bluuebunny/crossref_metadata_2025 huggingface dataset:
A characterization of flat metrics on tori by ergodicity
Authors: Innami Nobuhiro
Published: September 2008

The Riemannian flat metrics on tori T2 are characterized by a weakly ergodic property of the geodesic flows.
On minimal self-joinings in topological dynamics
Authors: del Junco Andrés
Published: September 2008

If X is a compact metric space and T a homeomorphism of X we say (X, T) has almost minimal power joinings (AMPJ) if there is a dense GδX* in X such that for each finite set k, x∈(X*)k and l:k → ℤ−{0}, the orbit closure cl {} is a product of off-diagonals (POOD) on Xk. By an offdiagonal on Xk′, k′k we mean a set of the form (⊗,j∈k′Tm(j))Δ, Δ the diagonal in Xk′, m:k′→ℤ any function, and by a POOD on Xk we mean that k is split into subsets k′, on each Xk′ we put an off-diagonal and then we take the product of these.

We show that examples of AMPJ exist and that this definition leads to a theory completely analogous to Rudolph's theory of minimal self-joinings in ergodic theory. In particular if (X, T) has AMPJ the automorphism group of T is {Tn}, T has only almost 1-1 factors (other than the trivial one) and the automorphism group and factors of ⊕i ∊ kT, k finite or countably infinite, can be very explicitly described. We also discuss ℝ-actions.
Expansive dynamics on zero-dimensional groups
Authors: Kitchens Bruce P.
Published: September 2008

If X is a compact, zero-dimensional group and T is an expansive, transitive automorphism then (X, T) is shown to be topologically conjugate to a full shift on finitely many symbols.

The problem of classifying such automorphisms up to simultaneous algebraic isomorphism and topological conjugacy is discussed but not solved. It is proved that for any entropy there are only finitely many such equivalence classes. When the entropy is log p for a prime p, there is only one equivalence class. All are then equivalent to
The rate of mixing for geodesic and horocycle flows
Authors: Ratner Marina
Published: September 2008

Using elementary methods we show that for every unitary representation π7 of G = SL(2, ℝ) with no non-zero invariant vectors the matrix coefficients 〈 π(a(t))v, w 〉 of

decay exponentially fast for any vectors v, w Holder continuous in the direction of the rotation subgroup of G.
An example of an amenable action from geometry
Authors: Spatzier R. J.
Published: September 2008

Let M be a compact manifold of not necessarily constant negative curvature. We observe that π1(M) acts amenably on the sphere at infinity of the universal cover of M with respect to a natural measure class. We also note that this action is of type III1.
A modulus of 3-dimensional vector fields
Authors: Togawa Y.
Published: September 2008

In this paper, we prove that μ/λ is a modulus for a Šilnikov system with eigenvalues λ and −μ ± iω. To prove this we define a number using knot and link invariants of periodic orbits, which is related to the ratio of eigenvalues μ/λ.
Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3
Authors: Dumortier F. , Roussarie R. , Sotomayor J.
Published: September 2008

A cusp type germ of vector fields is a C∞ germ at 0∈ℝ2, whose 2-jet is C∞ conjugate to

We define a submanifold of codimension 5 in the space of germs consisting of germs of cusp type whose 4-jet is C0 equivalent to

Our main result can be stated as follows: any local 3-parameter family in (0, 0) ∈ ℝ2 × ℝ3 cutting transversally in (0, 0) is fibre-C0 equivalent to
Diffeomorphisms on surfaces with a finite number of moduli
Authors: de Melo W. , van Strien S.J.
Published: September 2008

This paper generalises the well known structural stability theorem which says that a diffeomorphism is structurally stable if it is Axiom A and if all invariant manifolds are transversal to each other.

If these transversality conditions are not satisfied then the diffeomorphism not only fails to be stable, but also this gives rise to the appearance of moduli. That is, one needs several real parameters to parameterise all conjugacy classes of nearby diffeomorphisms. (The minimum number of parameters needed is called the number of moduli).

Here we deal with diffeomorphisms on two dimensional manifolds, whose asymptotic dynamics are well understood (the class of Axiom A diffeomorphisms). The main result characterises those Axiom A diffeomorphisms which have a finite number of moduli. This result can be regarded as a generalisation of the structural stability theorem. From the proofs it follows that the dynamics of these diffeomorphisms can also be well understood.

In the proof of our main theorem we need certain invariant foliations to be quite smooth. In an appendix we prove a differentiable version of the Lambda Lemma.
A bound for the fixed point index of area-preserving homeomorphisms of two-manifolds
Authors: Pelikan Stephan , Slaminka Edward E.
Published: September 2008

The study of area preserving maps of manifolds has an extensive history in the theory of dynamical systems. One interest has been in the behaviour of such maps near an isolated fixed point. In 1974 Carl Simon proved the existence of an upper bound for the index of an isolated fixed point for Ck area preserving diffeomorphisms of a surface. We extend his result to homeomorphisms of an orientable two manifold. The proof utilizes the notion of free modification, developed by Morton Brown, and enlarges the scope of the problem to the consideration of ‘nice’ measures, i.e. uniformly equivalent to Lebesgue measure on compact sets. By suitably modifying the homeomorphism and the measure, we obtain the following theorem.
Finitely presented dynamical systems
Authors: Fried David
Published: September 2008

We extend results of Bowen and Manning on systems with good symbolic dynamics. In particular we identify the class of dynamical systems that admit Markov partitions. For these systems the Manning-Bowen method of counting periodic points is explained in terms of topological coincidence numbers. We show, in particular, that an expansive system with a finite cover by rectangles has a rational zeta function.
On the generic existence of homoclinic points
Authors: Oliveira Fernando
Published: September 2008

This work is concerned with the generic existence of homoclinic points for area preserving diffeomorphisms of compact orientable surfaces. We give a shorter proof of Pixton's theorem that shows that, Cr-generically, an area preserving diffeomorphism of the two sphere has the property that every hyperbolic periodic point has transverse homoclinic points. Then, we extend Pixton's result to the torus and investigate certain generic aspects of the accumulation of the invariant manifolds all over themselves in the case of symplectic diffeomorphisms of compact manifolds.
Generic properties of periodic reflecting rays
Authors: Stojanov Luchezar
Published: September 2008

It is shown that for generic domains D in n, n ≥ 2, every periodic billiard trajectory in D passes only once through each of its reflection points, and any two different periodic billiard trajectories in D have no common reflection point.
Entropy of expansive flows
Authors: Thomas Romeo F.
Published: September 2008

Let h(φ) be the topological entropy of a real continuous flow φ on a compact metric space X. Introducing an equivalent definition for the topological entropy on an expansive real flow enables us to investigate the topological entropies of mutually conjugate expansive flows and estimate the periodic orbits of an expansive flow which has the pseudo-orbit tracing property.
Dynamics induced on the ends of non-compact manifold
Authors: Alpern Steve , Prasad V. S.
Published: September 2008

Let ℋ denote the group of homeomorphisms of a σ-compact manifold M which preserve a locally positive non-atomic measure μ. Such a manifold can be compactified by adjoining ideal points called ‘ends’, collectively denoted by E. Every homeomorphism h in ℋ induces a measure preserving system (E, , μ*, h*) where is the algebra of clopen subsets of E, μ* is a 0-∞ measure induced on E by μ, and h*: E → E is a μ*-preserving homeomorphism. For any induced homeomorphism σ = f*, where f belongs to ℋ, define ℋσ = {h ∈ ℋ: h* = σ}. We prove that ergodicity is generic in ℋσ for the compact-open topology if and only if (E, , μ*, σ) is incompressible and ergodic. Furthermore ℋσ contains an ergodic homeomorphism if and only if (E, , μ*, σ) is incompressible. Since the identity on M induces the identity on E, which is incompressible, our results establish that every manifold (M, μ) supports an ergodic μ-preserving homeomorphism.
The speed interval: a rotation algorithm for endomorphisms of the circle
Authors: Barkmeijer Jan
Published: September 2008

Letfbe a continuous map of the circle into itself of degree one. We introduce the notion of rotation algorithms. One of these algorithms associates eachz∈S1with an interval, the so-called speed intervalS(z,f), which is contained in the rotation interval ρ(f) off. In contrast with the rotation set ρ(z,f), the intervalS(z,f) sometimes allows us to ascertain that ρ(f) is non-degenerate, by using only finitely many elements of {fn(z) |n≥ 0}. We further show that all choices for ρ(z,f) andS(z,f) occur, for certainz∈S1provided that ρ(z,f) ⊂S(z,f) ⊂ ρ(f).
An invariant for rigid rank-1 transformations
Authors: Friedman Nathaniel , Gabriel Patrick , King Jonathan
Published: September 2008

Associated to a rigid rank-1 transformationTis a semigroup ℒ(T) of natural numbers, closed under factors. If ℒ(S) ≠ ℒ(T) thenSandTcannot be copied isomorphically onto the same space so that they commute. If ℒ(S) ⊅ ℒ(T) thenScannot be a factor ofT. For each semigroupLwe construct a weak mixingSsuch that ℒ(S) =L. TheSwhere ℒ(S) = {l}, despite having uncountable commutant, has no roots.

Preceding and preparing for this example are two others: An uncountable abelian groupGof weak mixing transformations for which any two (non-identity) members have identical self-joinings of all orders and powers. The second example, to contrast with the rank-1 property that the weak essential commutant must be the trivial group, is of a rank-2 transformation with uncountable weak essential commutant.
Duality for automorphisms on a compact C*-dynamical system
Authors: Evans David E. , Kishimoto Akitaka
Published: September 2008

When considering an action α of a compact group G on a C*-algebra A, the notion of an α-invariant Hilbert space in A has proved extremely useful [1, 4, 8, 14, 17, 18]. Following Roberts [13] a Hilbert space in (a unital algebra) A is a closed subspace H of A such that x*y is a scalar for all x, y in H. For example if G is abelian, and α is ergodic in the sense that the fixed point algebra Aα is trivial, then A is generated as a Banach space by a unitary in each of the spectral subspaces

which are then invariant one dimensional Hilbert spaces. If G is not abelian, then Hilbert spaces (which are always assumed to be invariant) do not necessarily exist, even for ergodic actions. For non-ergodic actions, it is also desirable to relax the requirement to x*y being a constant multiple of some positive element of Aα. More generally, if γ is a finite dimensional matrix representation of G and n is a positive integer, we define to be the subspace

where d is the dimension d(γ) of γ, and Mnd denotes n×d complex matrices. (Usually we will denote the extended action of αg to αg ⊗ 1 on A⊗Mnd again by αg.) Let .
Some compact invariant sets for hyperbolic linear automorphisms of torii
Authors: Fathi Albert
Published: September 2008

If the action induced by a pseudo-Anosov map on the first homology group is hyperbolic, it is possible, by a theorem of Franks, to find a compact invariant set for the toral automorphism associated with this action. If the stable and unstable foliations of the Pseudo-Anosov map are orientable, we show that the invariant set is a finite union of topological 2-discs. Using some ideas of Urbański, it is possible to prove that the lower capacity of the associated compact invariant set is >2; in particular, the invariant set is fractal. When the dilatation coefficient is a Pisot number, we can compute the Hausdorff dimension of the compact invariant set.
Geodesic flows of negatively curved manifolds with smooth stable and unstable foliations
Authors: Kanai Masahiko
Published: September 2008

We are concerned with closed C∞ riemannian manifolds of negative curvature whose geodesic flows have C∞ stable and unstable foliations. In particular, we show that the geodesic flow of such a manifold is isomorphic to that of a certain closed riemannian manifold of constant negative curvature if the dimension of the manifold is greater than two and if the sectional curvature lies between − and −1 strictly.
Propriétés des attracteurs de Birkhoff
Authors: Le Calvez P.
Published: September 2008

We study dissipative twist maps of the annulus, following the ideas of G. D. Birkhoff explained in an article of 1932.

In the first part, we give complete and rigorous proofs of the results of this article. We define the Birkhoff attractor of a dissipative twist map which has an attracting bounded annulus, we give its main properties and we define its upper and lower rotation numbers.

In the second part we give further results on these sets, thus we show that they often coincide with the closure of a hyperbolic periodic point and that they can contain an infinite number of sinks. We also show that the Birkhoff attractors don't depend on a continuous way on the maps.
Rotation intervals for a family of degree one circle maps
Authors: Jonker Leo B.
Published: September 2008

Let f be a C0 circle map of degree one with precisely one local minimum and one local maximum, and let [ρ−(f), ρ+(f)] be the interval of rotation numbers of f. We study the structure of the function ρ(λ) = ρ+(Rλ∘f), where Rλ is the rotation through the angle λ.
Smooth chaotic maps with zero topological entropy
Authors: Misiurewicz M. , Smítal J.
Published: September 2008

We find a class of C∞ maps of an interval with zero topological entropy and chaotic in the sense of Li and Yorke.
Realizing symmetries of a shift
Authors: Wagoner J. B.
Published: September 2008

All subshifts of finite type are known to appear as basic parts of the non-wandering sets of Smale diffeomorphisms in dimensions three or more. This paper concerns the symmetries of subshifts of finite type; that is, the homeomorphisms of the shift space which commute with the shift. The group of symmetries is known to be very large for aperiodic shifts. For certain (structually stable) Smale diffeomorphisms of the sphere of dimension five or more, we show each symmetry can be extended to a homeomorphism of the sphere commuting with the diffeomorphism on the whole sphere.
Andrei Nikolaevich Kolmogorov
Authors: Tihomirov V. M.
Published: September 2008

On the 20th of October 1987 one of the greatest scholars of our time, Andrei Nikolaevich Kolmogorov, passed away.
About A. N. Kolmogorov's work on the entropy of dynamical systems
Authors: Sinai Ya. G.
Published: September 2008

In the fall of 1957 A. N. Kolmogorov started lecturing on the theory of dynamical systems and supervised a seminar on the same theme at the Mechanical–Mathematical Department of Moscow State University. He began his lectures with the theory of systems with a pure point spectrum, which he approached from a probabilistic point of view. This approach, undoubtedly, has many advantages. In the seminar we studied Ito's theory of multiple stochastic integrals and, under the supervision of A. N. Kolmogorov, I. V. Girsanov constructed an example of a Gaussian dynamical system with a simple continuous spectrum. At one of the meetings of the seminar, still before the advent of entropy, Kolmogorov suggested a proof of an assertion, which today would read: the unitary operator induced by a K-automorphism has a countable Lebesgue spectrum. At this point Kolmogorov was studying Shannon's theory of information and the concept of the capacity of functional spaces. Judging by his well known article, we can say that the first of these, and all that is related to it, played a big role in the development of information theory in our country. The investigation of capacity is connected with Kolmogorov's work on Hilbert's 13th problem and was summarized in his well-known survey co-authored by V. M. Tihomirov.
Infinite limits in the iteration of entire functions
Authors: Baker I. N.
Published: September 2008

If f is a transcendental entire function and D is a non-wandering component of the set of normality of the iterates of f such that fn → ∞ in D then log |fn(z)| = O(n) as n → ∞ for z in D. For a wandering component the convergence of fn to ∞ in D may be arbitrarily fast.
Rotation shadowing properties of circle and annulus maps
Authors: Barge Marcy , Swanson Richard
Published: September 2008

We define the notions of the pseudo-rotation set and rotation shadowing of pseudo-orbits for endomorphisms of the circle and for homeomorphisms of the annulus. The results include: the rotation shadowing property holds for all endomorphisms of the circle; the pseudo-rotation set equals the closure of the rotation set; the closure of the rotation set varies upper-semicontinuously.
Regular dependence of invariant curves and Aubry–Mather sets of twist maps of an annulus
Authors: Douady Raphaël
Published: September 2008

We prove that smooth enough invariant curves of monotone twist maps of an annulus with fixed diophantine rotation number depend on the map in a differentiable way. Partial results hold for Aubry-Mather sets.

Then we show that invariant curves of the same map with different rotation numbers ω and ω′ cannot approach each other at a distance less than cst. |ω−ω′|. By K.A.M. theory, this implies that, under suitable assumptions, the union of invariant curves has positive measure.

Analogous results are due to Zehnder and Herman (for the first part), and to Lazutkin and Pöschel (for the second one), in the case of Hamiltonian systems and area preserving maps.
Examples of discrete groups of hyperbolic motions conservative but not ergodic at infinity
Authors: Taniguchi Masahiko
Published: September 2008

For every n ≥ 2, a discrete subgroup of isometries of the hyperbolic n-space, which is conservative but not ergodic on the sphere at infinity, is constructed.
Dynamics of the geodesic flow of a foliation
Authors: Walczak Paweł G.
Published: September 2008

The geodesic flow of a foliated Riemannian manifold (M, F) is studied. The invariance of some smooth measure is established under some geometrical conditions on F. The Lyapunov exponents and the entropy of this flow are estimated. As an application, the non-existence of foliations with ‘short’ second fundamental tensors is obtained on compact negatively curved manifolds.
Generalized baker's transformations
Authors: Bose Christopher J.
Published: September 2008

A class of automorphisms of the unit square called generalized baker'stransformations (gbt) is defined in such a way that every stationary stochastic process may be represented as the movement of a simple partition of the square under a gbt. This extends the classical example of the representation of independent processes by the well-known baker's transformation.

Every ergodic, positive-entropy automorphism is measurably isomorphic to some gbt (again generalizing the classical result about Bernoulli shifts), and we show that a large class of gbt's satisfying certain continuity restrictions are actually measurably isomorphic to Bernoulli shifts.
Real analytic Bernoulli geodesic flows on S2
Authors: Burns Keith , Gerber Marlies
Published: September 2008

We obtain a family of metrics on the two-dimensional sphere whose geodesic flow is ergodic and Bernoulli. This family includes real analytic metrics.
Dynamical properties of plane polynomial automorphisms
Authors: Friedland Shmuel , Milnor John
Published: September 2008

This note studies the dynamical behavior of polynomial mappings with polynomial inverse from the real or complex plane to itself.
Un exemple de transformation dilatante et C1 par morceaux de l'intervalle, sans probabilité absolument continue invariante
Authors: Gora P. , Schmitt B.
Published: September 2008

We construct a transformation on the interval [0, 1] into itself, piecewiseC1 and expansive, which doesn't admit any absolutely continuous invariant probability measure (a.c.i.p.).

So in this case we give a negative answer to a question by Anosov: is C1 character sufficient for the existence of absolutely continuous measure?

Moreover, in our example,ƒ' has a modulus of type K/(|1+|log|x‖); it is known that a modulus of continuity of type K/(1+|log|x‖)1+γ, γ>0 implies the existence of a.c.i.p..
Théorèmes ergodiques pour une translation sur un nilvariété
Authors: Lesigne Emmanuel
Published: September 2008

Let T be a translation defined on a nil-manifold N/Γ, where N is a nilpotent group of order two.
Restricting a compact action to an injective subfactor
Authors: Longo Roberto
Published: September 2008

Suppose we are given an action α: G → Aut (M) of a group G on a factor M; α possible way to analyse α may be to look at the invariant components where the action becomes more tractable. This point of view naturally leads to the study of the injective invariant subalgebras (recall for instance the good properties shared by amenable discrete or compact actions in the hyperfinite case [14]).
On ergodic properties of restrictions of inner functions
Authors: Martin N.F.G.
Published: September 2008

We consider inner functions on the unit disk which have a finite number of singularities on the unit circle. The restriction of such functions to the circle are maps onto the circle. We give sufficient conditions that these restrictions are exact endomorphisms whose natural extensions are Bernoulli and that the entropy is given by Rohlin's formula, We also give the entropy in closed form if ƒ' is in the Nevalinna class N. An example is considered. In the last section we show that if two restrictions are metrically isomorphic, they are diffeomorphic.
Two-particle billiard system with arbitrary mass ratio
Authors: Simányi Nándor , Wojtkowski Maciej P.
Published: September 2008

We describe ergodic properties of the system of two hard discs with arbitrary masses moving on the two dimensional torus.
Endpoints of rotation intervals for maps of the circle
Authors: Świątek Grzegorz
Published: September 2008

For certain C2 one-parameter families of endomorphisms of the circle both endpoints of rotation intervals are rational except for a set of parameter values of zero Lebesgue measure.
Rigidity of horospherical foliations
Authors: Witte Dave
Published: September 2008

M. Ratner's theorem on the rigidity of horocycle flows is extended to the rigidity of horospherical foliations on bundles over finite-volume locally-symmetric spaces of non-positive sectional curvature, and to other foliations of the same algebraic form.
Compact metric spaces, Fredholm modules, and hyperfiniteness
Authors: Connes A.
Published: September 2008

We show that the existence of a finitely summable unbounded Fredholm module (h, D) on a C* algebra A implies the existence of a trace state on A and that no such module exists on the C* algebra of a non amenable discrete group. Both for the needs of non commutative differential geometry and of analysis in infinite dimension we are led to the better notion of the θ-summable Fredholm module.
Actions of lattices in Sp (1, n)
Authors: Cowling Michael , Zimmer Robert J.
Published: September 2008

We study questions concerning the ergodic theory, von Neumann algebras, geometry, and topology of actions of lattices in Sp (1, n).
An estimate for the measure theoretic entropy of geodesic flows
Authors: Ballmann Werner , Wojtkowski Maciej P.
Published: September 2008

A new proof and a generalization of the Osserman-Sarnak estimate for the measure theoretic entropy of geodesic flows is presented.
Topology and growth of a special class of holomorphic self-maps of ℂ*
Authors: Keen Linda
Published: September 2008

It is a general problem to find appropriate sets of moduli for families of functions that generate dynamical systems. In this paper we solve this problem for a specific family of holomorphic self-maps of ℂ* defined by

The main theorem states that any function topologically conjugate to a member of ℱ is holomorphically conjugate to some member of the family. It follows that the coefficients of the polynomials P(z) and Q(z) are a suitable set of moduli for the families of dynamical systems generated by these functions.

The moduli spaces of functions in ℱ are easy to study computationally and have been studied by many authors. (See references in the text.)
The conjugacy of Collet–Eckmann's map of the interval with the tent map is Holder continuous
Authors: Nowicki T. , Przytycki F.
Published: September 2008

It is proved that a homeomorphism h, which conjugates a smooth unimodal map of the interval with negative Schwarzian derivative and positive Lyapunov exponent along the forward trajectory of the critical value with a tent map, and its inverse h−1 are Hölder continuous.
Homoclinic points and moduli
Authors: Posthumus Rense A.
Published: September 2008

In this paper we study some conjugacy invariants (moduli) for discrete two dimensional dynamical systems, with a homoclinic tangency. We show that the modulus obtained by Palis in the heteroclinic case also turns up in the case considered here. We also present two new conjugacy invariants.
Discontinuity of Hausdorff dimension and limit capacity on arcs of diffeomorphisms
Authors: Diaz Lorenzo J. , Viana Marcelo
Published: September 2008

We consider one-parameter families of torus diffeomorphisms that bifurcate from global hyperbolic maps (Anosov) to DA maps (derived from Anosov). For an open set of these families, we show that the Hausdorff dimension and limit capacity of the nonwandering set are not continuous across the bifurcation. We also study the behaviour of equilibrium measures near the bifurcation.
Invariant tensor fields of dynamical systems with pinched Lyapunov exponents and rigidity of geodesic flows
Authors: Feres Renato , Katok Anatoly
Published: September 2008

We consider in this note smooth dynamical systems equipped with smooth invariant affine connections and show that, under a pinching condition on the Lyapunov exponents, certain invariant tensor fields are parallel. We then apply this result to a problem of rigidity of geodesic flows for Riemannian manifolds with negative curvature.
Propriétés ergodiques, en mesure infinie, de certains systèmes dynamiques fibrés
Authors: Guivarc'h Y.
Published: September 2008

We study the ergodic properties of a class of dynamical systems with infinite invariant measure. This class contains skew-products of Anosov systems with ℝd. The results are applied to theKproperty of skew-products and also to the ergodicity of the geodesic flow on abelian coverings of compact manifolds with constant negative curvature.
A new description of the Bowen—Margulis measure
Authors: Hamenstädt Ursula
Published: September 2008

The Bowen-Margulis measure on the unit tangent bundle of the universal covering of a compact manifold of negative curvature is determined by its restriction to the leaves of the strong unstable foliation. We describe this restriction to any strong unstable manifold W as a spherical measure with respect to a natural distance on W.
Decidability of epimorphisms of dimension groups and certain modules
Authors: Kim K. H. , Roush F. W.
Published: September 2008

We give an algorithm for deciding the existence of epimorphisms of finitely additively generated Z[t] modules and dimension groups. This shows that the existence of eventual right resolving shift factor maps for mixing finite type shifts is decidable.
Rokhlin's School in ergodic theory
Authors: Yuzvinsky Sergey
Published: September 2008

Ergodic theory is one of several fields of mathematics where the name Vladimir Abramovich Rokhlin (spelled also ‘Rochlin’ and ‘Rohlin’) is well known to the specialists. That name is attached to some very often used theorems, but the goal of this paper is not just to remind the reader of these theorems. I put them in the context of the general development of ergodic theory during the thirty years 1940–70. Most of all, I want to emphasize that Rokhlin was not only a researcher producing powerful results but also a founder of schools at the two best Universities in the USSR. For at least 10 years (1958–68) these schools dominated ergodic theory. This paper is not biographical. Rokhlin's life certainly deserves a better biographer. However, I mention certain circumstances of a non-mathematical nature where it seems to be appropriate.
Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems: 1. The case of negative Schwarzian derivative
Authors: Lyubich M. Yu.
Published: September 2008

It is proved that an arbitrary one dimensional dynamical system with negative Schwarzian derivative and non-degenerate critical points has no wandering intervals. This result implies a rather complete view of the dynamics of such a system. In particular, every minimal topological attractor is either a limit cycle, or a one dimensional manifold with boundary, or a solenoid. The orbit of a generic point tends to some minimal attractor.
Minimal self-joinings for nonsingular transformations
Authors: Rudolph Daniel J. , Silva Cesar E.
Published: September 2008

The notion of minimal self-joinings for conservative nonsingular actions is defined as a restriction on the nature of rational self-joinings. The need to consider rational joinings is demonstrated by showing that any two type II1actions whose Cartesian product is ergodic have type IIIλnonsingular joinings. Lastly, actions of all Krieger types with minimal self-joinings are constructed. Hence these actions are prime and commute only with their powers.
Monotone recurrence relations, their Birkhoff orbits and topological entropy
Authors: Angenent Sigurd B.
Published: September 2008

A generalization of the class of monotone twistmaps to maps ofs1×RNis proposed. The existence of Birkhoff orbits is studied, and a criterion for positive topological entropy is given. These results are then specialized to the case of monotone twist maps. Finally it is shown that there is a large class of symplectic maps to which the foregoing discussion applies.
Convergence for moving averages
Authors: Bellow Alexandra , Jones Roger , Rosenblatt Joseph
Published: September 2008

Assume T is an ergodic measure preserving point transformation from a probability space onto itself. Let be a sequence of pairs of positive integers, and define the sequence of averaging operators . Necessary and sufficient conditions are given forthis sequence of averages to converge almost everywhere. Weighted versions are also considered.
Regular isomorphism of Markov chains is almost topological
Authors: Boyle Mike , Tuncel Selim
Published: September 2008

Two Markov chains are regularly isomorphic if and only if they have a common Markov extension by right closing block codes of degree one. A certain ideal class in the integral group ring of the ratio group associated to a Markov chain is a new invariant of regular isomorphism and some other coding relations.
Stability of m-equivalence to the weak Pinsker property
Authors: Fieldsteel Adam , Rudolph Daniel J.
Published: September 2008

Let denote the class of transformations with the weak Pinsker property, and let []m denote the class of transformations m-equivalent to some member of , where m is an entropy-preserving size. We show that if T is a factor of an element of []m, then T ∈ []m, and if T is an m-limit of elements of []m, then T ∈ []m.
The growth rate of trajectories of a quadratic differential
Authors: Masur Howard
Published: September 2008

Supposeqis a holomorphic quadratic differential on a compact Riemann surface of genusg≥ 2. Thenqdefines a metric, flat except at the zeroes. A saddle connection is a geodesic joining two zeroes with no zeroes in its interior. This paper shows the asymptotic growth rate of the number of saddles of length at mostTis at most quadratic inT. An application is given to billiards.
Generic bifurcations of the twist coefficient
Authors: Moeckel Richard
Published: September 2008

We study the behavior of the twist coefficient near an elliptic fixed point for a one-parameter family of area-preserving diffeomorphisms. By looking at the singularities near resonance we can explain the sign changes which are typically found in such a family.
On the topological stable rank of certain transformation group C*-algebras
Authors: Putnam Ian F.
Published: September 2008

We consider the crossed product or transformation group C*-algebras arising from actions of the group of integers on a totally disconnected compact metrizable space. Under a mild hypothesis, we give a necessary and sufficient dynamical condition for the invertibles in such a C*-algebra to be dense. We also examine the property of residual finiteness for such C*-algebras.
Parabolic fixed points, invariant curves and action-angle variables
Authors: Aharonov Dov , Elias Uri
Published: September 2008

A fixed point of an area-preserving mapping of the plane is called elliptic if the eigenvalues of its linearization are of unit modulus but not ±1; it is parabolic if both eigenvalues are 1 or −1. The elliptic case is well understood by Moser's theory. Here we study when is a parabolic fixed point surrounded by closed invariant curves. We approximate our mapping T by the phase flow of an Hamiltonian system. A pair of variables, closely related to the action-angle variables, is used to reduce T into a twist mapping. The conditions for T to have closed invariant curves are stated in terms of the Hamiltonian.
Minimal geodesics
Authors: Bangert Victor
Published: September 2008

Motivated by the close relation between Aubry-Mather theory and minimal geodesies on a 2-torus we study the existence and properties of minimal geodesics in compact Riemannian manifolds of dimension ≥3. We prove that there exist minimal geodesics with certain rotation vectors and that there are restrictions on the rotation vectors of arbitrary minimal geodesics. A detailed analysis of the minimal geodesics of the ‘Hedlund examples’ shows that – to a certain extent – our results are optimal.
Invariant manifolds for near identity differentiable maps and splitting of separatrices
Authors: Fontich E. , Simó C.
Published: September 2008

We consider families of differentiable diffeomorphisms with hyperbolic points, close to the identity, which tend to it when the parameter goes to zero.

We study the asymptotic behaviour of the invariant manifolds. Then we consider the case when there are homo-heteroclinic points and we find that the maximum separation between the invariant manifolds is of the order of some power of the parameter which is related to the degree of differentiability.

Finally the analogous case for flows is considered.
×2 and ×3 invariant measures and entropy
Authors: Rudolph Daniel J.
Published: September 2008

Let p and q be relatively prime natural numbers. Define T0 and S0 to be multiplication by p and q (mod 1) respectively, endomorphisms of [0,1).

Let μ be a borel measure invariant for both T0 and S0 and ergodic for the semigroup they generate. We show that if μ is not Lebesgue measure, then with respect to μ both T0 and S0 have entropy zero. Equivalently, both T0 and S0 are μ-almost surely invertible.
Connecting ergodicity and dimension in dynamical systems
Authors: Cutler C. D.
Published: September 2008

In this paper we make precise the relationship between local or pointwise dimension and the dimension structure of Borel probability measures on metric spaces. Sufficient conditions for exact-dimensionality of the stationary ergodic distributions associated with a dynamical system are obtained. A counterexample is provided to show that ergodicity alone is not sufficient to guarantee exactdimensionality even in the case of continuous maps or flows.
A topological version of a theorem of Veech and almost simple flows
Authors: Glasner Eli
Published: September 2008

Almost simple (AS) minimal flows are defined and it is shown that any factor map of an AS flow is, up to almost 1−1 equivalence, a group factor. An analogous theorem for metric, regular, point distal extensions is proved. In particular a theorem of Gottschalk is strengthened to show that any regular, point distal, metric flow is equicontinuous. When the acting groupTis commutative it is shown that every proper minimal joining of an AS flowXand a minimal flowY, is, up to almost 1−1 extensions, the relative product ofXandYover a common factor which is a group factor ofX.
Un théorème de disjonction de systèmes dynamiques et une généralisation du théorème ergodique de Wiener—Wintner
Authors: Lesigne E.
Published: September 2008

If (Ω, μ, T) is a measure preserving system and if ƒ ∈ L1(μ) then, for almost all ω, for every real polynomial P, the sequence

To prove this result we use the concept of disjointness of dynamical systems and the fact that disjointness is preserved by isometric extensions.
The number of limit cycles of polynomial deformations of a Hamiltonian vector field
Authors: Mardešić P.
Published: September 2008

We prove that the lowest upper bound for the number of limit cycles of small nonconservative polynomial deformations of degree n of the Hamiltonian vector field is n − 1.

A consequence is that the lowest upper bound for the number of limit cycles of generic n-parameter deformations of cusps is n − 1.
On Cherry flows
Authors: Martens Marco , Strien Sebastian Van , Melo Welington De , Mendes Pedro
Published: September 2008

The purpose of this research is to describe all smooth vector fields on the torus T2 with a finite number of singularities, no periodic orbits and no saddleconnections. In this paper we are able to complete the description within the class of vector fields which are area contracting near all singularities. In particular we give a large class of analytic vector fields on the torus T2 which have non-trivial recurrence and also sinks.
Differentiability with respect to parameters of average values in probabilistic contracting dynamical systems
Authors: Withers W. Douglas
Published: September 2008

We consider a dynamical system consisting of a compact subset of RN or CN with several contracting maps chosen with prescribed probabilities, which may depend on position. We show that if the maps and the probabilities are Cl+α functions of the spatial variable and an external parameter, then the average value of a Cl+α function is a differentiate function of the parameter. One implication of this theorem is that for certain families of complex functions dependent on a parameter the reciprocal of the dimension of an invariant measure on the Julia set is a harmonic function of the parameter.
Bounded-to-1 factors of an aperiodic shift of finite type are 1-to-1 almost everywhere factors also
Authors: Ashley Jonathan
Published: September 2008

We show that if π: ΣG → ΣH is a bounded-to-1 factor map from an irreducible shift of finite type ΣG with period pG to a shift of finite type ΣH with period pH, then there is a factor map that is (pG/pH)-to-1 almost everywhere. Moreover, if π is right closing, then may be taken to be right closing also.
Anosov flows with smooth foliations and rigidity of geodesic flows on three-dimensional manifolds of negative curvature
Authors: Feres Renato , Katok Anatole
Published: September 2008

We consider Anosov flows on a 5-dimensional smooth manifold V that possesses an invariant symplectic form (transverse to the flow) and a smooth invariant probability measure λ. Our main technical result is the following: If the Anosov foliations are C∞, then either (1) the manifold is a transversely locally symmetric space, i.e. there is a flow-invariant C∞ affine connection ∇ on V such that ∇R ≡ 0, where R is the curvature tensor of ∇, and the torsion tensor T only has nonzero component along the flow direction, or (2) its Oseledec decomposition extends to a C∞ splitting of TV (defined everywhere on V) and for any invariant ergodic measure μ, there exists χμ > 0 such that the Lyapunov exponents are −2χμ, −χμ, 0, χμ, and 2χμ, μ-almost everywhere.

As an application, we prove: Given a closed three-dimensional manifold of negative curvature, assume the horospheric foliations of its geodesic flow are C∞. Then, this flow is C∞ conjugate to the geodesic flow on a manifold of constant negative curvature.
A map with topological minimal self-joinings in the sense of del Junco
Authors: King Jonathan L.
Published: September 2008

Andrés del Junco has proposed a definition of topological minimal self-joinings intended to parallel Dan Rudolph's measure-theoretic concept. By means of a rank-two ‘cutting and stacking’, this article constructs the first example of a system (a subshift) satisfying his proposed definition of 2-fold topological minimal self-joinings.

The second part of the article shows that 2-fold topological minimal self-joinings does not imply 3-fold and that no map has 4-fold topological minimal self-joinings. This latter result follows from a generalization of a theorem of Schwartzman.
Compact subgroups in the centralizer of natural factors of an ergodic group extension of a rotation determine all factors
Authors: Lemańczyk Mariusz , Mentzen Mieczyslaw K.
Published: September 2008

For ergodic group extensions of transformations with discrete spectra it is proved that each invariant sub-σ-algebra is determined by a compact subgroup in the centralizer of a natural factor.
KAM theory for particles in periodic potentials
Authors: Levi Mark
Published: September 2008

It is shown that the system of the form x + V′ (x) = p (t) with periodic V and p and with (p) = 0 is near-integrable for large energies. In particular, most (in the sense of Lebesgue measure) fast solutions are quasiperiodic, provided V ∈ C(5) and p ∈ L1; furthermore, for any solution x(t) there exists a velocity bound c for all time: |x(t)| < c for all t ∈ R. For any real number r there exists a solution with that average velocity, and when r is rational, this solution can be chosen to be periodic.
Lorenz attractors through Šil'nikov-type bifurcation. Part I
Authors: Rychlik Marek Ryszard
Published: September 2008

The main result of this paper is a construction of geometric Lorenz attractors (as axiomatically defined by J. Guckenheimer) by means of an Ω-explosion. The unperturbed vector field on ℝ3is assumed to have a hyperbolic fixed point, whose eigenvalues satisfy the inequalities λ1> 0, λ2> 0, λ3> 0 and |λ2|>|λ1|>|λ3|. Moreover, the unstable manifold of the fixed point is supposed to form a double loop. Under some other natural assumptions a generic two-parameter family containing the unperturbed vector field contains geometric Lorenz attractors.

A possible application of this result is a method of proving the existence of geometric Lorenz attractors in concrete families of differential equations. A detailed discussion of the method is in preparation and will be published as Part II.
A Class of examples of D-sequences
Authors: Bick T. A. , Coffey John
Published: September 2008

We use a new method to construct a class of strongly uniformly distributed D-sequences. Such sequences are used to determine ergodic measure-preserving transformations.
The ergodic theorem for additive cocycles of ℤd or ℝd
Authors: Boivin Daniel , Derriennic Yves
Published: September 2008

Let us consider (Ω, , μ, G) a measure-preserving dynamical system, (Ω, , μ) is a probability space. The group G, which is supposed to be either ℤd or ℝd (d ≥ 1), acts on Ω by measure-preserving transformations. This action is denned by a map

which is jointly measurable, such that Tx+y = TxTy and Txμ = μ
Floquet exponents for Jacobi fields
Authors: Craig Walter
Published: September 2008

This paper introduces a Riemannian invariant of a compact Riemannian manifold based on the spectral theory for the Jacobi field operator. It is the Floquet exponent for this operator, a purely dynamical quantity computable directly from the asymptotic behavior of Jacobi fields. We show that it is related to certain traces of the Green's function, and we derive further regularity and analyticity properties for the Green's function. In case the geodesic flow is ergodic, the Floquet exponent generalizes the measure entropy, and several entropy estimates follow. An asymptotic expansion of the Floquet exponent gives rise to a sequence of ‘Jacobi invariants’, which are related to the polynomial invariants of the K dV equation.
Uniqueness and ergodic properties of attractive g-measures
Authors: Hulse Paul
Published: September 2008

We consider g-measures for the shift on where S is a finite set. For a certain class of continuous g, two g-measures are identified; they are equal if and only if there is a unique g-measure. We prove that the natural extensions of these measures are Bernoulli. With a further restriction on g when S is a two-point set, we show that there is a unique g-measure. We also consider extensions of these results to the non-continuous case.
A method for proving that monotone twist maps have no invariant circles
Authors: Jungreis Irwin
Published: September 2008

We present an existence theorem for certain kinds of orbits of a monotone twist map and use it to obtain a criterion for proving that there are no invariant circles with a certain range of rotation numbers. We have used this criterion to prove (computer assisted) that the standard map has no invariant circles for several parameter values includingk= 0.9718.
The weight-per-symbol polytope and scaffolds of invariants associated with Markov chains
Authors: Marcus Brian , Tuncel Selim
Published: September 2008

We study Markov chains via invariants constructed from periodic orbits. Canonical extensions, based on these invariants, are used to establish a constraint on the degree of finite-to-one block homomorphisms from one Markov chain to another. We construct a polytope from the normalized weights of periodic orbits. Using this polytope, we find canonically-defined induced Markov chains inside the original Markov chain. Each of the invariants associated with these Markov chains gives rise to a scaffold of invariants for the original Markov chain. This is used to obtain counterexamples to the finite equivalence conjecture and to a conjecture regarding finitary isomorphism with finite expected coding time. Also included are results related to the problem of minimality (with respect to block homomorphism) of Bernoulli shifts in the class of Markov chains with beta function equal to the beta function of the Bernoulli shift.
Resolving factor maps for shifts of finite type with equal entropy
Authors: Ashley Jonathan
Published: September 2008

We sharpen a result of Boyle, Marcus and Trow as follows. An aperiodic shift of finite type ΣAfactors onto another ΣBwith equal entropy by a 1-to-l almost everywhere right-closing map if and only if (1) the dimension group for ΣBis a quotient of that for ΣA; and (2) ΣAand ΣBsatisfy the trivial periodic point condition for existence of a shift-commuting map from ΣAto ΣB.
Iterates of meromorphic functions: I
Authors: Baker I. N. , Kotus J. , Yinian Lü
Published: September 2008

For functions meromorphic in the plane, apart from an exceptional case, the Julia set J is the closure of the set of all preimages of poles. The repelling periodic cycles are dense in J. In contrast with the case of transcendental entire functions, J may be a subset of a straight line and general classes of functions for which this is the case can be determined. J may also lie on a quasicircle through infinity which is not a straight line.
An extension of Šarkovskiư's Theorem to the n-od
Authors: Baldwin Stewart
Published: September 2008

The n-od is defined to be the set of all complex numbers z such that zn is a real number in the interval [0,1], i.e., a central point with n copies of the unit interval attached at their endpoints. Given a space X and a function f:X → X, Per (f) is defined to be the set {k: f has for a point of (least) period k, k a positive integer}. The main result of this paper is to give, for each n, a complete characterization of all possible sets Per (f), where f ranges over all continuous functions on the n-od.