Ergodic properties of skew products in infinite measure

Let (Ω, µ) be a shift of finite type with a Markov probability, and (Y, ν) a non-atomic standard measure space. For each symbol i of the symbolic space, let Φ_i be a non-singular automorphism of (Y, ν). We study skew products of the form (ω, y) ? (σω, Φ_ω0 (y)), where σ is the shift map on (Ω, µ). We prove that, when the skew product is recurrent, it is ergodic if and only if the Φ_i’s have no common non-trivial invariant set.     

Ergodic properties where order 4 implies infinite order

For a family of dynamical properties, knowing that the condition holds for order 4 implies that it holds for all orders. Here we establish this for the properties minimal self-joinings, simplicity and for cartesiandisjointness. An application of the first yields an analog to Kalikow’s celebrated result that for rank-1 transformations, 2-fold mixing implies 3-fold mixing. Via a joining argument we show that for any rank-1 ℤ ^D -action, 4-fold mixing implies mixing of all orders. Indeed, the rank need only be sufficiently close to 1 for the implication to hold and so this result is new even when the acting group is ℤ. By means of limit-joinings, we settle affirmatively an old open question by establishing, for anyM, thatM-fold Rényi-mixing impliesM-fold mixing. Partially supported by National Science Foundation Postdoctoral Research Fellowship.     

Ergodic automorphisms of the infinite torus are bernoulli

We show that ergodic algebraic automorphisms of the infinite torus are measure isomorphic to Bernoulli shifts. Using the same techniques, we also show that the existence of such an automorphism with finite entropy is equivalent to an open problem in algebraic number theory.     

Ergodic properties of flows of classes of positive binary quadratic forms in Gaussian genera

The present paper is devoted to further development and refinement of previous results due to A. V. Malyshev and the author concerning the so-called discrete ergodic method of Yu. V. Linnik. An ergodic theorem and a mixing theorem for flows of positive binary quadratic forms are proved; these theorems describe the asymptotic distribution of the coefficients of these forms over the residue classes and over the corresponding surface. Bibliography: 12titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 149–161.     

Some extensions of weakly mixing flows

Using a technique of R. Ellis we prove the existence of many weakly mixing (w.m.) flows which are distal extensions of a given w.m. flow. Then we indicate two w.m. minimal flows whose product has a minimal non-w.m. subflow. This work was done under the helpful supervision of Professor H. Furstenberg as a part of the author’s Ph.D. thesis to be submitted to the Hebrew University of Jerusalem.     

Suborbits and group extensions of flows

Given a pair of an ergodic measured discrete equivalence relationR and a subrelationS ⊂R of finite index, a classification of the inclusion up to orbit equivalence will be discussed. In case of amenable and type III₀ relations, the orbit equivalence classes of inclusions will be completely classified in terms of a collection of a subgroupH and a normal subgroupG ₀ of a finite groupG and an ergodic group (G/G ₀) extension of a nonsingular flow. This is a generalization of Krieger’s theorem by which orbit equivalence classes of single relations were classified. Due to this result, essential type III inclusions will be made clear. Supported by the Japan Ministry of Education, Grant-in-Aid for Scientific Research No. (C)07640223. An erratum to this article is available at .     

Existence of infinite non-Birkoff periodic orbits for area-preserving monotone twist maps of cylinders

The exact monotone twist map of infinite cylinders in the Birkhoff region of instability is studied. A variational method based on Aubry-Mather theory is used to discover infinitely many non-Birkhoff periodic orbits of fixed rotation number sufficiently close to some irrational number for which the angular invariant circle does not exist.     

Ergodic measures of geodesic flows on compact Lie groups

Let Ψ be the geodesic flow associated with a two-sided invariant metric on a compact Lie group. In this paper, we prove that every ergodic measure μ of Ψ is supported on the unit tangent bundle of a flat torus. As an application, all Lyapunov exponents of μ are zero hence μ is not hyperbolic. Our underlying manifolds have nonnegative curvature (possibly strictly positive on some sections), whereas in contrast, all geodesic flows related to negative curvature are Anosov hence every ergodic measure is hyperbolic.     

Ergodic properties of crystallization processes

We consider a birth and growth process with germs which are born according to a Poisson point process whose intensity rneasure is invariant under trunslations of the space. The germs can be born in the unoccupied space; then they grow until they occupy the available space. In this general framework, the crystallization process can be characterized by a random field, which assigns to any point of the state space the first time at which this point is reached by a crigstal. Under general conditions on the growth speed and geometric shape of free crystals, we prone that the random field is mixing in the sense of ergodic theory. This result is illustrated by applications to the problem of parameter estimation. Bibliography: 7 titles.     

Ergodic properties of triangle partitions

We study the ergodic properties of a map called the Triangle Sequence. We prove that the algorithm is weakly convergent almost surely, and ergodic. As far as we know, it is the first example of a 2-dimensional algorithm where a surprising diophantine phenomenon happens: there are sequences of nested cells whose intersection is a segment, although no vertex is fixed. Examples of n-dimensional algorithms presenting this behavior were known for n ≥ 3.      

Ergodic properties of triangle partitions

We study the ergodic properties of a map called the Triangle Sequence. We prove that the algorithm is weakly convergent almost surely, and ergodic. As far as we know, it is the first example of a 2-dimensional algorithm where a surprising diophantine phenomenon happens: there are sequences of nested cells whose intersection is a segment, although no vertex is fixed. Examples of n-dimensional algorithms presenting this behavior were known for n ≥ 3.     

Self-dual normal bases for infinite galois field extensions

Let L ? K be an infinite Galois field extension with the property that every finite Galois extension M ? K, where L ? M, has a self-dual normal basis. We prove a self-dual normal basis theorem for L ? K when char (K) ≠2.     

Self-dual normal integral bases for infinite unramified extensions

We prove a generalization to infinite Galois extensions of local fields, of a classical result by Noether on the existence of normal integral bases for finite tamely ramified Galois extensions. We also prove a self-dual normal integral basis theorem for infinite unramified Galois field extensions of local fields with finite residue fields of characteristic different from 2. This generalizes a result by Fainsilber for the finite case. To do this, we obtain an injectivity result concerning pointed cohomology sets, defined by inverse limits of norm-one groups of free finite-dimensional algebras with involution over complete discrete valuation rings.     

Semigroups for flows in infinite networks

Inspired by previous work of M. Kramar and E. Sikolya (Math. Z. 249, 139–162, [2005]), we study transport processes on infinite networks. These “flows” can be modeled by operator semigroups on a suitable Banach space. Using functional analytical and graph theoretical methods, we investigate its spectral properties to determine the long time behavior of the system, and finally characterize uniform convergence of the semigroup to a periodic group under appropriate assumptions on the network.