We define a criterion called Følner Independence for a stationary process over an amenable group. Intuitively, a process satisfies the criterion if, for sufficiently invariant Følner sets, the process in the Følner set is nearly independent of the process outside. We show that Følner Independence implies Finitely Determined. As an application, we show that, in its extreme Gibbs states, the amenable attractive Ising model is Følner Independent (hence Finitely Determined, hence Bernoulli).

Let ([0, 1]z, ℬ([0, 1]Z), μz, ϕ) be the dynamical system where ϕ is the shift on the product of the unit interval with Lebesgue measure. We show that this dynamical system has the following properties: (1) there exists ƒ ∈ L1(μz) (in fact in L(Log Log L)β) where 0 <β< 1 for which the following Wiener-Wintner property does not hold: (W-W). There exists a single null set N ⊂ X off which for all x ∈ X/N the sequence converges for all ε ⊂ [0,1).

(2) The property (W-W) holds in all Lp(μz), 1< p≤∞. Added to a continuity property of the helical transform, (W-W) is equivalent to the Carleson-Hunt result on the pointwise convergence of Fourier series.

We construct an area preserving homeomorphism ƒ: T2→T2 that is isotopic to the identity and fixed point free, but has the property that every essential simple closed curve C satisfies ƒ(C)∩C ≠ Ø. This answers a question of Guillou.

Let ƒ be an orientation-preserving homeomorphism of the plane such that ƒ-Id is contracting. Under these hypotheses, we establish the existence for every periodic orbit , of a fixed point which has a nonzero linking number with .

This paper considers the closing problem, r≥2, for flows on T2 with finitely many singularities. We study the action of such flows on circles transverse to the flow, using Katznelson and Ornstein's picture of rigid rotations as Rokhlin towers. We are able to obtain closing on purely topological grounds for a large class of flows by means of perturbations which simply twist a flow along an embedded transverse circle (twist-perturbations). However, we give an example of a flow (with finitely many singularities) for which no small twist-perturbation yields closing; indeed such perturbations either leave the non-wandering set unchanged or else collapse it to the set of singularities.

In 1960 Schmidt [S] showed that if p and q are not powers of the same integer, i.e., if log q/log p is irrational, then for certain special measures μ on [0,1), invariant under S:x ↦ px (mod 1), μ-almost every x is normal to the base q. The measures considered in [S] were similar to Cantor-Lebesgue measure: namely, under μ the p-digit process was a special i.i.d. process where for some k ≥ 2 the elements of a certain k-element subset of the p-digits assumed probability 1/k each. The proof was fairly complicated, and did not seem to yield much more (see Keane and Pearce [K] for another proof).

We characterize numbers having finite β-expansions where β belongs to a certain class of Pisot numbers: when the β-expansion of 1 is equal to a1a2…am, where a1≥a2≥…≥am≥1 and when the β-expansion of 1 is equal to t1t2…tm(tm+1)ω where t1≥t2≥…≥tm>tm+1≥1.

First, the class of id-finitely fixed actions of ℤd on a Lebesgue space is defined. Then, it is demonstrated that this property is stable under id-Kakutani equivalence and that, conversely, any two id-finitely fixed ℤd-actions of the same (finite) positive entropy are id-Kakutani equivalent. By id-Kakutani equivalence we mean that element in A. del Junco and D. Rudolph's family of relations on ℤd-actions corresponding to the identity d × d matrix.

In this paper the results of Shub and Sacksteder are extended to the following theorem: let ƒ1 and ƒ2 be two commuting, expansive, orientation-preserving maps of the circle with a common fixed point and with both in C1+ε or Cr, r ≥2. Assume ƒ1 is p-to-1 and ƒ2 is q-to-1 where p and q generate a nonlacunary semigroup. Then there exists a diffeomorphism g of the same class such that gƒ1g−1 = TP and gƒ2g−1 = Tq.

The paper is devoted to the triangular maps of the square into itself. The results presented were recently obtained by the author and are briefly stated (in Russian) in a difficult paper as well as those (jointly published with A. N. Sharkovsky) published in ECIT-89 (abstract). All these results are systematized and extended by the new ones. The more detailed proofs of all statements are given. It is shown, for example, that triangular maps exist such that their minimal attraction centres do not coincide with the centres, as well as such ones exist that the Milnor attractor is not contained in the closure of the set of periodic points.

We prove that for each ergodic automorphism T:(X, ℬ, μ)→(X, ℬ, μ) for which we can find an element S∈C(T) such that the corresponding Z2-action (S, T) on (X, ℬ, μ) is free, there exists a circle valued cocycle φ such that the group extension Tφ is ergodic but is not coalescent. In particular, the existence of such a cocycle is proved for all ergodic rigid automorphisms. As a corollary, in the class of ergodic transformations of [0,1) × [0,1) given by

for each irrational α we find φ such that Tφ is not coalescent. In some special cases the group law of the centralizer is given.

For interval maps, we define the entropy of a periodic orbit as the smallest topological entropy of a continuous interval map having this orbit. We consider the problem of computing the limit entropy of longer and longer periodic orbits with the same ‘pattern’ repeated over and over (one example of such orbits is what we call ‘snakes’). We get an answer in the form of a variational principle, where the supremum of metric entropies is taken only over those ergodic measures for which the integral of a certain function is zero. In a symmetric case, this gives a very easy method of computing this limit entropy. We briefly discuss applications to topological entropy of countable chains.

Bounded irreducible local Siegel disks include classical Siegel disks of polynomials, bounded irreducible Siegel disks of rational and entire functions, and the examples of Herman and Moeckel. We show there are only two possibilities for the structure of the boundary of such a disk: either the boundary admits a nice decomposition onto a circle or it is an indecomposable continuum.

For a finite Abelian group G define the two-dimensional Markov shift for all . Let μG be the Haar measure on the subgroup . The group ℤ2 acts on the measure space (XG, MG) by shifts. We prove that if G1, and G2 are p-groups and E(G1,) ≠ E(G2), where E(G) is the least common multiple of the orders of the elements of G, then the shift actions on and are not measure-theoretically isomorphic. For any finite Abelian groups G1 and G2 the shift actions on and are topologically conjugate if and only if G1 and G2 are isomorphic.