209 research outputs found

### A toral diffeomorphism with a non-polygonal rotation set

We construct a diffeomorphism of the two-dimensional torus which is isotopic
to the identity and whose rotation set is not a polygon

### Rotation sets of billiards with one obstacle

We investigate the rotation sets of billiards on the $m$-dimensional torus
with one small convex obstacle and in the square with one small convex
obstacle. In the first case the displacement function, whose averages we
consider, measures the change of the position of a point in the universal
covering of the torus (that is, in the Euclidean space), in the second case it
measures the rotation around the obstacle. A substantial part of the rotation
set has usual strong properties of rotation sets

### A Classification of Minimal Sets of Torus Homeomorphisms

We provide a classification of minimal sets of homeomorphisms of the
two-torus, in terms of the structure of their complement. We show that this
structure is exactly one of the following types: (1) a disjoint union of
topological disks, or (2) a disjoint union of essential annuli and topological
disks, or (3) a disjoint union of one doubly essential component and bounded
topological disks. Periodic bounded disks can only occur in type 3. This result
provides a framework for more detailed investigations, and additional
information on the torus homeomorphism allows to draw further conclusions. In
the non-wandering case, the classification can be significantly strengthened
and we obtain that a minimal set other than the whole torus is either a
periodic orbit, or the orbit of a periodic circloid, or the extension of a
Cantor set. Further special cases are given by torus homeomorphisms homotopic
to an Anosov, in which types 1 and 2 cannot occur, and the same holds for
homeomorphisms homotopic to the identity with a rotation set which has
non-empty interior. If a non-wandering torus homeomorphism has a unique and
totally irrational rotation vector, then any minimal set other than the whole
torus has to be the extension of a Cantor set.Comment: Published in Mathematische Zeitschrift, June 2013, Volume 274, Issue
1-2, pp 405-42

### Strictly Toral Dynamics

This article deals with nonwandering (e.g. area-preserving) homeomorphisms of
the torus $\mathbb{T}^2$ which are homotopic to the identity and strictly
toral, in the sense that they exhibit dynamical properties that are not present
in homeomorphisms of the annulus or the plane. This includes all homeomorphisms
which have a rotation set with nonempty interior. We define two types of
points: inessential and essential. The set of inessential points $ine(f)$ is
shown to be a disjoint union of periodic topological disks ("elliptic
islands"), while the set of essential points $ess(f)$ is an essential
continuum, with typically rich dynamics (the "chaotic region"). This
generalizes and improves a similar description by J\"ager. The key result is
boundedness of these "elliptic islands", which allows, among other things, to
obtain sharp (uniform) bounds of the diffusion rates. We also show that the
dynamics in $ess(f)$ is as rich as in $\mathbb{T}^2$ from the rotational
viewpoint, and we obtain results relating the existence of large invariant
topological disks to the abundance of fixed points.Comment: Incorporates suggestions and corrections by the referees. To appear
in Inv. Mat

### Statistical stability of equilibrium states for interval maps

We consider families of multimodal interval maps with polynomial growth of
the derivative along the critical orbits. For these maps Bruin and Todd have
shown the existence and uniqueness of equilibrium states for the potential
$\phi_t:x\mapsto-t\log|Df(x)|$, for $t$ close to 1. We show that these
equilibrium states vary continuously in the weak$^*$ topology within such
families. Moreover, in the case $t=1$, when the equilibrium states are
absolutely continuous with respect to Lebesgue, we show that the densities vary
continuously within these families.Comment: More details given and the appendices now incorporated into the rest
of the pape

### Piecewise Linear Models for the Quasiperiodic Transition to Chaos

We formulate and study analytically and computationally two families of
piecewise linear degree one circle maps. These families offer the rare
advantage of being non-trivial but essentially solvable models for the
phenomenon of mode-locking and the quasi-periodic transition to chaos. For
instance, for these families, we obtain complete solutions to several questions
still largely unanswered for families of smooth circle maps. Our main results
describe (1) the sets of maps in these families having some prescribed rotation
interval; (2) the boundaries between zero and positive topological entropy and
between zero length and non-zero length rotation interval; and (3) the
structure and bifurcations of the attractors in one of these families. We
discuss the interpretation of these maps as low-order spline approximations to
the classic ``sine-circle'' map and examine more generally the implications of
our results for the case of smooth circle maps. We also mention a possible
connection to recent experiments on models of a driven Josephson junction.Comment: 75 pages, plain TeX, 47 figures (available on request

### Multicomponent dynamical systems: SRB measures and phase transitions

We discuss a notion of phase transitions in multicomponent systems and
clarify relations between deterministic chaotic and stochastic models of this
type of systems. Connections between various definitions of SRB measures are
considered as well.Comment: 13 pages, LaTeX 2

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