bluuebunny/crossref_metadata_2025 huggingface dataset:
Equivariant images of projective space under the action of SL (n, ℤ)
Authors: Zimmer Robert J.
Published: September 2008

The point of this note is to answer in the affirmative a question of G. A. Margulis. In the course of his proof of the finiteness of either the cardinality or the index of a normal subgroup of an irreducible lattice in a higher rank semi-simple Lie group [3], [4], Margulis proves that if Γ = SL (n, ℤ),n≥3, (X, μ) is a measurable Γ-space, μ quasi-invariant, and φ: ℙn−1→Xis a measure class preserving Γ-map, then either φ is a measure space isomorphism or μ is supported on a point. Margulis then asks whether the topological analogue of this result is true. This is answered in the following.
A proof of the estimation from below in Pesin's entropy formula
Authors: Ledrappier Fran¸ois , Strelcyn Jean-Marie
Published: September 2008

We give a proof of Pesin entropy formula in a very general setting.
Perturbations and transitivity for certain maps of an interval
Authors: Misiurewicz MichaŁ
Published: September 2008

We consider perturbations of certain transitive maps of an interval into itself and estimate how far from the transitivity the perturbed maps are. The distance turns out not to be of greater order than the square of the size of the perturbation.
On the ergodicity of geodesic flows
Authors: Ballmann W. , Brin M.
Published: September 2008

In this paper we study the ergodic properties of the geodesic flows on compact manifolds of non-positive curvature. We prove that the geodesic flow is ergodic and Bernoulli if there exists a geodesic γ such that there is no parallel Jacobi field along γ orthogonal to γ. In particular, this is true if there exists a tangent vector v such that the sectional curvature is strictly negative for all two-planes containing v, or if there exists a tangent vector v such that the second fundamental form of the horosphere determined by v is definite at the support of v.
Invariant measures for some one-dimensional attractors
Authors: Jacobson M. V.
Published: September 2008

We consider certain non-invertible maps of the square which are extensions of the quadratic maps of the interval and their small perturbations. We show that several maps of the type possess attractors which are not hyperbolic but have invariant measures similar to Bowen-Ruelle measures for hyperbolic attractors.
Perturbations of random matrix products in a reducible case
Authors: Kifer Yuri , Slud Eric
Published: September 2008

It is known that for any sequence X1, X2…, of identically distributed independent random matrices with a common distribution μ. the limit

exists with probability 1. We study some conditions under which Λ(μk)→Λ(μ) provided μk → μ in the weak sense.
Gibbs measures for partially hyperbolic attractors
Authors: Pesin Ya. B. , Sinai Ya. G.
Published: September 2008

We consider iterates of absolutely continuous measures concentrated in a neighbourhood of a partially hyperbolic attractor. It is shown that limit points can be measures which have conditional measures of a special form for any partition into subsets of unstable manifolds.
Hyperbolic behaviour of geodesic flows on manifolds with no focal points
Authors: Burns Keith
Published: September 2008

It is shown that the unit tangent bundle of a compact uniform visibility manifold with no focal points contains a subset of positive Liouville measure on which all the characteristic exponents of the geodesic flow (except in the flow direction) are non-zero. This completes Pesin's proof that the geodesic flow of such a manifold is Bernoulli.
Positive Liapunov exponents and absolute continuity for maps of the interval
Authors: Collet P. , Eckmann J.-P.
Published: September 2008

We give a sufficient condition for a unimodal map of the interval to have an invariant measure absolutely continuous with respect to the Lebesgue measure. Apart from some weak regularity assumptions, the condition requires positivity of the forward and backward Liapunov exponent of the critical point.
Rigidity of lattices of non-positive curvature
Authors: Eberlein P.
Published: September 2008

Let M, M* denote compact, connected manifolds of non-positive sectional curvature whose fundamental groups are isomorphic and whose Euclidean de Rham factors are trivial. We prove that: if M is a compact irreducible quotient of a reducible symmetric space H, then M and M* are isometric up to a constant multiple of the metric; and that the number and dimensions of the local de Rham factors are the same for M and M*. Gromov has independently proved the first result in the more general case that M is locally symmetric and globally irreducible with rank at least two.
Characteristic exponents of dynamical systems in metric spaces
Authors: Kifer Yuri
Published: September 2008

We introduce for dynamical systems in metric spaces some numbers which in the case of smooth dynamical systems turn out to be the maximal and the minimal characteristic exponents. These numbers have some properties similar to the smooth case. Analogous quantities are defined also for invariant sets.
Dynamical semigroups commuting with compact abelian actions
Authors: Bratteli Ola , Evans David E.
Published: September 2008

Let be a C*-algebra and τ:G → Aut a compact abelian action such that the fixed point algebra τ is simple. Denote by F the *-subalgebra of G-finite elements. Let H: F → be a *-operator commuting with τ such that and the matrix inequality

holds for all finite sequences X1, …, Xn in F. Then H is closable, and the closure is the generator of a strongly continuous semigroup {exp (−t): t ≥ 0} of completely positive contractions. Furthermore, there exists a convolution semigroup {μt: t ≥ 0} of probability measures on G such that

.

This result has various extensions and refinements.
Periodic points of surface homeomorphisms with zero entropy
Authors: Smillie John
Published: September 2008

This paper deals with the question of which periods can occur as periods of periodic points of zero entropy surface homeomorphisms in a given isotopy class. We give new examples of isotopy classes for which there are non-trivial restrictions and describe how the possible periods can be computed. Certain phenomena occur only for surfaces of large genus. These results have applications to the periodic data question for Morse–Smale maps.
A stochastic ergodic theorem for superadditive processes
Authors: Akcoglu M. A. , Sucheston L.
Published: September 2008

An elementary proof is given of Krengel's stochastic ergodic theorem in the setting of multiparameter superadditive processes.
Geometry, electrostatic measure and orthogonal polynomials on Julia sets for polynomials
Authors: Barnsley M. F. , Geronimo J. S. , Harrington A. N.
Published: September 2008

The Julia set B for an N'th degree polynomial T and its equilibrium electrostatic measure μ are considered. The unique balanced measure on B is shown to be μ. Integral properties of μ and of the monic polynomials orthogonal with respect to μ, Pn, n = 0, 1, 2, …, are derived. Formulae relating orthogonal polynomials of the second kind of different degrees are displayed. The measure μ is recovered both in the limit from the zeros and from the poles of the [Nn − 1/Nn] Padé approximant to the moment generating function to μ. For infinitely many polynomials of each degree N the zeros and poles all lie on an increasing sequence of trees of analytic arcs contained in B. The properties of these Padé approximant sequences support conjectures of George Baker which have not previously been tested on measures supported on sets nearly as complicated as Julia sets spread out in the complex plane.
Homoclinic and non-wandering points for maps of the circle
Authors: Block Louis , Coven Ethan , Mulvey Irene , Nitecki Zbigniew
Published: September 2008

For continuous maps ƒ of the circle to itself, we show: (A) the set of nonwandering points of ƒ coincides with that of ƒn for every odd n; (B) ƒ has a horseshoe if and only if it has a non-wandering homoclinic point; (C) if the set of periodic points is closed and non-empty, then every non-wandering point is periodic.
Structure de certains produits semi directs
Authors: Lamotte A.
Published: September 2008

We consider the skew product of a dynamical system with a Bernoulli flow and we prove that under additional conditions this skew product is isomorphic to a direct product. We use this result to show that the minimal diffeomorphism with strictly positive entropy (constructed by M. R. Herman) is isomorphic to a direct product of a 0-entropy dynamical system with a Bernoulli shift.
Flots d'Anosov sur les 3-variétés fibrées en cercles
Authors: Ghys Etienne
Published: September 2008

We consider Anosov flows on closed 3-manifolds which are circle bundles. Our main result is that, up to a finite covering, these flows are topologically equivalent to the geodesic flow of a suface of constant negative curvature. The same method shows that, if M is a closed hyperbolic manifold of any dimension, all the geodesic flows which correspond to different metrics on M and which are of Anosov type are topologically equivalent.
Kakutani equivalence of ergodic ℤn actions
Authors: Junco Andres Del , Rudolph Daniel J.
Published: September 2008

We define two families of relations between ergodic ℤn actions, both indexed equivariantly by non-singular n × n matrices. The first is to be Katok cross-sections of the same flow, indexed in a natural way by the matrices. The second is determined by the existence of an orbit preserving injection with an extra asymptotic linearity condition. We demonstrate that these two families are identical. In one dimension this is the classical theory of Kakutani equivalence.
Bowen's equidistribution theory and the Dirichlet density theorem
Authors: Parry William
Published: September 2008

Let φ be an Axiom A flow restricted to a basic set, let g be a C∞ function and let , where λg(τ) is the g length of the closest orbit τ, λ(τ) is the period of τ and h is the topological entropy of φ. We obtain an asymptotic formula for πg which includes the ‘prime number’ theorem for closed orbits. This result generalizes Bowen's theorem on the equidistribution of closed orbits. After establishing an analytic extension result for certain zeta functions the proofs proceed by orthodox number theoretical techniques.
A complex Ruelle-Perron-Frobenius theorem and two counterexamples
Authors: Pollicott Mark
Published: September 2008

In this paper a new proof of a theorem of Ruelle about real Perron-Frobenius type operators is given. This theorem is then extended to complex Perron-Frobenius type operators in analogy with Wielandt's theorem for matrices. Finally two questions raised by Ruelle and Bowen concerning analyticity properties of zeta functions for flows are answered.
Skew products and minimal dynamical systems on separable Hilbert manifolds
Authors: Fathi A.
Published: September 2008

We prove that any locally compact, non-compact, second countable group acts minimally on any metrizable connected manifold modelled on the separable Hilbert space.
A return time invariant for finitary isomorphisms
Authors: Fiebig U. R.
Published: September 2008

Poincare's recurrence theorem says that, given a measurable subset of a space on which a finite measure-preserving transformation acts, almost every point of the subset returns to the subset after a finite number of applications of the transformation. Moreover, Kac's recurrence theorem refines this result by showing that the average of the first return times to the subset over the subset is at most one, with equality in the ergodic case. In particular, the first return time function to any measurable set is integrable. By considering the supremum over all p ≥ 1 for which the first return time function is p-integrable for all open sets, we obtain a number for each almost-topological dynamical system, which we call the return time invariant. It is easy to show that this invariant is non-decreasing under finitary homomorphism. We use the invariant to construct a continuum number of countable state Markov shifts with a given entropy (and hence measure-theoretically isomorphic) which are pairwise non-finitarily isomorphic.
Non-existence of invariant circles
Authors: Mather John N.
Published: September 2008

The dynamical system associated to the difference equation

has been studied numerically by several authors. On the basis of numerical evidence, they conclude that there exists a numberk0≈ 0.97 such that there are homotopically non-trivial invariant circles for |k|≤k0and there are none for |k|>k0. In this note, we give a simple rigorous proof that there are none for |k|>.
Ergodic and mixing sequences of transformations
Authors: Berend Daniel , Bergelson Vitaly
Published: September 2008

The notions of ergodicity, strong mixing and weak mixing are defined and studied for arbitrary sequences of measure-preserving transformations of a probability space. Several results, notably ones connected with mean ergodic theorems, are generalized from the case of the sequence of all powers of a single transformation to this case. The conditions for ergodicity, strong mixing and weak mixing of sequences of affine transformations of compact groups are investigated.
Stability of the weak Pinsker property for flows
Authors: Fieldsteel Adam
Published: September 2008

An ergodic flow is said to have the weak Pinsker property if it admits a decreasing sequence of factors whose entropies tend to zero and each of which has a Bernoulli complement. We show that this property is preserved under taking factors and d-limits. In addition, we show that a flow has the weak Pinsker property whenever one ergodic transformation in the flow has this property.
Unfoldings of discrete dynamical systems
Authors: Robbin Joel W.
Published: September 2008

A universal unfolding of a discrete dynamical system f0 is a manifold F of dynamical systems such that each system g sufficiently near f0 is topologically conjugate to an element f of F with the conjugacy φ and the element f depending continuously on f0. An infinitesimally universal unfolding of f0 is (roughly speaking) a manifold F transversal to the topological conjugacy class of f0. Using Nash-Moser iteration we show infinitesimally universal unfoldings are universal and (in part II) give a class of examples relating to moduli of stability introduced by Palis and De Melo.
Rotation intervals of endomorphisms of the circle
Authors: Bamon R. , Malta I. P. , Pacifico M. J. , Takens F.
Published: September 2008

The rotation number of a diffeomorphism f: S1 → S1 with lift is defined as . We investigate the case where f is an endomorphism. Then this limit may not exist and may depend on x. We investigate the set of limit points of , as a function of x.
Non-integrability of the 1:1:2–resonance
Authors: Duistermaat J. J.
Published: September 2008

A Hamiltonian system of n degrees of freedom, defined by the function F, with an equilibrium point at the origin, is called formally integrable if there exist A A formal power series , functionally independent, in involution, and such that the Taylor expansion of F is a formal power series in the .

Take n = 3, , F(k) homogeneous of degree k, F(2) > 0 and the eigenfrequencies in ratio 1:1:2. If F(3) avoids a certain hypersurface of ‘symmetric’ third order terms, then the F system is not formally integrable. If F(3) is symmetric but F(4) is in a non-void open subset, then homoclinic intersection with Devaney spiralling occurs; the angle decays of order 1 when approaching the origin.
Billiards on almost integrable polyhedral surfaces
Authors: Gutkin E.
Published: September 2008

The phase space of the geodesic flow on an almost integrable polyhedral surface is foliated into a one-parameter family of invariant surfaces. The flow on a typical invariant surface is minimal. We associate with an almost integrable polyhedral surface its holonomy group which is a subgroup of the group of motions of the Euclidean plane. We show that if the holonomy group is discrete then the flow on an invariant surface is ergodic if and only if it is minimal.
A topological version of a theorem of Mather on twist maps
Authors: Richard Hall Glen
Published: September 2008

In this report we show that a twist map of an annulus with a periodic point of rotation number p/q must have a Birkhoff periodic point of rotation number p/q. We use topological techniques so no assumption of area-preservation or circle intersection property is needed. If the map is area-preserving then this theorem andthe fixed point theorem of Birkhoff imply a recent theorem of Aubry and Mather. We also show that periodic orbits of (significantly) smallest period for a twist map must be Birkhoff.
Finite group actions on shifts of finite type
Authors: Adler R. L. , Kitchens B. , Marcus B. H.
Published: September 2008

A continuous ℤ⊗TG action on a subshift of finite type consists of a subshift of finite type with its shift transformation, together with a group, G, of homeomorphisms of the subshift and a group automorphism T, so that the commutation relation σ ° g = Tg ° ∑A is any positive entropy subshift of finite type, G is any finite group and T is any automorphism of G then there is a non-trivial ℤ⊗TG action on ∑A. We then classify all such actions up to ‘almost topological‘ conjugacy.
On generic points in the Cartesian square of Chacón's transformation
Authors: Del Junco A. , Keane M.
Published: September 2008

We give an example of a compact metric space X and a strictly ergodic homeomorphism T of X with invariant probability μ such that for every x ∈ X the set is not generic for μ × μ} is countable.
An invariant for bounded-to-one factor maps between transitive sofic subshifts
Authors: Nasu Masakazu
Published: September 2008

We define a ‘core-matrix’ of a transitive sofic subshift, which is unique up to similarity for each transitive sofic subshift. We prove that if there exists a bounded-to-one factor map from one transitive sofic subshift to another, the block of the Jordan form of a core-matrix of this second subshift with non-zero eigenvalues is a principal submatrix of the Jordan form of a core-matrix of the first. We also prove that the subshifts that are almost of finite type are ‘spectrally of finite type’.
Expanding endomorphisms of the circle revisited
Authors: Shub Michael , Sullivan Dennis
Published: September 2008

Two Cr, r ≥ 2, expanding maps of the circle which are absolutely continuously conjugate are Cr conjugate. Here ƒ and g: S1 → S1 are expanding if they stretch tangent vectors in some metric, and a conjugacy is an isomorphism h: S1 → S1 such that fh = hg.
On non-additive processes
Authors: Wacker Ulrich
Published: September 2008

The aim of this paper is to introduce and study the class of boundedly non-additive processes. The main result is the decomposition in theorem (2.1) and theorem (3.1), which says that a boundedly non-additive process is the sum of a non-positive subadditive, a non-negative superadditive and an additive process. By this decomposition we can extend the mean and local ergodic theorems for superadditive processes of M. A. Akcoglu and U. Krengel to boundedly non-additive processes. At the end of this paper some examples are given.
Actions of lattices in semisimple groups preserving a G-structure of finite type
Authors: Zimmer Robert J.
Published: September 2008

In this paper we study actions of lattices in semisimple groups preserving a G-structure of finite type.
A quantitative version of the Kupka-Smale theorem
Authors: Yomdin Y.
Published: September 2008

Let Mm be a compact, m-dimensional smooth manifold. The n-periodic point x of a diffeomorphism f: M → M is called γ-hyperbolic, for γ≥O, if the eigenvalues λj, of dfn(x) satisfy . We prove that any Ck-diffeomorphism f: M → M, k≥3, for any ε>0 can be ε-approximated in Ck-norm by fe: M → M such that for any n each n-periodic point of fe is (a(ε))nα - hyperbolic. Here and ao>0 depends on f
On infinite tensor products of factors of type I2
Authors: Giordano T. , Skandalis G.
Published: September 2008

It is proved, using Krieger's theorem, that ITPFI's of bounded type are ITPFI2. This answers a question asked by E. J. Woods.
An Anosov action on the bundle of Weyl chambers
Authors: Hof Hans-Christoph Im
Published: September 2008

We introduce an Anosov action on the bundle of Weyl chambers of a riemannian symmetric space of non-compact type, which for rank one spaces coincides with the geodesic flow.
Anosov foliations and cohomology
Authors: Fried David
Published: September 2008

The cohomology action of an Anosov diffeomorphism on a nilmanifold resembles that of a Cartesian product map. Corresponding results hold for infranilmanifolds, giving an invariant bigrading of the cohomology and a fourfold symmetry that extends Poincaré duality. Holonomy invariant cocycles are applied to the action on first cohomology.
Chaos and topological entropy in dimension n>1
Authors: García Covadonga Blanco
Published: September 2008

In this paper we present a generalization to higher dimensions of the techniques for computation of the entropy of graphs in dimension one. Following these methods, we obtain a lower bound for the topological entropy of a differentiable map F:ℝn→ℝn possessing a snap-back repeller.
Resolving maps which commute with a power of the shift
Authors: Trow Paul
Published: September 2008

In this paper, we prove an extension of a theorem of Marcus, which says that every subshift of finite type of entropy log n, n an integer, factors onto the full n-shift. Let p(x) be a monic polynomial, irreducible over ℚ, whose coefficients (except for the leading coefficient) are non-positive integers. Suppose C(λ) is the companion matrix of p(x), where λ is the largest real root of p(x) (λ exists, by the Perron-Frobenius theorem). Then for any aperiodic, non-negative, integral matrix A, with Perron value λ, we give necessary and sufficient conditions for the existence of a positive integer n and a right-closing map f: ΣA→ΣC(λ) satisfying fσn = σnf (where σ is the shift map).
Errata to ‘Transitivity and invariant measures for the geometric model of the Lorenz equations’
Authors: Robinson Clark
Published: September 2008

All the theorems stated in [2] are true. In particular, theorem B, on the existence of an invariant measure, is true. However, in the proof of theorem B, it is necessary to use the result of G. Keller [1] on the existence of an invariant measure for a C1+α function. The attempt, in § 4 of this paper, to modify the proof of S. Wong so that it applies to this general case of a C1+α function, is faulty. These calculations do not overcome the real difficulties involved in the proof and leave major gaps in the estimates needed.
Chains, entropy, coding
Authors: Petersen Karl
Published: September 2008

Various definitions of the entropy for countable-state topological Markov chains are considered. Concrete examples show that these quantities do not coincide in general and can behave badly under nice maps. Certain restricted random walks which arise in a problem in magnetic recording provide interesting examples of chains. Factors of some of these chains have entropy equal to the growth rate of the number of periodic orbits, even though they contain no subshifts of finite type with positive entropy; others are almost sofic – they contain subshifts of finite type with entropy arbitrarily close to their own. Attempting to find the entropies of such subshifts of finite type motivates the method of entropy computation by loop analysis, in which it is not necessary to write down any matrices or evaluate any determinants. A method for variable-length encoding into these systems is proposed, and some of the smaller subshifts of finite type inside these systems are displayed.
Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable
Authors: Host B.
Published: September 2008

Parmi les systèmes dynamiques définis par des substitutions, les mieux connus sont ceux qui proviennent de substitutions à longueur constante. F. M. Dekking [1] a déterminé les valeurs propres de ces systèmes, et, plus récemment, M. Queffelec a étudié leur type spectral maximal. On se propose ici de déterminer les valeurs propres de ces systèmes dans le cas général des substitutions à longueur variable; dans cette direction, un résultat partiel a été obtenu par J. C. Martin [3] dans le cas des substitutions de 2 lettres, pour les valeurs propres correspondant à des fonctions propres continues.
The Connes spectrum for actions of Abelian groups on continuous-trace algebras
Authors: Hurder Steven , Olesen Dorte , Raeburn Iain , Rosenberg Jonathan
Published: September 2008

We study the various notions of spectrum for an action α of a locally compact abelian groupGon a typeIC*-algebraA, and discuss how these are related to the structure of the crossed productA⋊αG. In the case whereAhas continuous trace and the action ofGon Â is minimal, we completely describe the ideal structure of the crossed product. A key role is played by the restriction of α to a certain ‘symmetrizer subgroup’Sof the common stabilizer inGof the points of Â. We show by example that, contrary to a conjecture of Bratteli, it is possble forA⋊Gto be primitive but not simple, provided thatSis not discrete. In such cases, the Connes spectrum Γ(α) differs from the strong Connes spectrumof Kishimoto. The counterexamples come from subtle phenomena in topological dynamics.
The spectra of topological Markov shifts
Authors: Lind D. A.
Published: September 2008

For every Perron number λ we construct an infinite collection of topological Markov shifts with entropy log λ whose spectra are disjoint except for the necessary conjugates of λ. This is used to show that Marcus' theorem about every Markov shift of entropy log n factoring onto the full n-shift does not extend to certain entropy values.
ℤn and ℝn cocycle extensions and complementary algebras
Authors: Rudolph Daniel J.
Published: September 2008

We present here an investigation of the degree to which mixing properties can be shown to lift to cocycle extensions of an ergodic map by ℤn and ℝn weakly mixing actions. A number of other results on such extensions are also included.
Random perturbations of matrix cocycles
Authors: Young L.-S.
Published: September 2008

Lyapunov exponents for two-dimensional matrix cocycles are shown to be stable under certain stochastic perturbations.
The algebraic-geometric AKNS potentials
Authors: de Concini Corrado , Johnson Russell A.
Published: September 2008

We characterize the algebraic-geometric potentials for the Schrödinger and AKNS operators using the Weyl m-functions and the Floquet exponent for these operators. The characterization is this: among random ergodic Schrödinger operators, the alebraic-geometric potentials are those for which (i) the spectrum is a union of finitely many intervals (or bands); (ii) the Lyapounov exponent vanishes on the spectrum.
Topological dynamics and ergodic theory
Authors: Ellis Robert
Published: September 2008

It is shown that when viewed properly some concepts in topological dynamics and ergodic theory are not merely analogous but equivalent. Also the Mackey-Halmos-von Neumann theorem on ergodic processes with discrete spectrum is generalized and an account of the Mackey-Zimmer theory of minimal cocycles is given in a more general setting.
Semi-rigidity of horocycle flows over compact surfaces of variable negative curvature
Authors: Feldman J. , Ornstein D.
Published: September 2008

Let g be the geodesic flow on the unit tangent bundle of a C3 compact surface of negative curvature. Let μ be the g-invariant measure of maximal entropy. Let h be a uniformly parametrized flow along the horocycle foliation, i.e., such a flow exists, leaves μ invariant, and is unique up to constant scaling of the parameter (Margulis). We show that any measure-theoretic conjugacy: (h, μ) → (h′, μ′) is a.e. of the form θ, where θ is a homeomorphic conjugacy: g → g′. Furthermore, any homeomorphic conjugacy g → g′; must be a C1 diffeomorphism.
On Gibbs and equilibrium states
Authors: Haydn Nicolai T. A.
Published: September 2008

Ruelle and Capocaccia gave a new definition of Gibbs states on Smale spaces. Equilibrium states of suitable functions thereon are known to be Gibbs states. The converse is discussed in this paper, where the problem is reduced to shift spaces and solved there by constructing suitable conjugating homeomorphisms in order to verify the conditions for Gibbs states which Bowen gave for shift spaces, where the equivalence to equilibrium states is known.
The C1,1 conclusions in Gromov's theory
Authors: Pugh Charles C.
Published: September 2008

According to M. Gromov, any sequence of Riemann manifolds with uniformly bounded geometry has a subsequence that converges to a limit. It is shown here that this limit Riemann structure is Lipschitz, generates a Lipschitz geodesic flow, and consequently, as Gromov asserted, the limit distance function is of class C1,1. Sharpness of the results is discussed. A simple, extrinsic proof of Gromov's Theorem is included.
The ordering on permutations induced by continuous maps of the real line
Authors: Bernhardt Chris
Published: September 2008

Continuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This ordering restricted to cycles is studied.

Necessary and sufficient conditions are given for a cycle to have an immediate predecessor. When a cycle has an immediate predecessor it is unique; it is shown how to construct it. Every cycle has immediate successors; it is shown how to construct them.
Orbit types for maps of the interval
Authors: Block Louis , Hart David
Published: September 2008

Relationships between the permutations associated to periodic orbits of maps of the interval and the topology of the space of these maps are given. These extend the author's previous work on periodic orbits to orbit types.
Eventually positive matrices with rational eigenvectors
Authors: Handelman David
Published: September 2008

Let A be an n × n real matrix; sufficient conditions were previously worked out, assuming non-commensurability of eigenvectors, for A to be SL(n, )-conjugate to a matrix all sufficiently large powers of which have strictly positive entries. We show that when the ‘large’ eigenvectors are commensurable and satisfy the obvious necessary conditions, then A is also going to be so conjugate. In particular, we deduce, if A is a rational matrix with large eigenvalue exceeding 1 and of multiplicity one, then A is algebraically shift equivalent to an eventually positive matrix, using only integer rectangular matrices.
Necessary and sufficient conditions for a maximal ergodic theorem along subsequences
Authors: Jones Roger L.
Published: September 2008

Let T be an ergodic measure preserving point transformation from a probability space X onto itself. Assume that is an increasing sequence of subsets of the positive integers. Conditions are given which are sufficient for the ergodic maximal function associated with these subsets to be weak type (p, p). These conditions are shown to be both necessary and sufficient for a larger two-sided maximal function. The conditions are in the form of covering lemmas for the integers.
A rank-one, rigid, simple, prime map
Authors: del Junco A. , Rudolph D. J.
Published: September 2008

We construct a rank-1 mapTsimilar to Chacon's map [C], [JRS], but with 2n+1n-blocks in an (n + 1)-block and a single spacer in the middle. HenceTis rigid and the centralizer ofT, C(T), is uncountable. We showTis simple [V], [JR], and hence any factor ofTis the algebra of invariant sets of some weakly compact subgroup ofC(T). We showC(T) has no such subgroups and henceTis prime. Lastly we show thatTis graphic in the sense of [AM].
Markov measures determine the zeta function
Authors: Tuncel Selim
Published: September 2008

With the purpose of understanding when two subshifts of finite type are equivalent from the point of view of their spaces of Markov measures we propose the notion of Markov equivalence. We show that a Markov equivalence must respect the cycles (periodic orbits) of the subshifts. In particular, Markov equivalent subshifts of finite type have the same zeta function.
Periodic attractors as a result of diffusion
Authors: Barkmeijer Jan
Published: September 2008

We present a dynamical system in ℝ2 with a global point attractor but so that two such systems, when coupled by linear diffusion, produce a system in ℝ4 with no point attractors and yet with all solutions bounded in the positive time direction.
Weakly mixing PET
Authors: Bergelson V.
Published: September 2008

Suppose that (X, ℬ, μ) is a probability measure space and T is an invertible measure perserving transformation of (X, ℬ, μ). T is called weakly mixing if for any two sets A1A2 ∈ ℬ one has:
An ergodic theorem for iterated maps
Authors: Elton John H.
Published: September 2008

Consider a Markov process on a locally compact metric space arising from iteratively applying maps chosen randomly from a finite set of Lipschitz maps which, on the average, contract between any two points (no map need be a global contraction). The distribution of the maps is allowed to depend on current position, with mild restrictions. Such processes have unique stationary initial distribution [BE], [BDEG].

We show that, starting at any point, time averages along trajectories of the process converge almost surely to a constant independent of the starting point. This has applications to computer graphics.
Invariant subsets of expanding mappings of the circle
Authors: Urbański Mariusz
Published: September 2008

The continuity of Hausdorff dimension of closed invariant subsetsKof aC2-expanding mappinggof the circle is investigated. Ifg/Ksatisfies the specification property then the equilibrium states of Hölder continuous functions are studied. It is proved that iffis a piecewise monotone continuous mapping of a compact interval and φ a continuous function withP(f,φ)> sup(φ), then the pressureP(f,φ) is attained on one-dimensional ‘Smale's horseshoes’, and some results of Misiurewicz and Szlenk [M−Sz] are extended to the case of pressure.
Cohomology of chain recurrent sets
Authors: Hirsch Morris W. , Pugh Charles C.
Published: September 2008

Let ϕ be a flow on a compact metric space Λ and let p ∈ Λ be chain recurrent. We show that (Λ; ℝ) ≠ 0 if dimp Λ = 1 or if p belongs to a section of ϕ. Applications to planar flows and to smooth flows are given.
Another proof of Jakobson's Theorem and related results
Authors: Rychlik Marek Ryszard
Published: September 2008

The author shows that any family C2-close to fα(x) = 1 − αx2(2 − ε ≤ α ≤ 2) satisfies Jakobson's theorem: For a positive measure set of α the transformation fα has an absolutely continuous invariant measure. He also indicates some generalizations.
Asymptotic behaviour of iterated piecewise monotone maps
Authors: Willms Jürgen
Published: September 2008

In this paper the asymptotic behaviour of piecewise monotone functions f: I → I with a finite number of discontinuities is studied (where I ⊆ ℝ is a compact interval). It is shown that there is a finite number of f-almost-invariant subsets C1,…, Cr, R1,…, Rs, where each Ci is a disjoint union of closed intervals and each Rj is a Cantor-like subset of I, such that if x is a ‘typical’ point in I (in a topological sense) then exactly one of the following three possibilities will happen:

(1) {fn (x)}n ≥ 0 eventually ends up in some Ci.

(2) {fn (x)}n ≥ 0 is attracted to some Rj.

(3) {fn (x): n ≥ 0} is contained in an open, invariant set Z ⊆ I, which is such that for each n ≥ 1 fn is monotone and continuous on each connected component of Z.

Moreover, f acts topologically transitively on each Ci and minimally on each Rj. Furthermore, it is shown how the sets C1,…, Cr, R1,…, Rs can be constructed. Finally, our results are applied to some examples.
Measure theoretic entropy of the system of hard spheres
Authors: Wojtkowski Maciej P.
Published: September 2008

We estimate from below the measure theoretic entropy of the system of spheres interacting by elastic collisions. We find the asymptotics of the entropy in the case of two disks on atoms as the radius increases. We establish that high density leads to large entropy. We introduce a general estimate for Lyapunov exponents in the case of special symplectic matrices.
Minimal flows of finite almost periodic rank
Authors: Auslander Joseph , Markley Nelson
Published: September 2008

The totally minimal flow (X, T) is said to have finite almost periodic rank if there is a positive integer n such that whenever (x1, x2,…, xn+1) is an almost periodic point of the product flow (Xn+1, T×…×T) then, for some i≠j, xi, and xj are in the same orbit. The rank of (X, T) is the smallest such integer. If (Y, S) is a graphic flow, (Y, Sn) has rank |n| and it is shown that every finite rank flow has, modulo a proximal extension, a graphic power factor. Various classes of finite rank flows are defined, and characterized in terms of their Ellis groups. There are four disjoint types which have basic structural differences.
Metric properties of ε-trajectories of dynamical systems with stochastic behaviour
Authors: Blank M. L.
Published: September 2008

A highly developed branch of the modern theory of dynamical systems is the study of deterministic ones with statistical properties in behaviour. During the last decade such systems were discovered in various domains of physics, chemistry, biology and technology. It is due to their complexity, that only the simplest of such systems have been analytically investigated (Lorenz system, Rikitake dynamo, billiard systems), and that is why numerical methods are widely used, especially in applied investigations. In numerical modelling we have no true trajectory of a dynamical system f, but an approximation = (x1, x2,…) such that the sequence of distances is small in some sense. For the case of round off errors in computer modelling, such a sequence is uniformly small, i.e. there exists some ε > 0, such that supnρ(xn+1, fxn)<ε. The sequence in this case is called an ε-trajectory of the dynamical system f[1]. In a series of investigations [1–14] a study was made of the properties and applications of ε-trajectories.
Rank two interval exchange transformations
Authors: Boshernitzan Michael D.
Published: September 2008

We consider interval exchange transformations T for which the lengths of the exchanged intervals have linear rank 2 over the field of rationals. We prove that, for such T, minimality implies unique ergodicity. We also provide an algorithm which tests T for aperiodicity and minimality.
On ergodic foliations
Authors: Park Kyewon
Published: September 2008

We define an ergodic ℤ-foliation and show that it can be realized as a quotient space of the ‘covering space’. The covering space has two actions, T and S, where T is a ℤ-action, S is a map of order two, and S and T skew-commute; that is, STS = T−1. We study the isometry between two foliations via the isomorphism between two bigger group actions in the covering spaces. Properties of an ergodic foliation are studied in a way similar to the study of an ergodic action. We construct a counterexample of a K-automorphism to show that, unlike Bernoulli automorphisms, ℤ-actions do not completely determine ℤ-foliations.
Erratum to ‘Anosov Foliations and Cohomology’
Authors: Fried David
Published: September 2008

We regret to say that there is an error in the proof of theorem 1 of [F]. The algebras E*(U), E*(S) used there are not preserved by d, as claimed, save in the restricted case when U and S commute. A counterexample is the nilmanifold automorphism of [AA], Appendix 23. Thus the results of §1 and §2 are dubious, although the third section is unaffected.
Geodesic flow on the two-sphere, Part I: Positive measure entropy
Authors: Donnay Victor J.
Published: September 2008

A C∞ metric is constructed on S2 whose geodesic flow has positive measure entropy.
Subrelations of ergodic equivalence relations
Authors: Feldman J. , Sutherland C. E. , Zimmer R. J.
Published: September 2008

We introduce a notion of normality for a nested pair of (ergodic) discrete measured equivalence relations of type II1. Such pairs are characterized by a groupQwhich serves as a quotient for the pair, or by the ability to synthesize the larger relation from the smaller and an action (modulo inner automorphisms) ofQon it; in the case whereQis amenable, one can work with a genuine action. We classify ergodic subrelations of finite index, and arbitrary normal subrelations, of the unique amenable relation of type II1. We also give a number of rigidity results; for example, if an equivalence relation is generated by a free II1-action of a lattice in a higher rank simple connected non-compact Lie group with finite centre, the only normal ergodic subrelations are of finite index, and the only strongly normal, amenable subrelations are finite.
A new construction of the Margulis measure for Anosov flows
Authors: Hasselblatt Boris
Published: September 2008

The Margulis measure for Anosov flows arises from a Hausdorff measure for a natural distance on unstable leaves. This generalizes work of Ursula Hamenstädt for the case of geodesic flows.
A remark on R. Moeckel's paper ‘Geodesies on modular surfaces and continued fractions’
Authors: Nakanishi Toshihiro
Published: September 2008

It is shown that a result by Moeckel holds not only for admissible subgroups of SL (2, ℤ), but also for arbitrary subgroups of finite index.
The singularity spectrumf(α) for cookie-cutters
Authors: Rand D. A.
Published: September 2008

I use a thermodynamic formalism to study the spectrumf(α) which characterises the large fluctuations of pointwise dimension in a Gibbs state supported on a hyperbolic cookie-cutter. Amongst other things, it is proved thatf(α) is the Hausdorff dimension of the set of points with pointwise dimension α, thatf(α) is real-analytic and that its Legendre transform τ(q) is related to the Renyi dimensionDqof the Gibbs state by the formula (1 −q)Dq= τ(q).
Vladimir Abramovich Rokhlin (23 August 1919 to 3 December 1984): Introductory note
Authors: Katok A.
Published: September 2008

The Ergodic Theory and Dynamical Systems journal pays tribute to V. A. Rokhlin, one of the founders of ergodic theory, a world-renowned topologist and geometer, and a man of tragic fate and exceptional courage. Rokhlin's mathematical heritage splits rather sharply into the ergodic theory – measure theory and topology – geometry parts. This fact has to do with a natural evolution of his interests but also with the keen sense of style in mathematics that Rokhlin possessed to an unusual degree. Naturally, we will concentrate on Rokhlin's contributions to ergodic theory and measure theory, his influence on other mathematicians working in those fields, and the development of some of his ideas. Fortunately, the topology part of Rokhlin's heritage has been superbly presented in Part I of the book ‘A la recherche de la topologie perdue’ published by Birkhaüser in Progress in Mathematics series (v. 62, 1986). The same cannot be said about the work on real algebraic geometry, Rokhlin's last big achievement. The impact of that work, carried out by his students, however, is very much felt now, and is very unlikely to be forgotten or neglected. On the other hand, his work in ergodic theory and measure theory, was restricted primarily to two relatively short time periods, 1947–1950 and 1959–1964, and for the most part was not followed by Rokhlin's immediate students. Hence, it runs a certain risk of being underestimated. We hope that the articles by A. Vershik; S. Yuzvinsky and B: Weiss, published in this issue, will put Rokhlin's work and his influence in ergodic theory into the right perspective.
Vladimir Abramovich Rokhlin—A biographical tribute (23.8.1919–3.12.1984)
Authors: Vershik A. M.
Published: September 2008

Vladimir Abramovich Rokhlin was born on 23 August 1919 in Baku (Azerbaijan). His parents, Abram Beniaminovich Rokhlin and Henrietta Emmanuilovna Levenson, came from Jewish families, who lived in the Ukraine and in Byelorussia and then moved to Baku. Rokhlin's mother was the sister of the well-known literary figure and children's writer Kornei Chukovsky; Rokhlin's grandmother, Klara Levenson, was one of the first women doctors in Russia. Rokhlin's mother graduated from a medical school in France and was a doctor in Baku. She died tragically in 1923. His father was a broadly educated man and took an active part in political activity before the revolution (he was a social democrat) and in the early years of the revolution. Later he was involved in administrative work in Baku, in the Ukraine, in Central Asia and in Moscow. Not surprisingly he did not escape the Stalinist repressions: in 1939 he was arrested and on 13.7.1941 was sentenced to be shot. In 1957 his relatives received a certificate of rehabilitation (‘the case is closed due to insufficient evidence’); it is clear from this certificate that it was still impossible to obtain reliable information about the last years of his life; in particular even the exact date of his death is not precisely established. Rokhlin's family was exiled to Siberia and remained there. Fortunately Rokhlin, who was at that time a student at the University of Moscow, escaped with comparatively minor unpleasantnesses, and was not expelled from the University.
The absolute continuity of the conjugation of certain diffeomorphisms of the circle
Authors: Katznelson Y. , Ornstein D.
Published: September 2008

Let f be an orientation preserving ℋ-diffeomorphism of the circle. If the rotation number α = ρ(f) is irrational and log Df is of bounded variation then, by a wellknown theorem of Denjoy, f is conjugate to the rigid rotation Rα. The conjugation means that there exists an essentially unique homeomorphism h of the circle such that f = h−lRαh. The general problem of relating the smoothness of h to that of f under suitable diophantine conditions on α has been studied extensively (cf. [H1], [KO], [Y] and the references given there). At the bottom of the scale of smoothness for f there is a theorem of M. Herman [H2] which states that if Df is absolutely continuous and D log Df ∈ Lp, p > 1, α = ρ (f) is of ‘constant type’ which means ‘the coefficients in the continued fraction expansion of α are bounded’, and if f is a perturbation of Rα, then h is absolutely continuous. Our purpose in this paper is to give a different proof and an improved version of Herman's theorem. The main difference in the result is that we do not need to assume that f is close to Rα; the proof is very different from Herman's and is very much in the spirit of [KO].
Automorphisms of compact groups
Authors: Kitchens Bruce , Schmidt Klaus
Published: September 2008

We study finitely generated, abelian groups Γ of continuous automorphisms of a compact, metrizable group X and introduce the descending chain condition for such pairs (X, Γ). If Γ acts expansively on X then (X, Γ) satisfies the descending chain condition, and (X, Γ) satisfies the descending chain condition if and only if it is algebraically and topologically isomorphic to a closed, shift-invariant subgroup of GΓ, where G is a compact Lie group. Furthermore every such subgroup of GΓ is a (higher dimensional) Markov shift whose alphabet is a compact Lie group. By using the descending chain condition we prove, for example, that the set of Γ-periodic points is dense in X whenever Γ acts expansively on X. Furthermore, if X is a compact group and (X, Γ) satisfies the descending chain condition, then every ergodic element of Γ has a dense set of periodic points. Finally we give an algebraic description of pairs (X, Γ) satisfying the descending chain condition under the assumption that X is abelian.
Trees and amenable equivalence relations
Authors: Adams Scot
Published: September 2008

Let R be a Borel equivalence relation with countable equivalence classes on a measure space M. Intuitively, a ‘treeing’ of R is a measurably-varying way of makin each equivalence class into the vertices of a tree. We make this definition rigorous. We prove that if each equivalence class becomes a tree with polynomial growth, then the equivalence relation is amenable. We prove that if the equivalence relation is finite measure-preserving and amenable, then almost every tree (i.e., equivalence class) must have one or two ends.