institut de Math\'ematiques de Jussieu, Paris in 2010.
They serve to illustrate some techniques and results from the dynamics on
low-dimensional Berkovich spaces and to exhibit the structure of these spaces.

Meminductors and memcapacitors do not allow a Lagrangian formulation in the
classical sense since these elements are nonconservative in nature and the
associated energies are not state functions. To circumvent this problem, a
different configuration space is considered that, instead of the usual loop
charges, consist of time-integrated loop charges. As a result, the
corresponding Euler-Lagrange equations provide a set of integrated Kirchhoff
voltage laws in terms of the element fluxes. Memristive losses can be included
via a second scalar function that has the dimension of action.

Consider in R^2 the semi-planes N={y>0} and S={y<0}$ having as common
boundary the straight line D={y=0}$. In N and S are defined polynomial vector
fields X and Y, respectively, leading to a discontinuous piecewise polynomial
vector field Z=(X,Y). This work pursues the stability and the transition
analysis of solutions of Z between N and S, started by Filippov (1988) and
Kozlova (1984) and reformulated by Sotomayor-Teixeira (1995) in terms of the
regularization method.

In this paper, a new rigorous numerical method to compute fundamental matrix
solutions of non-autonomous linear differential equations with periodic
coefficients is introduced. Decomposing the fundamental matrix solutions
$\Phi(t)$ by their Floquet normal forms, that is as product of real periodic
and exponential matrices $\Phi(t)=Q(t)e^{Rt}$, one solves simultaneously for
$R$ and for the Fourier coefficients of $Q$ via a fixed point argument in a
suitable Banach space of rapidly decaying coefficients.

We prove that the Hausdorff dimension of the set of three-period orbits in
classical billiards is at most one. Moreover, if the set of three-period orbits
has Hausdorff dimension one, then it has a tangent line at almost every point.

We compute the Cech cohomology with integer coefficients of one-dimensional
tiling spaces arising from not just one, but several different substitutions,
all acting on the same set of tiles. These calculations involve the
introduction of a universal version of the Anderson-Putnam complex. We show
that, under a certain condition on the substitutions, the projective limit of
this universal Anderson-Putnam complex is isomorphic to the tiling space, and
we introduce a simplified universal Anderson-Putnam complex that can be used to
compute Cech cohomology.

We prove existence of absolutely continuous invariant probability measures
for skew-products with arbitrary base dynamics and asymptotic expansion along
the one-dimensional fibers. We also prove a similar result for skew-product
maps having higher-dimensional fibers with essentially arbitrary base dynamics
and non-uniform expansion along the fibers. In both cases either critical or
singular points on the dynamics along the fibers are admitted.

A pentagon in the plane with fixed side-lengths has a two-dimensional shape
space. Considering the pentagon as a mechanical system with point masses at the
corners we answer the question of how much the pentagon can rotate with zero
angular momentum. We show that the shape space of the equilateral pentagon has
genus 4 and find a fundamental region by discrete symmetry reduction with
respect to symmetry group D_5.

If the stable, center, and unstable foliations of a partially hyperbolic
system are quasi-isometric, the system has Global Product Structure. This
result also applies to Anosov systems and to other invariant splittings.

If a partially hyperbolic system on a manifold with abelian fundamental group
has quasi-isometric stable and unstable foliations, the center foliation is
without holonomy. If, further, the system has Global Product Structure, then
all center leaves are homeomorphic.

Systems biology of plants offers myriad opportunities and many challenges in
modeling. A number of technical challenges stem from paucity of computational
methods for discovery of the most fundamental properties of complex dynamical
systems. In systems engineering, eigen-mode analysis have proved to be a
powerful approach.

In this paper we investigate multifractal decompositions based on values of
Birkhoff averages of functions from a class of symbolically continuous
functions. This will be done for an expanding interval map with infinitely many
branches and is a generalisation of previous work for expanding maps with
finitely many branches. We show that there are substantial differences between
this case and the setting where the expanding map has only finitely many
branches.

We investigate the invariant probability measures for Cherry flows, i.e.
flows on the two-torus which have a saddle, a source, and no other fixed
points, closed orbits or homoclinic orbits. In the case when the saddle is
dissipative or conservative we show that the only invariant probability
measures are the Dirac measures at the two fixed points, and the Dirac measure
at the saddle is the physical measure.

We prove the existence of at least two geometrically different periodic
solution with winding number N for the forced relativistic pendulum. The
instability of a solution is also proved. The proof is topological and based on
the version of the Poincar\'e-Birkhoff theorem by Franks. Moreover, with some
restriction on the parameters, we prove the existence of twist dynamics.

We establish the existence of new rigidity and rationality phenomena in the
theory of nonabelian group actions on the circle, and introduce tools to
translate questions about the existence of actions with prescribed dynamics
into finite combinatorics. A special case of our theory gives a very short new
proof of Naimi's theorem (i.e. the conjecture of Jankins-Neumann) which was the
last step in the classification of taut foliations of Seifert fibered spaces.

We show that if $f \colon S^1 \times S^1 \to S^1 \times S^1$ is $C^2$, with
$f(x, t) = (f_t(x), t)$, and the rotation number of $f_t$ is equal to $t$ for
all $t \in S^1$, then $f$ is topologically conjugate to the linear Dehn twist
of the torus $(1&1 0&1)$. We prove a differentiability result where the
assumption that the rotation number of $f_t$ is $t$ is weakened to say that the
rotation number is strictly monotone in $t$.

Following [6,12], we study coupled map networks over arbitrary finite graphs.
An estimate from below for a topological entropy of a perturbed coupled map
network via a topological entropy of an unperturbed network by making use of
the covering relations for coupled map networks is obtained. The result is
quite general, particularly no assumptions on hyperbolicity of a local dynamics
or linearity of coupling are made.

An analog of the Baumslag-Solitar group BS(1,k) naturally acts on the sphere
by conformal transformations. The action is not locally rigid in higher
dimension, but exhibits a weak form of local rigidity. More precisely, any
perturbation preserves a smooth conformal structure.

There are several different common definitions of a property in topological
dynamics called "topological transitivity," and it is part of the folklore of
dynamical systems that under reasonable hypotheses, they are equivalent.
Various equivalences are proved in different places, but the full story is
difficult to find. This note provides a complete description of the
relationships among the different properties.

We investigate the use of iterated function system (IFS) models for data
analysis. An IFS is a collection of dynamical systems that switches between
deterministic regimes. An algorithm is developed to detect the regime switches
under the assumption of continuity. This method is tested on a simple IFS and
applied to an experimental computer performance data set. This methodology has
a wide range of potential uses: from change-point detection in time-series
data, to the field of digital communications.

We consider a ground state (soliton) of a Hamiltonian PDE. We prove that if
the soliton is orbitally stable then it is also asymptotically stable. The main
assumptions are transversal nondegeneracy of the manifold of ground states,
linear dispersion (in the form of Strichartz estimates) and nonlinear Fermi
Golden Rule. We allow for an arbitrary number of eigenvalues of the
linearization of the equations at the soliton. The theory is modeled on the
application to the translational invariant NLS in space dimension 3.

The arithmetics of the frequency and of the rotation number play a fun-
damental role in the study of reducibility of analytic quasi-periodic cocycles
which are sufficiently close to a constant. In this paper we show how to
generalize previous works by L.H.Eliasson which deal with the diophantine case
so as to implement a Brjuno-Russmann arithmetical condition both on the
frequency and on the rotation number. Our approach adapts the Poschel-Russmann
KAM method, which was previously used in the problem of linearization of vector
fields, to the problem of reducing cocycles.

We investigate the existence of Arnold diffusion-type orbits for systems
obtained by iterating in any order the flows of a family of Tonelli
Hamiltonians. Our approach is close to the one of Bernard in [3]. When
specialized to families of twist maps, our results are similar to those of
Moeckel [20] and Le Calvez [15], and generalize the connecting results of
Mather for a single twist map in [18].

In this paper we consider the Stochastic isothermal, nonlinear,
incompressible bipolar viscous fluids driven by a genuine cylindrical
fractional Bronwnian motion with Hurst parameter $H \in (1/4,1/2)$ under
Dirichlet boundary condition on 2D square domain. First we prove the existence
and regularity of the stochastic convolution corresponding to the stochastic
non-Newtonian fluids. Then we obtain the existence and uniqueness results for
the stochastic non-Newtonian fluids. Under certain condition, the random
dynamical system generated by non-Newtonian fluids has a random attractor.

The periodic wind-tree model is a family T(a,b) of billiards in the plane in
which identical rectangular scatterers of size axb are disposed at each integer
point. It was proven by P. Hubert, S. Leli\`evre and S. Troubetzkoy
(arXiv:0912.2891v1) that for a residual set of parameters (a,b) the billiard
flow in T(a,b) is recurrent in almost every direction. We prove that for many
parameters (a,b) there exists a set S of angles of positive Hausdorff dimension
such that every billiard trajectory in T(a,b) with initial angle in S is
self-avoiding.

Let $\bS=\{S_1,...,S_K\}$ be a finite set of complex $d\times d$ matrices and
$\varSigma_{K}^+$ the compact space of all one-sided infinite sequences
$i_{\bcdot}\colon\mathbb{N}\rightarrow\{1,...,K\}$. An ergodic probability
$\mu_*$ of the Markov shift
$\theta\colon\varSigma_{K}^+\rightarrow\varSigma_{K}^+;\ i_{\bcdot}\mapsto
i_{\bcdot+1}$, is called "extremal" for $\bS$, if
${\rho}(\bS)=\lim_{n\to\infty}\sqrt[n]{\norm{S_{i_1}...S_{i_n}}}$ holds for
$\mu_*$-a.e. $i_{\bcdot}\in\varSigma_{K}^+$, where $\rho(\bS)$ denotes the
generalized/joint spectral radius of $\bS$.

We prove that the Lyapunov exponent of quasi-periodic cocyles with
singularities behaves continuously over the analytic category. We thereby
generalize earlier results, where singularities were either excluded completely
or constrained by additional hypotheses. Applications are one-parameter
families of analytic Jacobi operators, such as extended Harper's model
describing crystals subject to external magnetic fields.

We study quantitatively a dynamically weighted asymptotic Feketeness of
pullbacks of points under a rational function of degree $d>1$ on the projective
line over a possibly non-archimedean algebraically closed field which is more
general than that of complex numbers.

We consider $(M,d)$ a connected and compact manifold and we denote by $X$ the
Bernoulli space $M^{\mathbb{N}}$. The shift acting on $X$ is denoted by
$\sigma$.

Deviation of ergodic sums is studied for substitution dynamical systems with
a matrix that admits eigenvalues of modulus 1. We consider the corresponding
eigenfunctions, and in Theorem 1.1 we prove that the limit inferior of the
ergodic sums is bounded for every point in the phase space. In Theorem 1.2, we
prove existence of limit distributions along certain exponential subsequences
of times for substitutions of constant length. Under additional assumptions, we
prove that ergodic integrals satisfy the Central Limit Theorem (Theorem 1.3,
Theorem 1.9).

In this paper we study the differential equations in $D\subseteq \R^{2N}$
having a complete set of independent first integrals. In particular we study
the case when the first integrals are
\[f_\nu=(Ax_\nu+By_\nu)^2+\displaystyle\sum_{j=1}^{N}\dfrac{(x_\nu
y_j-x_jy_\nu)^2}{a_\nu-a_j},\]for $\nu=1,...,N,$ where $A,B$ and
$a_1<a_2...<a_N$ are constants.

For a normal subgroup $N$ of the free group $\F_{d}$ with at least two
generators we introduce the radial limit set $L_{r}(N,\Phi)$ of $N$ with
respect to a graph directed Markov system $\Phi$ associated to $\F_{d}$. These
sets are shown to provide fractal models of radial limit sets of normal
subgroups of Kleinian groups of Schottky type.

In late sixties, Mihail Budyko and William Sellers, a Russian and an American
climate scientists, independently introduced the concept of Energy Balance
Model with ice albedo feedback. Since then many have followed in their
footsteps to establish various versions of this model. In this paper, a novel
equation is introduced to account for the dynamics of the ice line, and is
coupled to Budyko's model. We found that the coupled temperature profile-ice
line system has a one dimensional center stable manifold.

The no invariant line fields conjecture is one of the main outstanding
problems in traditional complex dynamics. In this paper we consider
non-autonomous iteration where one works with compositions of sequences of
polynomials with suitable bounds on the degrees and coefficients. We show that
the natural generalization of the no invariant line fields conjecture to this
setting is not true.

Let $f:M\rightarrow M$ be a biholomorphisms on two--dimensional a complex
manifold, and let $X\subseteq M$ be a compact $f$--invariant set such that
$f|X$ is asymptotically dissipative and without sinks periodic points. We
introduce a solely dynamical obstruction to dominated splitting, namely
critical point. Critical point is a dynamical object and capture many of the
dynamical properties of their one--dimensional counterpart.

We study an intermittent map which has exactly two ergodic invariant
densities. The densities are supported on two subintervals with a common
boundary point. Due to certain perturbations, leakage of mass through subsets,
called holes, of the initially invariant subintervals occurs and forces the
subsystems to merge into one system that has exactly one invariant density. We
prove that the invariant density of the perturbed system converges in the
$L^1$-norm to a particular convex combination of the invariant densities of the
intermittent map.

The main results of this paper are limit theorems for horocycle flows on
compact surfaces of constant negative curvature. One of the main objects of the
paper is a special family of horocycle-invariant finitely-additive Hoelder
measures on rectifiable arcs. An asymptotic formula for ergodic integrals for
horocycle flows is obtained in terms of the finitely-additive measures, and
limit theorems follow as a corollary of the asymptotic formula. The objects and
results of this paper are similar to those in [15], [16], [4] and [5] for
translation flows on flat surfaces.

E. G. Effros and C-L. Shen constructed the dimension group of free rank 2
from the simple continued fraction algorithm. The notion of negative slope
algorithm was introduced by S. Ferenczi, C. Holton, and L. Zamboni in their
study of 3-interval exchange transformations. The negative slope algorithm is
the 2-dimensional continued fraction algorithm. Then the author succeed to
construct the dimensional group of free rank 3 by the similar method which E.
G. Effros and C-L. Shen used.

We show that there are no normally contracting actions of unimodular Lie
groups on closed manifolds.

We consider the standard family of area-preserving twist maps of the annulus
and the corresponding KAM curves. Inspired by Kolmogorov, Arnold and Herman, we
show that, instead of viewing these invariant curves as separate objects, each
of which having its own Diophantine frequency, one can encode them in a single
function of the frequency which is monogenic in the sense of Borel; this
implies a remarkable property of quasianalyticity.

We study the effects of noise in two models of spiny dendrites. Through the
introduction of different types of noise to both the Spike-diffuse-spike (SDS)
and Baer-Rinzel (BR) models we investigate the change in behaviour of the
travelling wave solutions present in the deterministic systems, as noise
intensity increases. We show that the speed of wave propagation in the SDS and
BR models respectively decreases and increases as the noise intensity in the
spine heads increases.

Time-delayed control in a balancing problem may be a nonsmooth function for a
variety of reasons. In this paper we study a simple model of the control of an
inverted pendulum by either a connected movable cart or an applied torque for
which the control is turned off when the pendulum is located within certain
regions of phase space. Without applying a small angle approximation for
deviations about the vertical position, we see structurally stable periodic
orbits which may be attracting or repelling.

Helmut Hofer introduced in '93 a novel technique based on holomorphic curves
to prove the Weinstein conjecture. Among the cases where these methods apply
are all contact 3--manifolds $(M,\xi)$ with $\pi_2(M) \ne 0$. We modify Hofer's
argument to prove the Weinstein conjecture for some examples of higher
dimensional contact manifolds. In particular, we are able to show that the
connected sum with a real projective space always has a closed contractible
Reeb orbit.

We generalize recent developments on normal forms and the spectral sequences
method to make a foundation for parametric normal forms. We further introduce a
new style and costyle to obtain unique parametric normal forms. The results are
applied to systems of generalized Hopf singularity with multiple parameters. A
different (new) version of this paper has been submitted for a possible
publication in a refereed journal.

In this paper, we consider the family of rational maps $$\F(z) = z^n +
\frac{\la}{z^d},$$ where $n \geq 2$, $d\geq 1$, and$\la \in \bbC$. We consider
the case where $\la$ lies in the main cardioid of one of the $n-1$ principal
Mandelbrot sets in these families. We show that the Julia sets of these maps
are always homeomorphic. However, two such maps $\F$ and $F_\mu$ are conjugate
on these Julia sets only if the parameters at the centers of the given
cardioids satisfy $\mu = \nu^{j(d+1)}\la$ or $\mu = \nu^{j(d+1)}\bar{\la}$
where $j \in \bbZ$ and $\nu$ is an $n-1^{\rm st}$ root of unity.

Among the topological conjugacy classes of the continuous flows $\{\phi^t\}$
whose orbit foliations are the planar Reeb foliation, there is one class called
the standard Reeb flow. We show that $\{\phi^t\}$ is conjugate to the standard
Reeb flow if and only if $\{\phi^t\}$ is conjugate to $\{\phi^{\lambda t}\}$
for any $\lambda>0$.

Approaching a dangerous bifurcation, from which a dynamical system such as
the Earth's climate will jump (tip) to a different state, the current stable
state lies within a shrinking basin of attraction. Persistence of the state
becomes increasingly precarious in the presence of noisy disturbances. We
consider an underlying potential, as defined theoretically for a saddle-node
fold and (via averaging) for a Hopf bifurcation. Close to a stable state, this
potential has a parabolic form; but approaching a jump it becomes increasingly
dominated by softening nonlinearities.

This paper is concerned with the dynamics of an infinite-dimensional gradient
system under small almost periodic perturbations. Under the assumption that the
original autonomous system has a global attractor given as the union of
unstable manifolds of a finite number of hyperbolic equilibrium solutions, we
prove that the perturbed non-autonomous system has exactly the same number of
almost periodic solutions. As a consequence, the pullback attractor of the
perturbed system is given by the union of unstable manifolds of these finitely
many almost periodic solutions.

We prove positive characteristic versions of the logarithm laws of Sullivan
and Kleinbock-Margulis and obtain related results in Metric Diophantine
Approximation.

We consider volume-preserving flows $(\Phi^f_t)_{t\in\mathbb{R}}$ on $S\times
\mathbb{R}$, where $S$ is a closed connected surface of genus $g\geq 2$ and
$(\Phi^f_t)_{t\in\mathbb{R}}$ has the form $\Phi^f_t(x,y)=(\phi_tx,y+\int_0^t
f(\phi_sx)ds)$, where $(\phi_t)_{t\in\mathbb{R}}$ is a locally Hamiltonian flow
of hyperbolic periodic type on $S$ and $f$ is a smooth real valued function on
$S$.

In this long paper we give a fairly complete analysis of outer billiards on
the Penrose kite. Our analysis reveals that this 2-dimensional non-compact
system has a 3-dimensional compactification, a certain polyhedron exchange map,
and that this compactification has a renormalization scheme. These two features
allow us to make some sharp statements concerning the distribution, large-scale
geometry, fine-scale geometry, and hidden algebraic symmetries of the orbits.
For instance, one of our results is that the union of the unbounded orbits has
Hausdorff dimension 1.

Many interesting physical systems have mathematical descriptions as
finite-dimensional or infinite-dimensional Hamiltonian systems. Poincare who
started the modern theory of dynamical systems and symplectic geometry
developed a particular viewpoint combining geometric and dynamical systems
ideas in the study of Hamiltonian systems. After Poincare the field of
dynamical systems and the field of symplectic geometry developed separately.
Both fields have rich theories and the time seems ripe to develop the common
core with highly integrated ideas from both fields.

In this paper, we develop numerical algorithms that use small requirements of
storage and operations for the computation of hyperbolic cocycles over a
rotation. We present fast algorithms for the iteration of the quasi-periodic
cocycles and the computation of the invariant bundles, which is a preliminary
step for the computation of invariant whiskered tori.

In this article, we study the global dynamics of a discrete two dimensional
competition model. We give sufficient conditions on the persistence of one
species and the existence of local asymptotically stable interior period-2
orbit for this system. Moreover, we show that for a certain parameter range,
there exists a compact interior attractor that attracts all interior points
except a Lebesgue measure zero set. This result gives a weaker form of
coexistence which is referred to as relative permanence.

We formulate general plant-herbivore interaction models with monotone plant
growth functions (rates). We study the impact of monotone plant growth
functions in general plant-herbivore models on their dynamics. Our study shows
that all monotone plant growth models generate a unique interior equilibrium
and they are uniform persistent under certain range of {parameters} values.
However, if the attacking rate of herbivore is too small or the quantity of
plant is not enough, then herbivore goes extinct.

We show that expanding toral endomorphisms, together with their respective
Lebesgue measure are isomorphic to 1-sided Bernoulli shifts. This result is
then extended to smooth perturbations of expanding toral endomorphisms,
together with their respective measures of maximal entropy. Also we study group
extensions of expanding toral endomorphisms and show that under certain, not
too restrictive conditions on the extension cocycle, these skew products are
1-sided Bernoulli as well.

By varying a parameter of a one-dimensional piecewise smooth map, stable
periodic orbits are observed. In this paper, complete analytic characterization
of these stable periodic orbits is obtained. An interesting relationship
between the cardinality of orbits and their period is established. It is proved
that for any $n$, there exist $\phi(n)$ distinct admissible patterns of
cardinality $n$. An algorithm to obtain these distinct admissible patterns is
outlined. Additionally, a novel algorithm to find the range of parameter for
which the orbit exists is proposed.

We study periodic linear trajectories in the double pentagon and periodic
billiard trajectories in the regular pentagon.

In a previous work, we prove the existence of weak solutions to an
initial-boundary value problem, with $H^1(\Omega)$ initial data, for a system
of partial differential equations, which consists of the equations of linear
elasticity and a nonlinear, degenerate parabolic equation of second order.
Assuming in this article the initial data is in $H^2(\Omega)$, we investigate
the regularity of weak solutions that is difficult due to the gradient term
which plays a role of a weight. The problem models the behavior in time of
materials with martensitic phase transitions.

We prove the global in time existence of spherically symmetric solutions to
an initial-boundary value problem for a system of partial differential
equations, which consists of the equations of linear elasticity and a
nonlinear, non-uniformly parabolic equation of second order. The problem models
the behavior in time of materials in which martensitic phase transitions,
driven by configurational forces, take place, and can be considered to be a
regularization of the corresponding sharp interface model.

We study the problem of lifting various mixing properties from a base
automorphism $T\in {\rm Aut}\xbm$ to skew products of the form $\tfs$, where
$\va:X\to G$ is a cocycle with values in a locally compact Abelian group $G$,
$\cs=(S_g)_{g\in G}$ is a measurable representation of $G$ in ${\rm Aut}\ycn$
and $\tfs$ acts on the product space

There are two main reasons why relative equilibria of N point masses under
the influence of Newton attraction are mathematically more interesting to study
when space dimension is at least 4: On the one hand, in a higher dimensional
space, a relative equilibrium is determined not only by the initial
configuration but also by the choice of a complex structure on the space where
the motion takes place; in particular, its angular momentum depends on this
choice; On the other hand, relative equilibria are not necessarily periodic: if
the configuration is "balanced" but not central, the motion is

We investigate the set of biaccessible points for connected polynomial Julia
sets of arbitrary degrees $d\geq 2$. We prove that the Hausdorff dimension of
the set of external angles corresponding to biaccessible points is less than 1,
unless the Julia set is an interval. This strengthens theorems of Stanislav
Smirnov and Anna Zdunik: they proved that the same set of external angles has
zero 1-dimensional measure.

We prove that if $X|_\Lambda$ has the weak specification property robustly,
where $\Lambda$ is an isolated set, then $\Lambda$ is a sectional hyperbolic
topologically mixing set and if $X|_\Lambda$ has the specification property
robustly then $\Lambda$ is a hyperbolic topologically mixing set .

We construct open sets of Ck (k bigger or equal to 2) vector fields with
singularities that have robust exponential decay of correlations and satisfy
the central limit theorem with respect to the unique physical measure. In
particular we prove that the geometric Lorenz attractor has exponential decay
of correlations with respect to the unique physical measure.

Bifurcations can cause dynamical systems with slowly varying parameters to
transition to far-away attractors. The terms "critical transition" or "tipping
point" have been used to describe this situation. Critical transitions have
been observed in an astonishingly diverse set of applications from ecosystems
and climate change to medicine and finance. The goal of this paper is to bring
together a variety of techniques from dynamical systems theory to analyze
critical transitions. In particular, we shall focus on identifying indicators
for catastrophic shifts in the dynamics.

Stability of Wardrop equilibria is analyzed for dynamical transportation
networks in which the drivers' route choices are influenced by information at
multiple temporal and spatial scales. The considered model involves a continuum
of indistinguishable drivers commuting between a common origin/destination pair
in an acyclic transportation network.

We survey an area of recent development, relating dynamics to theoretical
computer science. We discuss the theoretical limits of simulation and
computation of interesting quantities in dynamical systems. We will focus on
central objects of the theory of dynamics, as invariant measures and invariant
sets, showing that even if they can be computed with arbitrary precision in
many interesting cases, there exists some cases in which they can not.

A renormalization approach has been used in \cite{EKW1} and \cite{EKW2} to
prove the existence of a \textit{universal area-preserving map}, a map with
hyperbolic orbits of all binary periods. The existence of a horseshoe, with
positive Hausdorff dimension, in its domain was demonstrated in \cite{GJ1}. In
this paper the coexistence problem is studied, and a computer-aided proof is
given that no elliptic islands with period less than $18$ exist in the domain.
It is also shown that the area enclosed by elliptic islands is less than
$0.046$.

We give sufficient conditions for a shift space $(\Sigma,\sigma)$ to be
intrinsically ergodic, along with sufficient conditions for every subshift
factor of $\Sigma$ to be intrinsically ergodic. As an application, we show that
every subshift factor of a beta-shift is intrinsically ergodic, which answers
an open question included in Mike Boyle's article "Open problems in symbolic
dynamics''. We obtain the same result for S-gap shifts, and describe an
application of our conditions to more general coded systems.

Let $\nu_\lambda^p$ be the distribution of the random series
$\sum_{n=1}^\infty i_n \lambda^n$, where $i_n$ is a sequence of i.i.d. random
variables taking the values 0,1 with probabilities $p,1-p$. These measures are
the well-known (biased) Bernoulli convolutions.

In this paper we study the multifractal spectrum of $\nu_\lambda^p$ for
typical $\lambda$. Namely, we investigate the size of the sets

\[ \Delta_{\lambda,p}(\alpha) = \left\{x\in\R: \lim_{r\searrow 0} \frac{\log
\nu_\lambda^p(B(x,r))}{\log r} =\alpha\right\}. \]

We study the global dynamics of integrate and fire neural networks composed
of an arbitrary number of identical neurons interacting by inhibition and
excitation. We prove that if the interactions are strong enough, then the
support of the stable asymptotic dynamics consists of limit cycles. We also
find sufficient conditions for the synchronization of networks containing
excitatory neurons. The proofs are based on the analysis of the equivalent
dynamics of a piecewise continuous Poincar\'e map associated to the system.

We investigate mushroom billiards, a class of dynamical systems with sharply
divided phase space. For typical values of the control parameter of the system
$\rho$, an infinite number of marginally unstable periodic orbits (MUPOs) exist
making the system sticky in the sense that unstable orbits approach regular
regions in phase space and thus exhibit regular behaviour for long periods of
time. The problem of finding these MUPOs is expressed as the well known problem
of finding optimal rational approximations of a real number, subject to some
system-specific constraints.

In this paper we present a general procedure allowing for the reduction or
expansion of any network (considered as a weighted graph) which maintains the
spectrum of the network's adjacency matrix up to a set of eigenvalues known
beforehand from its graph structure. This procedure can be used to establish
new equivalence relations on the class of all weighted graphs (networks) where
two graphs are equivalent if they can be reduced to the same graph.

What input signals will lead to synchrony vs. desynchrony in a group of
biological oscillators? This question connects with both classical dynamical
sys- tems analyses of entrainment and phase locking and with emerging studies
of stimulation patterns for controlling neural network activity. Here, we focus
on the response of a population of uncoupled, elliptically bursting neurons to
a com- mon pulsatile input. We extend a phase reduction from the literature to
capture inputs of varied strength, leading to a circle map with discontinuities
of various orders.

We prove that if $G\subset\text{Diff}^{1}(\mathbb{R}^2)$ is an Abelian
subgroup generated by a family of commuting diffeomorphisms of the plane, all
of which are $C^{1}$-close to the identity in the strong $C^{1}$-topology, and
if there exist a point $p\in\mathbb{R}^2$ whose orbit is bounded under the
action of $G$, then the elements of $G$ have a common fixed point in the convex
hull of $\bar{\mathcal{O}_{p}(G)}$. Here, $\bar{\mathcal{O}_{p}(G)}$ denotes
the topological closure of the orbit of $p$ by $G$.

In the present paper we study interval identification systems of order three.
We prove that Rauzy induction preserves symmetry: for any symmetric interval
identification system of order three after a finite number of iterations of
Rauzy induction we always obtain a symmetric system. We also provide an example
of symmetric interval identification system of thin case.

We apply the Gradient-Newton-Galerkin-Algorithm (GNGA) of Neuberger & Swift
to find solutions to a semilinear elliptic Dirichlet problem on the region
whose boundary is the Koch snowflake. In a recent paper, we described an
accurate and efficient method for generating a basis of eigenfunctions of the
Laplacian on this region. In that work, we used the symmetry of the snowflake
region to analyze and post-process the basis, rendering it suitable for input
to the GNGA. The GNGA uses Newton's method on the eigenfunction expansion
coefficients to find solutions to the semilinear problem.

In this paper we prove two results. First we show that dynamical systems with
an $\alpha$-mixing have in the limit Poisson distributed return times almost
everywhere. We use the Chen-Stein method to also obtain rates of convergence.
Our theorem improves on previous results by allowing for infinite partitions
and dropping the requirement that the invariant measure have finite entropy
with respect to the given partition.

We demonstrate that the question whether or not a given postcritically finite
topological ramified covering map of the 2-sphere is Thurston equivalent to a
rational map is algorithmically decidable.

This paper is devoted to the investigation of the nonnegative solutions and
the stability and asymptotic properties of the solutions of fractional
differential dynamic systems involving delayed dynamics with point delays. The
obtained results are independent of the sizes of the delays.

We extend the theory of transience to general dynamical systems with no
Markov structure assumed. This is linked to the theory of phase transitions. We
also provide examples of new kinds of transient behaviour.

If $\vf_1, ... \vf_m\colon\Z\to\Z^\ell$ are polynomials with zero constant
terms and $E\subset\Z^\ell$ has positive upper Banach density, then we show
that the set $E\cap (E-\vf_1(p-1))\cap\...\cap (E-\vf_m(p-1))$ is nonempty for
some prime $p$. We also prove mean convergence for the associated averages
along the prime numbers, conditional to analogous convergence results along the
full integers. This generalizes earlier results of the authors, of Wooley and
Ziegler, and of Bergelson, Leibman and Ziegler.

We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in
a class of four-dimensional systems which may be Hamiltonian or not. Only one
parameter is enough to treat these types of bifurcations in Hamiltonian systems
but two parameters are needed in general systems. We apply a version of
Melnikov’s method due to Gruendler to obtain saddle-node and pitchfork types of
bifurcation results for homoclinic orbits.

This paper is concerned with the asymptotic spreading of a Lotka-Volterra
cooperative system. Utilizing the theory developed by Berestycki et al.
[Asymptotic spreading in heterogeneous diffusive excitable media, J. Funct.
Anal. \textbf{255} (2008), 2146-2189] for nonautonomous scalar equations, the
lower bounds of spreading speeds of unknown functions formulated by a coupled
system are estimated.

This paper defines the pressure for asymptotically subadditive potentials
under a mistake function, including the measuretheoretical and the topological
versions. Using the advanced techniques of ergodic theory and topological
dynamics, we reveals a variational principle for the new defined topological
pressure without any additional conditions on the potentials and the compact
metric space.

We establish the background for the study of geodesics on noncompact
polygonal surfaces. For illustration, we study the recurrence of geodesics on
$Z$-periodic polygonal surfaces. We prove, in particular, that almost all
geodesics on a topologically typical $Z$-periodic surface with boundary are
recurrent.

Traditionally, rotation numbers for toroidal billiard flows are defined as
the limiting vectors of average displacements per time on trajectory segments.
The billard trajectories, being curves, oftentimes getting very close to closed
loops, quite naturally define elements of the fundamental group of the billiard
table. The simplest non-trivial fundamental group obtained this way belongs to
the classical Sinai billiard, i.e., the billiard flow on the 2-torus with a
single, convex obstacle removed.

We study the transport properties of nonautonomous chaotic dynamical systems
over a finite time duration. We are particularly interested in those regions
that remain coherent and relatively non-dispersive over finite periods of time,
despite the chaotic nature of the system. We develop a novel probabilistic
methodology based upon transfer operators that automatically detects maximally
coherent sets. The approach is very simple to implement, requiring only
singular vector computations of a matrix of transitions induced by the
dynamics.

A cyclic cover over the Riemann sphere branched at four points inherits a
natural flat structure from the "pillow" flat structure on the basic sphere. We
give an explicit formula for all individual Lyapunov exponents of the Hodge
bundle over the corresponding arithmetic Teichmuller curve. The key technical
element is evaluation of degrees of line subbundles of the Hodge bundle,
corresponding to eigenspaces of the induced action of deck transformations.

A cyclic cover of the projective plane branched at four points has a natural
structure of a square-tiled surface. We describe the combinatorics of such a
square-tiled surface, the geometry of the corresponding Teichm\"uller curve,
and compute the Lyapunov exponents of the determinant bundle over the
Teichm\"uller curve with respect to the geodesic flow.

We study pseudo-Abelian integrals associated with polynomial deformations of
slow-fast Darboux integrable systems. Under some assumptions we prove local
boundedness of the number of their zeros.

A stability analysis is presented of the hydrolysis of methyl isocyanate
(MIC) using a homogeneous flow reactor paradigm. The results simulate the
thermal runaway that occurred inside the storage tank of MIC at the Bhopal
Union Carbide plant in December 1984. The stability properties of the model
indicate that the thermal runaway may have been due to a large amplitude, hard
thermal oscillation initiated at a subcritical Hopf bifurcation. This type of
thermal misbehavior cannot be predicted using conventional thermal diagrams,
and may be typical of liquid thermoreactive systems.

In this article we consider two properties of cellular automata semigroups
which relate to "largeness". The first property is ID and the other is maximal
commutativity. A semigroup has the ID property if the only infinite invariant
set (with respect to the semigroup's action) is the entire space. We shall
consider two examples of semigroups of cellular automata acting on the one
sided shift space. One semigroup is spanned by cellular automata that represent
multiplications in R/Z and the other semigroup consists of all the linear
cellular automata (when taking the symbol set to be Z/nZ.

We prove that a polynomial Julia set which is a finitely irreducible
continuum is either an arc or an indecomposable continuum. For the more general
case of rational functions, we give a topological model for the dynamics when
the Julia set is an irreducible continuum and all indecomposable subcontinua
have empty interior.

We construct Anosov diffeomorphisms on manifolds that are homeomorphic to
infranilmanifolds yet have exotic smooth structures. These manifolds are
obtained from standard infranilmanifolds by connected summing with certain
exotic spheres. Our construction produces Anosov diffeomorphisms of high
codimension.

The purpose of this review-and-research paper is twofold: (i) to review the
role played in climate dynamics by fluid-dynamical models; and (ii) to
contribute to the understanding and reduction of the uncertainties in future
climate-change projections. To illustrate the first point, we focus on the
large-scale, wind-driven flow of the mid-latitude oceans which contribute in a
crucial way to Earth's climate, and to changes therein.

A precise description of the singularities of the Borel transform of
solutions of a level-one linear differential system is deduced from a proof of
the summable-resurgence of the solutions by the perturbative method of J.
\'Ecalle. Then we compare the meromorphic classification (Stokes phenomenon)
from the viewpoint of the Stokes cocycle and the viewpoint of alien
derivatives. We make explicit the Stokes-Ramis matrices as functions of the
connection constants in the Borel plane and we develop two examples. No
assumption of genericity is made.

We consider the homotopical dynamics on compact orientable surfaces of
positive genus g. We establish a sufficient and necessary algebraic criterion
for homotopy classes with infinitely many periodic points of maps on such
surfaces in terms of the characteristic polynomial of the matrix representing
the correspondig homomorphism of the first homology group.

We prove the existence of periodic bounce orbits of prescribed energy on an
open bounded domain in Euclidean space. We derive explicit bounds on the period
and the number of bounce points.

Consider a difference equation which takes the k-th largest output of m
functions of the previous m terms of the sequence. If the functions are also
allowed to change periodically as the difference equation evolves this is
analogous to a differential equation with periodic forcing. A large class of
such non-autonomous difference equations are shown to converge to a periodic
limit which is independent of the initial condition. The period of the limit
does not depend on how far back each term is allowed to look back in the
sequence, and is in fact equal to the period of the forcing.

Let $\phi(x) = |1 - 1/x|$ for all $x > 0$. Then $\phi(x)$ is an unbounded
continuous map from $(0, \infty)$ onto $[0, \infty)$ which maps the set
$\mathbb {R_+} \setminus \mathbb {Q_+}$ of irrational points in $(0, \infty)$
onto itself.

We investigate the behavior of an infinite array of (reverse) von K'arm'an
streets. Our primary motivation is to model the wake dynamics in large fish
schools. We ignore the fish and focus on the dynamic interaction of multiple
wakes where each wake is modeled as a reverse von K'arm'an street. There exist
configurations where the infinite array of vortex streets is in relative
equilibrium, that is, the streets move together with the same translational
velocity.

Szalai et al. (SIAM J. on Sci. Comp. 28(4), 2006) gave a general construction
for characteristic matrices for systems of linear delay-differential equations
with periodic coefficients. First, we show that matrices constructed in this
way can have a discrete set of poles in the complex plane, which may possibly
obstruct their use when determining the stability of the linear system. Then we
modify and generalize the original construction such that the poles get pushed
into a small neighborhood of the origin of the complex plane.

The topological classification of the inner mappings on the fully invariant
regular components of the wandering set with a special attracting boundary up
to the topological conjugacy is defined in terms of distinguishing graph. Two
inner mappings of the same degree are topologically equivalent on their fully
invariant regular components of a certain class if and only if their
distinguishing graphs are equivalent.

We consider, on a compact manifold, the group of diffeomorphisms that are
isotopic to the identity. We show that every recurrent element is a distorsion
element. This generalizes Avila's theorem on circle diffeomorphisms. The method
also provides a new proof of a result by Calegari and Freedman: on a compact
manifold, in the group of homeomorphisms that are isotopic to the identity,
every element is distorted.

The billiard motion inside an ellipsoid $Q \subset \Rset^{n+1}$ is completely
integrable. Its phase space is a symplectic manifold of dimension $2n$, which
is mostly foliated with Liouville tori of dimension $n$. The motion on each
Liouville torus becomes just a parallel translation with some frequency
$\omega$ that varies with the torus. Besides, any billiard trajectory inside
$Q$ is tangent to $n$ caustics $Q_{\lambda_1},...,Q_{\lambda_n}$, so the
caustic parameters $\lambda=(\lambda_1,...,\lambda_n)$ are integrals of the
billiard map.

We study discrete dynamical systems through the topological concepts of limit
set, which consists of all points that can be reached arbitrarily late, and
asymptotic set, which consists of all adhering values of orbits. In particular,
we deal with the case when each of these are a singleton, or when the
restriction of the system is periodic on them, and show that this is equivalent
to some simple dynamics in the case of subshifts or cellular automata.
Moreover, we deal with the stability of these properties with respect to some
simulation notions.

Using a formula of Kozlovski (An integral formula for topological entropy of
$C^{\infty}$ maps, ETDS 18, 405-424, 1998) for the topological pressure, we
give a way of constructing measures close to an equilibrium state for
$\mathcal{C}^{\infty}$ maps of a smooth compact manifold. We also prove large
deviations results, upper and lower bound, for such maps. We apply the results
to the time-one map of the geodesic flow of a smooth compact Riemannian
manifold.

We investigate the distribution of orbits of a geometrically finite group
acting on a hyperbolic space and its geometric boundary. In particular we show
that the orbit of a non-elementary geometrically finite subgroup of the
(orientation-preserving) isometry group of hyperbolic space in the geometric
boundary is equidistributed with respect to the Patterson-Sullivan measure
supported on the limit set.

We show that there are no Anosov actions by (n-1)-dimensional unimodular Lie
groups on closed n-dimensional manifolds.

While compactness is an essential assumption for many results in dynamical
systems theory, for many applications the state space is only locally compact.
Here we provide a general theory for compactifying such systems, i.e. embedding
them as invariant open subsets of compact systems. In the process we don't want
to introduce recurrence which was not there in the original system. For example
if a point lies on an orbit which remains in any compact set for only a finite
span of time then the point becomes non-wandering if we use the one-point
compactification.

We generalize Bourgain-Lindenstrauss-Michel-Venkatesh's recent
one-dimensional quantitative density result to abelian algebraic actions on
higher dimensional tori. Up to finite index, the group actions that we study
are conjugate to the action of $U_K$, the group of units of some non-CM number
field $K$, on a compact quotient of $K\otimes_{\mathbb Q}\mathbb R$. In such a
setting, we investigate how fast the orbit of a generic point can become dense
in the torus.

We define a diophantine condition for interval exchange transformations
(i.e.t.). When the number of intervals is two, that is for rotations on the
circle, our condition coincides with the one in the classical Khinchin theorem,
modulo the identification of a rotation with its rotation number. We prove that
for i.e.t.s we have the same dichotomy of Khinchin theorem.

In many natural synchronization phenomena, communication between individual
elements occurs not directly, but rather through the environment. One of these
instances is bacterial quorum sensing, where bacteria release signaling
molecules in the environment which in turn are sensed and used for population
coordination. Extending this motivation to a general non- linear dynamical
system context, this paper analyzes synchronization phenomena in networks where
communication and coupling between nodes are mediated by shared dynamical quan-
tities, typically provided by the nodes' environment.

The Kuramoto model, which describes synchronization phenomena, is a system of
ordinary differential equations on $N$-torus defined as coupled harmonic
oscillators. The order parameter is often used to measure the degree of
synchronization. In this paper, a few properties of the continuous model for
the Kuramoto model are investigated. In particular, the moments systems are
introduced for both of the Kuramoto model and its continuous model.

First, we study p-adic matrices and their discrete dynamics over p-adic
numbers C_p. We prove that if p-adic absolute value of every eigenvalue of a
p-adic matrix is less than 1 then every solution of v_{n+1}=Av_n converges to 0
as n tends to infinity. Second, we study the periodicity of solutions of the
system over finite fields.

We study various solution behaviors of scale equations which are recently
proposed in \cite{Kim}. On the contrary to conventional mathematical tools,
scale equations are capable to accommodate various behaviors at different scale
levels into one integrated solution. Some solutions of scale equations often
retain strong stochastic properties such as fractional Brownian Motion,
although constructing those solutions is a deterministic process.

We embed multidimensional Farey fractions in large horospheres and explain
under which conditions they become uniformly distributed in the ambient
homogeneous space. This question has recently been investigated in the case of
SL(d,Z) to prove the asymptotic distribution of Frobenius numbers. The present
paper extends these studies to general lattices in SL(d,R).

We study the topological dynamics by iterations of a piecewise continuous,
non linear and locally contractive map in a real finite dimensional compact
ball. We consider those maps satisfying the "separation property": different
continuity pieces have disjoint images. The continuity pieces act as stable
topological manifolds while the points in the discontinuity lines, separating
different continuity pieces, act as topological saddles with an infinite
expanding rate.

We consider interval exchange transformations of periodic type and construct
different classes of recurrent ergodic cocycles of dimension $\geq 1$ over this
special class of IETs. Then using Poincar\'e sections we apply this
construction to obtain recurrence and ergodicity for some smooth flows on
non-compact manifolds which are extensions of multivalued Hamiltonian flows on
compact surfaces.

A standard interval exchange map is a one-to-one map of the interval which is
locally a translation except at finitely many singularities. We define for such
maps, in terms of the Rauzy-Veech continuous fraction algorithm, a diophantine
arithmetical condition called restricted Roth type which is almost surely
satisfied in parameter space. Let $T_0$ be a standard interval exchange map of
restricted Roth type, and let $r$ be an integer $\geq 2$.