Ergodic Theory and Dynamical Systems - Current Issue

Volume 28 Issue 06

Mon, 01 Dec 2008 00:00:00 GMT

Ergodic Theory and Dynamical Systems, Volume 28 Issue 06

Ergodic Theory and Dynamical Systems focuses on a rich variety of research areas which, although diverse, employ as common themes global dynamical methods. The journal provides a focus for this important and flourishing area of mathematics and brings together many major contributions in the field. The journal acts as a forum for central problems of dynamical systems and of interactions of dynamical systems with areas such as differential geometry, number theory, operator algebras, celestial and statistical mechanics, and biology.

Mixing with staircase multiplicity functions

Research Articles
OLEG AGEEV,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1687-1700

Abstract
Every subgroup of the symmetric group defines a natural factor of the Cartesian power of a transformation. We calculate the set of values of the spectral multiplicity function of such factors (under certain conditions on the transformation) in terms of the number of orbits of diagonal actions of these subgroups. An analogous statement also holds on the unitary level for operators that preserve 1. In particular, we prove that for every positive integer n, there exists a transformation which is mixing of all orders and has a staircase multiplicity function of length n; that is, the essential values of the spectral multiplicity function are {1,2, ,n}.

On the Ruelle eigenvalue sequence

Research Articles
OSCAR F. BANDTLOW, OLIVER JENKINSON,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1701-1711

Abstract
For certain analytic data, we show that the eigenvalue sequence of the associated transfer operator is insensitive to the holomorphic function space on which acts. Explicit bounds on this eigenvalue sequence are established.

Hedgehogs of Hausdorff dimension one

Research Articles
KINGSHOOK BISWAS,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1713-1727

Abstract
We present a construction of hedgehogs for holomorphic maps with an indifferent fixed point. We construct, for a family of commuting nonlinearizable maps, a common hedgehog of Hausdorff dimension one, the minimum possible.

Topological dichotomy and strict ergodicity for translation surfaces

Research Articles
YITWAH CHEUNG, PASCAL HUBERT, HOWARD MASUR,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1729-1748

Abstract
In this paper the authors find examples of translation surfaces that have infinitely generated Veech groups, satisfy the topological dichotomy property that for every direction the flow in that direction is either completely periodic or minimal, and yet have minimal but non-uniquely ergodic directions.

Axiom A polynomial skew products of 2 and their postcritical sets

Research Articles
LAURA DEMARCO, SUZANNE LYNCH HRUSKA,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1749-1779

Abstract
A polynomial skew product of 2 is a map of the form f(z,w)=(p(z),q(z,w)), where p and q are polynomials, such that f extends holomorphically to an endomorphism of 2 of degree at least two. For polynomial maps of , hyperbolicity is equivalent to the condition that the closure of the postcritical set is disjoint from the Julia set; further, critical points either iterate to an attracting cycle or infinity. For polynomial skew products, Jonsson [Dynamics of polynomial skew products on C2. Math. Ann. 314(3) (1999), 403 447] established that f is Axiom A if and only if the closure of the postcritical set is disjoint from the right analog of the Julia set. Here we present an analogous conclusion: critical orbits either escape to infinity or accumulate on an attracting set. In addition, we construct new examples of Axiom A maps demonstrating various postcritical behaviors.

Elliptic isles in families of area-preserving maps

Research Articles
P. DUARTE,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1781-1813

Abstract
We prove that every one-parameter family of area-preserving maps unfolding a homoclinic tangency has a sequence of parameter intervals, approaching the bifurcation parameter, where the dynamics exhibits wild hyperbolic sets accumulated by elliptic isles. This is a parametric conservative analogue of a famous theorem of Newhouse on the abundance of wild hyperbolic sets.

Analytic invariants associated with a parabolic fixed point in 2

Research Articles
V. GELFREICH, V. NAUDOT,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1815-1848

Abstract
It is well known that in a small neighbourhood of a parabolic fixed point a real-analytic diffeomorphism of (2,0) embeds in a smooth autonomous flow. In this paper we show that the complex-analytic situation is completely different and a generic diffeomorphism cannot be embedded in an analytic flow in a neighbourhood of its parabolic fixed point. We study two analytic invariants with respect to local analytic changes of coordinates. One of the invariants was introduced earlier by one of the authors. These invariants vanish for time-one maps of analytic flows. We show that one of the invariants does not vanish on an open dense subset. A complete analytic classification of the maps with a parabolic fixed point in 2 is not available at the present time.

Lyapunov optimizing measures for C 1 expanding maps of the circle

Research Articles
OLIVER JENKINSON, IAN D. MORRIS,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1849-1860

Abstract
For a generic C1 expanding map of the circle, the Lyapunov maximizing measure is unique and fully supported, and has zero entropy.

Strongly singular MASAs and mixing actions in finite von Neumann algebras

Research Articles
PAUL JOLISSAINT, YVES STALDER,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1861-1878

Abstract
Let 0 be an infinite abelian subgroup of , 0) is a singular MASA in M=L( Neumann Neumann extensions and semidirect products, and in particular we exhibit examples of singular MASAs that satisfy the weak mixing condition but not the strong mixing one.

Existence of critical invariant tori

Research Articles
HANS KOCH,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1879-1894

Abstract
We consider analytic Hamiltonian systems with two degrees of freedom, and prove that every Hamiltonian on the strong local stable manifold of the renormalization group fixed point obtained in Koch [A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete Contin. Dyn. Syst. A 11 (2004), 881 909] has a non-differentiable golden invariant torus (conjugacy to a linear flow).

Free curves and periodic points for torus homeomorphisms

Research Articles
ALEJANDRO KOCSARD, ANDRES KOROPECKI,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1895-1915

Abstract
We study the relationship between free curves and periodic points for torus homeomorphisms in the homotopy class of the identity. By free curve we mean a homotopically non-trivial simple closed curve that is disjoint from its image. We prove that every rational point in the rotation set is realized by a periodic point provided that there is no free curve and the rotation set has empty interior. This gives a topological version of a theorem of Franks. Using this result, and inspired by a theorem of Guillou, we prove a version of the Poincar Birkhoff theorem for torus homeomorphisms: in the absence of free curves, either there is a fixed point or the rotation set has non-empty interior.

The spectrum of Poincaré recurrence

Research Articles
KA-SING LAU, LIN SHU,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1917-1943

Abstract
We investigate the relationship between Poincar , and 85] for symbolic spaces.

Module shifts and measure rigidity in linear cellular automata

Research Articles
MARCUS PIVATO,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1945-1958

Abstract
Suppose is a finite commutative ring of prime characteristic, is a finite -module, :=D is an -linear cellular automaton on . If -invariant measure which is multiply must be the Haar measure on a coset of some submodule shift of . Under certain conditions, this means that must be the uniform Bernoulli measure on .

Veech’s dichotomy and the lattice property

Research Articles
JOHN SMILLIE, BARAK WEISS,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1959-1972

Abstract
Veech showed that if a translation surface has a stabilizer which is a lattice in SL(2,), then any direction for the corresponding constant slope flow is either completely periodic or uniquely ergodic. We show that the converse does not hold: there are translation surfaces that satisfy Veech s dichotomy but for which the corresponding stabilizer subgroup is not a lattice. The construction relies on work of Hubert and Schmidt.

The numbers of periodic orbits hidden at fixed points of n -dimensional holomorphic mappings

Research Articles
GUANG YUAN ZHANG,
Ergodic Theory and Dynamical Systems, Volume 28 Issue 06 , pp 1973-1989

Abstract
Let M be a positive integer and let f be a holomorphic mapping from a ball } into n such that the origin 0 is an isolated fixed point of both f and fM, the Mth iteration of f. Then one can define the number M(f,0), interpreted as the number of periodic orbits of f with period M that are hidden at the fixed point 0. For an n n 2.