bluuebunny/crossref_metadata_2025 huggingface dataset:
Odometer actions on G-measures
Authors: Brown Gavin , Dooley Anthony H.
Published: September 2008

The introduction of results from harmonic analysis leads to new methods in the study of the ergodic properties of measures under the action of the direct sum of finite groups. We take the first steps in a systematic development of part of ergodic theory based on the formalism of the Riesz product construction.
The classification of non-singular actions, revisited
Authors: Katznelson Y. , Weiss B.
Published: September 2008

A simpler treatment is given of the theorems of Dye and Krieger concerning the classification of non-singular ergodic transformations up to orbit equivalence.
The structure of automorphisms of real suspension flows
Authors: Keynes Harvey B. , Markley Nelson G. , Sears Michael
Published: September 2008

This paper is motivated by the connections between automorphisms of real suspension flows and ℝ2 suspension actions. Automorphisms which naturally lead to ℤ2-cocyles are examined from the viewpoint of covering theory in terms of an associated cylinder flow. A natural type of automorphisms (called simple) is analyzed via ergodic methods. It is shown that all automorphisms of suspensions built over minimal rotations on tori satisfy this condition. A more general approach using eigenfunctions extends this result to minimal affines, Furstenberg-type distal flows, certain nilmanifolds and a class of non-distal flows on the 2-torus.
On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems
Authors: Nair R.
Published: September 2008

In this paper we prove the following theorem.

Theorem 1. For a measure-preserving system (X, β, μ, T) and a positive integer k, if f ∈ L2(X, β, μ), the averages

,

converge μ almost everywhere. Here p runs over the rational primes and πN denotes their number in [1, N].
New estimates of Weyl sums and the remainder term in the law of distribution of the fractional part of a polynomial
Authors: Pustyl'Nikov L. D.
Published: September 2008

By using ergodic theory new estimates of Weyl sums and new estimates of the remainder term in the law of distribution of fractional parts of the polynomial are obtained.
The action at infinity of conservative groups of hyperbolic motions need not have atoms
Authors: Velling John A. , Matsuzaki Katsuhiko
Published: September 2008

Herein the authors show that discrete groups of motions on Hn+1 may be conservative on Sn but have no positive measure ergodic components for this boundary action. An explicit example of such a group is given for H3 using the Apollonian circle packing of R2.
Local complexity growth for iterations of real analytic mappings and semicontinuity moduli of the entropy
Authors: Yomdin Y.
Published: September 2008

We consider some ways in which regularity of a mapping influences dynamics of its iterations and growth of various complexity-type invariants.
Chain recurrence and attraction in non-compact spaces
Authors: Hurley Mike
Published: September 2008

In the study of a dynamical systemf:X→Xgenerated by a continuous mapfon a compact metric spaceX, thechain recurrent setis an object of fundamental interest. This set was defined by C. Conley, who showed that it has two rather different looking, but equivalent, definitions: one given in terms of ‘approximate orbits’ through individual points (pseudo-orbits, or ε-chains), and the other given in terms of the global structure of the class of ‘attractors’ and ‘basins of attraction’ off. The first of these definitions generalizes directly to dynamical systems on any metric space, compact or not. The main purpose of this paper is to extend the second definition to non-compact spaces in such a way that it remains equivalent to the first.
Absorbing cantor sets and trapping structures
Authors: Johnson Stewart D.
Published: September 2008

It it shown that a minimal attractor for a continuous, lebesgue non-singular transformation on an interval with no wandering intervals is either a periodic orbit, a finite collection of intervals, a simply attracting cantor set, or an absorbing cantor set.
Lattice invariants for sofic shifts
Authors: Williams Susan
Published: September 2008

To a factor map φ from an irreducible shift of finite type ΣAto a sofic shiftS, we associate a subgroup of the dimension group (GA, Â) which is an invariant of eventual conjugacy for φ. This invariant yields new necessary conditions for the existence of factor maps between equal entropy sofic shifts.
Meromorphic multifunctions in complex dynamics
Authors: Baribeau L. , Ransford T. J.
Published: September 2008

Let {RA} be an analytic family of rational maps and denote by j(λ) the Julia set of Rλ. We prove that the upper semicontinuous regularization j(λ) of j(λ) (which coincides with j(λ) for all λ in a dense open set) is a meromorphic multifunction, and give applications that illustrate the instability of Julia sets. In a similar vein, we also consider forward orbits of critical points and limit sets of Kleinian groups.
On the topology of manifolds with completely integrable geodesic flows
Authors: Paternain Gabriel P.
Published: September 2008

We show that if M is a compact simply connected Riemannian manifold whose geodesic flow is completely integrable with periodic integrals, then M is rationally elliptic, i.e. the rational homotopy of M is finite dimensional. We also show that rational ellipticity is shared by simply connected compact manifolds whose cotangent bundle admits a multiplicity free compact action that leaves invariant the Hamiltonian associated with some Riemannian metric. In particular it follows that if M is a Riemannian manifold whose geodesic flow is completely integrable by the Thimm method, then M is rationally elliptic. Other questions concerning the global behaviour of geodesics on homogeneous spaces are discussed.
A Wiener—Wintner property for the helical transform
Authors: Assani I.
Published: September 2008

Let (X,ℱ,μ,ϕ) be a dynamical system ϕ is an invertible measure-preserving transformation on the measure space (X,ℱ,μ). We show that for each p, 1<p<∞,f ∈ Lp(μ) we can find a single null set off which exists for all ε ℝ.
Théorème ergodique pour les opérateurs positifs à moyennes bornées sur les espaces Lp(1 < p < ∞)
Authors: Brunel Antoine
Published: September 2008

The main result is a dominated ergodic theorem for a linear positive operator T on Lp(1 > p > ∞); the theorem holds if, and only if, T is Cesaro-bounded.
The topological entropy of cellular automata is uncomputable
Authors: Hurd Lyman P. , Kari Jarkko , Culik Karel
Published: September 2008

There is no algorithm which will take a description of a celluar automaton and determine whether it has zero topological entropy, or for any fixed ε > 0 compute its topological entropy to a tolerance e. Furthermore a set of aperiodic Wang tiles arising from Penrose's kite and dart tiles is used to demonstrate specific examples of cellular automata with a single periodic point but non-trivial non-wandering sets, which furthermore can be constructed to have arbitrarily high topological entropy.
Every convex polygon with rational vertices is a rotation set
Authors: Kwapisz Jaroslaw
Published: September 2008

We prove that every convex polygon in ℝ2with vertices in ℚ2is a rotation set for some isotopic to identity homeomorphism of the two dimensional terms.
Applications of the asymptotic range to analytic subalgebras of groupoid C*-algebras
Authors: Solel Baruch
Published: September 2008

For a 1-cocycle c on a principal r-discrete groupoid G, that vanishes only on the unit space of G, we show that the asymptotic range of c, , is an invariant for the total order c−1([0, ∞]). It follows that is also an invariant (with respect to isometric isomorphisms) of the triangular analytic algebra supported on c−1([0, ∞]). We also prove that if and only if the analytic algebra has a certain maximality property.
D-function of a minimal set and an extension of Sharkovskii's theorem to minimal sets
Authors: Ye Xiangdong
Published: September 2008

LetXbe a compact Hausdorff space,f∈C0(X, X) andA⊂Xa minimal set off. We first introduce a new topological invariant, the D-function of a minimal set, by the investigation of the decomposition of the minimal set A under the action offn,n∈N. Then important properties about the invariant and the existence of minimal set with a given D-function in some subshift of finite type are discussed. Finally Sharkovskii's theorem is generalized to minimal sets of continuous mappings from the interval into itself.
Accessible saddles on fractal basin boundaries
Authors: Alligood Kathleen T. , Yorke James A.
Published: September 2008

For a homeomorphism of the plane, the basin of attraction of a fixed point attractor is open, connected, and simply-connected, and hence is homeomorphic to an open disk. The basin boundary, however, need not be homeomorphic to a circle. When it is not, it can contain periodic orbits of infinitely many different periods.
Bifurcations of dynamic rays in complex polynomials of degree two
Authors: Atela Pau
Published: September 2008

In the study of bifurcations of the family of degree-two complex polynomials, attention has been given mainly to parameter values within the Mandelbrot set M (e.g., connectedness of the Julia set and period doubling). The reason for this is that outside M, the Julia set is at all times a hyperbolic Cantor set. In this paper weconsider precisely this, values of the parameter in the complement of M. We find bifurcations occurring not on the Julia set itself but on the dynamic rays landing on itfrom infinity. As the parameter crosses the external rays of M, in the dynamic plane the points of the Julia set gain and lose dynamic rays. We describe these bifurcations with the aid of a family of circle maps and we study in detail the case of the fixed points.
Homeomorphic restrictions of smooth endomorphisms of an interval
Authors: Brucks Karen M. , Otero-Espinar Maria Victoria , Tresser Charles
Published: September 2008

We describe the asymptotic dynamics of homeomorphisms obtained as restrictions of generic C2 endomorphisms of an interval with finitely many critical points, all of which are non-flat, and with all periodic points hyperbolic. The ω -limit set of such a restricted endomorphism cannot be infinite, except when the restriction of the endomorphism to the closure of the orbit of some critical point is a minimal homeomorphism of an infinite set.
Random circle homeomorphisms
Authors: Downarowicz Tomasz , Mauldin R. Daniel , Warnock Tony T.
Published: September 2008

We investigate the behaviour of random homeomorphisms of the circle induced by composing a random homeomorphism of the interval with a randomly chosen rotation. These maps and their iterates are a.s. singular and for each rational number r in [0,1) it is shown that there is a positive probability of obtaining a map with rotation number r. For a ‘canonical’ method of producing these maps, bounds on the probability of obtaining a fixed point are obtained. We estimate this probability via computer simulations in three different ways. Simulations are also carried out for two periods. It remains unknown for this method whether a rational rotation number is obtained a.s.
An ergodic transformation with trivial Kakutani centralizer
Authors: Fieldsteel Adam , Rudolph Daniel J.
Published: September 2008

Generalizing from the centralizer of a measure-preserving dynamical system (X, ℬ, μ,T), one defines the Kakutani centralizerKC(T) of all ‘even Kakutani factor maps’ ϕ fromTto itself. Such a ϕ is a composition φ2ϕ1of an even Kakutani orbit equivalence ϕ1and a factor map ϕ2. We construct here an ergodicTacting on a nonatomic Lebesgue space (X,ℬ,μ) with the property that any ϕ ∈ KC(T) is invertible and of the form

All invertible maps of this form are automatically in KC(T) and hence for thisTthe Kakutani centralizer is as small as possible.
Markov partitions and shadowing for non-uniformly hyperbolic systems with singularities
Authors: Krüger Tyll , Troubetzkoy Serge
Published: September 2008

We show the existence of countable Markov partitions for a large class of non-uniformly hyperbolic systems with singularities including dispersing billiards in any dimension.
Matings of quadratic polynomials
Authors: Lei Tan
Published: September 2008

We apply Thurston's equivalence theory between dynamical systems of post-critically finite branched coverings and rational maps, to try to construct, from a pair of polynomials, a rational map. We prove that given two post-critically finite quadratic polynomials fc: z→z2+c and fc:z→ z2+c′, one can get a rational map if and only if c, c′ are not in conjugate limbs of the Mandelbrot set.
Existence of invariant tori in volume-preserving diffeomorphisms
Authors: Xia Zhihong
Published: September 2008

In this paper we consider certain volume-preserving diffeomorphisms on I × Tn, where I ∈ ℝ is a closed interval and Tn is an n-dimensional torus. We show that under certain non-degeneracy conditions, all of the maps sufficiently close to the integrable maps preserve a large set of n-dimensional invariant tori.
Normal numbers from independent processes
Authors: Feldman J. , Smorodinsky M.
Published: September 2008

In 1960 Schmidt [S] showed that if p and q are not powers of the same integer, i.e., if log q/log p is irrational, then for certain special measures μ on [0,1), invariant under S:x ↦ px (mod 1), μ-almost every x is normal to the base q. The measures considered in [S] were similar to Cantor-Lebesgue measure: namely, under μ the p-digit process was a special i.i.d. process where for some k ≥ 2 the elements of a certain k-element subset of the p-digits assumed probability 1/k each. The proof was fairly complicated, and did not seem to yield much more (see Keane and Pearce [K] for another proof).
Finite beta-expansions
Authors: Frougny Christiane , Solomyak Boris
Published: September 2008

We characterize numbers having finite β-expansions where β belongs to a certain class of Pisot numbers: when the β-expansion of 1 is equal to a1a2…am, where a1≥a2≥…≥am≥1 and when the β-expansion of 1 is equal to t1t2…tm(tm+1)ω where t1≥t2≥…≥tm>tm+1≥1.
Entropy of snakes and the restricted variational principle
Authors: Misiurewicz M. , Tolosa J.
Published: September 2008

For interval maps, we define the entropy of a periodic orbit as the smallest topological entropy of a continuous interval map having this orbit. We consider the problem of computing the limit entropy of longer and longer periodic orbits with the same ‘pattern’ repeated over and over (one example of such orbits is what we call ‘snakes’). We get an answer in the form of a variational principle, where the supremum of metric entropies is taken only over those ergodic measures for which the integral of a certain function is zero. In a symmetric case, this gives a very easy method of computing this limit entropy. We briefly discuss applications to topological entropy of countable chains.
Singularities in the boundaries of local Siegel disks
Authors: Rogers James T.
Published: September 2008

Bounded irreducible local Siegel disks include classical Siegel disks of polynomials, bounded irreducible Siegel disks of rational and entire functions, and the examples of Herman and Moeckel. We show there are only two possibilities for the structure of the boundary of such a disk: either the boundary admits a nice decomposition onto a circle or it is an indecomposable continuum.
Variation of Hausdorff dimension of Julia sets
Authors: Ransford T. J.
Published: September 2008

Let (Rλ)λ∈D be an analytic family of rational maps of degree d ≥ 2, where D is a simply connected domain in ℂ, and each Rλ is hyperbolic. Then the Hausdorff dimension δ(λ) of the Julia set of Rλ satisfies

where ℋ is a collection of harmonic functions u on D. We examine some consequences of this, and show how it can be used to obtain estimates for the Hausdorff dimension of some particular Julia sets.
Rigidity of centralizers of real analytic diffeomorphisms
Authors: Rocha Jorge
Published: September 2008

We define a topology, denoted by Cω, on the space of real analytic diffeomorphisms and show that the elements of a residual (Baire second category) subset of diffeomorphisms that satisfy Axiom A and the Transversality Condition have a trivial centralizer; they only commute with their integer powers.
On the C2-creation of links of critical points
Authors: Contreras Gonzalo
Published: September 2008

We give necessary conditions for an endomorphism of the interval not to be C2+α-approximated by endomorphisms having a link between critical points. We apply these techniques to show that on a C2+α-generic set of unimodal maps the endomorphisms are either Axiom A or there exists an ergodic measure with zero Lyapunov exponent. Applying this result we prove the formula conjectured by Eckmann and Ruelle for the rate of escape of the complement of the basin of the sinks. We add some remarks on the C2-stability conjecture.
Almost everywhere exponential convergence of the modified Jacobi—Perron algorithm
Authors: Ito S. , Keane M. , Ohtsuki M.
Published: September 2008

We prove that there exists a constant δ > 0 such that for almost every pair of numbers α and β there existsn0=n0(α,β) such that for anyn≥n0

where the integerspn,qn,rnare provided by the modified Jacobi-Perron algorithm.
Some open sets of nonuniformly hyperbolic cocycles
Authors: Young L.-S.
Published: September 2008

We consider some very simple examples ofSL(2, ℝ)-cocycles and prove that they have positive Lyapunov exponents. These cocycles form an open set in theC1topology.
Periodic points and finite group actions on shifts of finite type
Authors: Fiebig Ulf-Rainer
Published: September 2008

Let G be an abstract finite group. For an action α of G on a shift of finite type (SFT) S we introduce the periodic data of α, a computable finite-ordered set of complex polynomials. We show that two actions of G on possibly different SFTs are conjugate on periodic points iff their periodic data coincide. For each subgroup H of G the points fixed by α|H (the restriction of α to H) form a subsystem of S, which is of finite type. Our result shows that the zeta functions of these subsystems determine the conjugacy class (on periodic points) of α up to finitely many possibilities.

The orbit space of a finite skew action on an SFT S, endowed with the homeomorphism induced by S, is shown to have a zeta function equal to the zeta function of an SFT which is a left-closing quotient of S. We show that this zeta function equals the zeta function of S iff the skew action is inert with respect to a certain power of S.

Finally we consider functions of the periodic data as for example gyration numbers.
Orbital factor map
Authors: Hamachi Toshihiro , Kosaki Hideki
Published: September 2008

For a certain pair of discrete measured ergodic equivalence relations, we specify how one is imbedded into the other and obtain a factor and its subfactor. For this pair of factors index theory is developed. We obtain an index formula, an ‘extended’ relation corresponding to the basic extension, and so on. Our formulas as well as construction are very explicit and are given in terms of ‘measure theoretic’ data. We also present examples showing the contrast between type III0 index theory and type II1 (or IIIλ) index theory.
Poisson law for Axiom A diffeomorphisms
Authors: Hirata Masaki
Published: September 2008

Let f be an Axiom A diffeomorphism, Ω its non wandering set, µ the Gibbs measure for the Lipschitz continuous potential. We consider the (suitably normalized) return times of the orbit to the ε-neighborhood of a point z ∈ Ω and prove that for µ-a.e. z the sequence of the normalized return times converges to the Poisson point process in finite dimensional distribution as ε → 0.
Cuntz—Krieger algebras associated with Fuchsian groups
Authors: Spielberg J. S.
Published: September 2008

For Fuchsian groups of the first kind containing parabolic elements, it is shown that the action on a suitable disconnection of the limit circle generates a Cuntz—Krieger C*-algebra. This clarifies and generalizes the situation of the subalgebra within O2, and provides a new proof of the simplicity and nuclearity of certain Cuntz—Krieger algebras. The proof relies on the Markov partition obtained from a suitable fundamental polygon for the group. Counter examples are given if an unsuitable fundamental polygon is used.
Weak disks of Denjoy minimal sets
Authors: Boyland Philip
Published: September 2008

We investigate the existence of Denjoy minimal sets and, more generally, strictly ergodic sets in the dynamics of iterated homeomorphisms. It is shown that for the full two-shift, the collection of such invariant sets with the weak topology contains topological balls of all finite dimensions. One implication is an analogous result that holds for diffeomorphisms with transverse homoclinic points. It is also shown that the union of Denjoy minimal sets is dense in the two-shift and that the set of unique probability measures supported on these sets is weakly dense in the set of all shift-invariant, Borel probability measures.
The enveloping semigroup of projective flows
Authors: Ellis Robert
Published: September 2008

The enveloping semigroup of a flow (X, T) has been used to study its dynamical properties. In this paper a detailed study is made of the class of enveloping semigroups which arise in the study of flows where T is a subgroup of GL(V) and X is the associated projective space, P(V).
Decomposing isometric extensions using group extensions
Authors: Forrest A. H.
Published: September 2008

This paper studies the structure of isometric extensions of compact metric topological dynamical systems with ℤ action and gives two decompositions of the general case to a more structured case. Suppose that Y → X is a M-isometric extension. An extension, Z, of Y is constructed which is also a G-isometric extension of X, where G is the group of isometries of M. The first construction shows that, provided that (X, T) is transitive, there are almost-automorphic extensions Y′ → Y and X′ → X, so that Y′ is homeomorphic to X′ × M and the natural projection Y′ → X′ is a group extension. The second shows that, provided that (X, T) is minimal, there is a G-action on Z which commutes with T and which preserves fibres and acts on each of them minimally. Each individual orbit closure, Za, in Z is a G′-isometric extension of X, where G′ is a subgroup of G, and there is a G′-action on Za which commutes with T, preserves fibres and acts minimally on each of them. Two illustrations are presented. Of the first: to reprove a result of Furstenberg; that every distal point is IP*-recurrent. Of the second: to describe the minimal subsets in isometric extensions of minimal topological dynamical systems.
A class ofC∞-stable foliations
Authors: Alaoui Aziz El Kacimi , Nicolau Marcel
Published: September 2008

We consider foliationsFobtained as the suspension of a linear foliationF0onnby means of a linear Anosov diffeomorphismAofnkeepingF0invariant. Under suitable conditions onAthe foliationsFare shown to beC∞-stable, i.e. any differentiable foliation which isC∞-close toFisC∞-conjugated toF. The proof relies on a criterium of stability stated by R. Hamilton.
Global stability of families of vector fields
Authors: Labarca R. , Plaza S.
Published: September 2008

We consider one-parameter families with simple recurrent set. Among these families we characterize the (structurally) stables ones.
Algorithmic complexity of points in dynamical systems
Authors: White Homer S.
Published: September 2008

This work is based on the author's dissertation. We examine the algorithmic complexity (in the sense of Kolmogorov and Chaitin) of the orbits of points in dynamical systems. Extending a theorem of A. A. Brudno, we note that for any ergodic invariant probability measure on a compact dynamical system, almost every trajectory has a limiting complexity equal to the entropy of the system. We use these results to show that for minimal dynamical systems, and for systems with the tracking property (a weaker version of specification), the set of points whose trajectories have upper complexity equal to the topological entropy is residual. We give an example of a topologically transitive system with positive entropy for which an uncountable open set of points has upper complexity equal to zero. We use techniques from universal data compression to prove a recurrence theorem: if a compact dynamical system has a unique measure of maximal entropy, then any point whose lower complexity is equal to the topological entropy is generic for that unique measure. Finally, we discuss algorithmic versions of the theorem of Kamae on preservation of the class of normal sequences under selection by sequences of zero Kamae-entropy.
Invariant measures for maps of the interval that do not have points of some period
Authors: Bobok Jozef , Kuchta Milan
Published: September 2008

We study invariant measures for a continuous function which maps a real interval into itself. We show that the ratio of the measures of the two subintervals into which it is divided by a fixed point is constrained by the the set of periods of periodic points. As a consequence of this we give a new forcing relation between periodic points.
Positivity conditions for polynomials
Authors: De Angelis Valerio
Published: September 2008

We study several notions of positivity for a class of real-valued functions of several variables that includes the Laurent polynomials. We show that Handelman's positivity condition is characterized by boundedness of the associated Legendre transformation, or boundedness of the entropy function, while another notion of positivity here introduced characterizes the mapping property of the Legendre transformation derived by Marcus and Tuncel for beta functions. Several examples are given to distinguish the various notions of positivity.
Ergodic sequences of averages of group representations
Authors: Lin Michael , Wittmann Rainer
Published: September 2008

Let G be a locally compact σ -compact group with right Haar measure λ. A sequence {μn} of probabilities on G is called ergodic if for every f ∈ L1(G, λ) and t ∈ G we have ‖μn* (f − δt* f) ‖1 → 0. If T (t) is a bounded continuous representation of G by linear operators in a Banach space X, we define the μ,-average of T(t) by .
A joinings proof of Bourgain's return time theorem
Authors: Rudolph Daniel J.
Published: September 2008

Let T be a measure-preserving transformation of the Lebesgue probability space (X, F, μ). We call (X, Fμ, T) a dynamical system. Bourgain has proven the following theorem.
Holomorphic families of quasi-Fuchsian groups
Authors: Astala K. , Zinsmeister M.
Published: September 2008

We produce a holomorphic family of infinitely generated quasi-Fuchsian groups such that the Hausdorff dimension of the limit set L (Гλ) is identical to 1 for small λ, but strictly greater than 1 for λ ˜ 1. In particular, this shows that Hausdorff dimension does not depend real analytically on the parameter λ, contrary to the case of finitely generated groups.
Non-uniqueness of measures of maximal entropy for subshifts of finite type
Authors: Burton Robert , Steif Jeffrey E.
Published: September 2008

It is known that in one dimension an irreducible subshift of finite type has a unique measure of maximal entropy, the so-called Parry measure. Here we give a counterexample to this in higher dimensions. For this example, we also describe the geometric structure of the measures of maximal entropy and show that there are exactly two extremal measures.
The zeta function of the beta transformation
Authors: Flatto Leopold , Lagarias Jeffrey C. , Poonen Bjorn
Published: September 2008

The β-transformation ƒβ(x) = βx(mod 1), for β > 1, has a symbolic dynamics generalizing radix expansions to an integer base. Two important invariants of ƒβ are the (Artin-Mazur) zeta function

where Pk counts the number of fixed points of , and the lap-counting function where Lk counts the number of monotonic pieces of the kth iterate . For β-transformations these functions are related by ζβ(z) = (1 − z)Lβ(z). The function ζβ(z) is meromorphic in the unit disk, is holomorphic in {z: |z| < 1/β}, has a simple pole at z = 1/β, and has no other singularities with |z| = 1/β. Let M(β) denote the minimum modulus of any pole of ζβ(z) in |z| < 1 other than z = 1/β, and set M(β) = 1 if no other pole exists with |z| < 1. Then Pk = βk + O((M(β)−1+ε)k) for any ε > 0. This paper shows that M(β) is a continuous function, that ( for all β, and that An asymptotic formula is derived for M(β) as β → 1+, which implies that M(β) < 1 for all β in an interval (1, 1 + c0). The set is shown to have properties analogous to the set of Pisot numbers. It is closed, totally disconnected, has smallest element ≥ 1 + C0 and contains infinitely many β falling in each interval [n, n + 1) for n ∈ ℤ+. All known members of are algebraic integers which are either Pisot or Salem numbers.
Analytic destruction of invariant circles
Authors: Forni Giovanni
Published: September 2008

We give destruction results under analytic small perturbations for invariant circles of exact area-preserving monotone twist maps, applying methods developed by M. Herman and J. Mather.
Anosov flows which are uniformly expanding at periodic points
Authors: Hamenstadt Ursula
Published: September 2008

A smooth transitive Anosov flow on a compact manifoldNwhich is uniformly (a, b)-expanding at periodic points for 1 <a 0.
Basins of attraction near homoclinic tangencies
Authors: Tatjer J.C. , Simó C.
Published: September 2008

We describe the behaviour of the basin of attraction of the attracting periodic points which appear near a non-degenerate tangential homoclinic point of a dissipative saddle fixed point for one-parameter families of planar diffeomorphisms. This behaviour depends on certain relations between the eigenvalues of the saddle point and on the geometry of the tangency.
Almost everywhere convergence of convolution powers
Authors: Bellow Alexandra , Jones Roger , Rosenblatt Joseph
Published: September 2008

Given an ergodic dynamical system (X,B,m, τ) and a probability measure μ on the integers, define for all f ∈ L1(X) The almost everywhere convergence of the convolution powers μnf(x) depends on the properties of μ. If μ has finite and then for all f ∈ Lp(X), 1< p < ∞, exists for a.e. x. However, if m2(μ) is finite and E(μ)≠0, then there exists E∈B such that a.e. and a.e. In the case when m2(μ) is infinite and E(μ)=0 we give examples for which we have divergence and other examples which show convergence is possible.
Topological conditions for the existence of invariant measures for unimodal maps
Authors: Bruin H.
Published: September 2008

We present a class of S-unimodal maps having an invariant measure which is absolutely continuous with respect to Lebesgue measure. This measure can often be proved to be finite. We give an example of a map which has such a finite measure and for which the lim inf of the derivatives of the iterates of the map in the critical value is finite. It will be shown that all topologically conjugate non-flat S-unimodal maps have the same properties.
One point intersections of middle-α Cantor sets
Authors: Kraft Roger L.
Published: September 2008

For α ∈ (0, 1), let Γα denote the middle-α Cantor set contained in the interval [0,1]. Let denote the set of all t∈[−1,1] such that Γα ∩(Γα+t)is a single point. Both a geometric and a symbolic description of each is presented. The symbolic description of each will be as a shift invariant subset of the one-sided two shift that is determined by inequalities. An equivalence relation is defined on the one-sided two shift and then a linear order relation is defined on the equivalence classes. This order relation on the equivalence classes describes how the sets shrink as a decreases from ⅓ to 0.
On Mayer's conjecture and zeros of entire functions
Authors: Levin G.
Published: September 2008

We study the spectrum of a transfer operator that gives a counterexample to the conjecture of Mayer. The effect of perturbations is considered.
Periodic orbits of a dynamical system in a compound central field and a perturbed billiards system
Authors: Morita Takehiko
Published: September 2008

We consider a compound central field in the Euclidean plane which is governed by a finite number of bell-shaped potential functions with finite range. The study of the qualitative behavior of the Hamilton flow in such a potential field can be reduced to that of the so-called perturbed billiards system. The main result in this paper is the construction of a symbolic dynamics of the Hamilton flow by using the perturbed billiards system provided that the energy E > 0 is small enough. We also try to show an analogue of the prime number theorem for the closed orbits of the flow following ideas presented by Morita and Parry and Pollicott.
Regularity of the Anosov splitting and of horospheric foliations
Authors: Hasselblatt Boris
Published: September 2008

‘Bunching’ conditions on an Anosov system guarantee the regularity of the Anosov splitting up toC2−ε. Open dense sets of symplectic Anosov systems and geodesic flows do not have Anosov splitting exceeding the asserted regularity. This is the first local construction of low-regularity examples.
The cusp horseshoe and its bifurcations in the unfolding of an inclination-flip homoclinic orbit
Authors: Homburg Ale Jan , Kokubu Hiroshi , Krupa Martin
Published: September 2008

Deng has demonstrated a mechanism through which a perturbation of a vector field having an inclination-flip homoclinic orbit would have a Smale horseshoe. In this article we prove that if the eigenvalues of the saddle to which the homoclinic orbit is asymptotic satisfy the condition 2λu > min{−λs, λuu} then there are arbitrarily small perturbations of the vector field which possess a Smale horseshoe. Moreover we analyze a sequence of bifurcations leading to the annihilation of the horseshoe. This sequence contains, in particular, the points of existence of n-homoclinic orbits with arbitrary n.
Metric properties of non-renormalizableS-unimodal maps. Part I: Induced expansion and invariant measures
Authors: Jakobson Michael , Światek Grzegorz
Published: September 2008

For an arbitrary non-renormalizable unimodal map of the interval,f:I→I, with negative Schwarzian derivative, we construct a related mapFdefined on a countable union of intervals Δ. For each interval Δ,Frestricted to Δ is a diffeomorphism which coincides with some iterate offand whose range is a fixed subinterval ofI. IfFsatisfies conditions of the Folklore Theorem, we callfexpansion inducing. Letcbe a critical point off. Forfsatisfyingf″(c) ≠ 0, we give sufficient conditions for expansion inducing. One of the consequences of expansion inducing is that Milnor's conjecture holds forf: the ω-limit set of Lebesgue almost every point is the interval [f2,f(c)]. An important step in the proof is a starting condition in the box case: if for initial boxes the ratio of their sizes is small enough, then subsequent ratios decrease at least exponentially fast and expansion inducing follows.
Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dyanmical systems
Authors: Katok Anatole , Burns Keith
Published: September 2008

We establish general criteria for ergodicity and Bernoulliness for volume preserving diffeormorphisms and flows on compact manifolds. We prove that every ergodic component with non-zero Lyapunov exponents of a contact flow is Bernoulli. As an application of our general results, we construct on every compact 3-dimensional manifold aC∞Riemannian metric whose geodesic flow is Bernoulli.
Birkhoff-Hénon attractors for dissipative perturbations of area-preserving twist maps
Authors: Mora Leonardo
Published: September 2008

We prove that an area-preserving twist map having an invariant curve, can be approximated by a twist map exhibiting a Birkhoff-Hénon attractor. This is done by showing that the invariant curve can be perturbed into a saddle-node cycle with criticalities and by using a recent result reported by Diaz, Rocha and Viana.
Group-theoretic obstructions to integrability
Authors: Churchill R. C. , Rod D. L. , Singer M. F.
Published: September 2008

Let V be a four-dimensional complex symplectic vector space. This paper classifies those connected linear algebraic subgroups of the symplectic group Sp(V) that admit two independent rational invariants. As an application we show the non integrability of a three degree of freedom Hamiltonian system.
Asymptotic stability of heteroclinic cycles in systems with symmetry
Authors: Krupa Martin , Melbourne Ian
Published: September 2008

Systems possessing symmetries often admit heteroclinic cycles that persist under perturbations that respect the symmetry. The asymptotic stability of such cycles has previously been studied on an ad hoc basis by many authors. Sufficient conditions, but usually not necessary conditions, for the stability of these cycles have been obtained via a variety of different techniques.

We begin a systematic investigation into the asymptotic stability of such cycles. A general sufficient condition for asymptotic stability is obtained, together with algebraic criteria for deciding when this condition is also necessary. These criteria are always satisfied in ℝ3 and often satisfied in higher dimensions. We end by applying our results to several higher-dimensional examples that occur in mode interactions with O(2) symmetry.
On the space of ergodic invariant measures of unipotent flows
Authors: Mozes Shahar , Shah Nimish
Published: September 2008

Let G be a Lie group and Γ be a discrete subgroup. We show that if {μn} is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotent one-parameter subgroups, then the limit μ of such a sequence is supported on a closed orbit of the subgroup preserving it, and is invariant and ergodic for the action of a unipotent one-parameter subgroup of G.
Sur une notion d'autonomie de systèmes dynamiques, appliquée aux ensembles invariants des flots d' Anosov algébriques
Authors: Zeghib A.
Published: September 2008

We introduce a notion of autonomous dynamical systems which generalizes algebraic dynamical systems. We show by giving examples and by describing some properties that this generalization is not a trivial one. We apply the methods then developed to algebraic Anosov systems. We prove that a C1-submanifold of finite volume, which is invariant by an algebraic Anosov system is ‘essentially’ algebraic.
Some new rigidity results for stable orbit equivalence
Authors: Adams Scot
Published: September 2008

Broadly speaking, we prove that an action of a group with very little commutativity cannot be stably orbit equivalent to an action of a group with enough commutativity, assuming both actions are free and finite measure preserving. For example, one group may be SL2(ℝ) and the other a group with infinite discrete center (e.g., the universal cover of SL2(ℝ)); I believe this is the first rigidity result of this type for a pair of simpleLie groups both of split rank one. Another example: one group may be any nonelementary word hyperbolic group, the other any group with infinite discrete center.
No division and the set of periods for tree maps
Authors: Alsedà Lluís , Ye Xiangdong
Published: September 2008

We extend the notion of no division for star maps to tree maps and prove that the set of periods of a tree map is cofinite if there exists some periodic orbit of the given map with period larger than one having no division. Using this result we obtain somesimple proofs of known results about the set of periods of a tree map. Also we show that this set is a union of initial segments of a finite number of linear orderings which depend only on the given tree minus a finite subset of ℕ.
Periods of maps on trees with all branching points fixed
Authors: Baldwin Stewart , Libre Jaume
Published: September 2008

We characterize all possible sets of periods for all continuous self—mapson a tree having all branching points fixed. This result solves a problem which was originally posed by Alsedà, Llibre and Misiurewicz.
Liapunov stability and adding machines
Authors: Buescu Jorge , Stewart Ian
Published: September 2008

Let X be a locally connected locally compact metric space and f: X → X a continuous map. Let A be a compact transitive set under f. If A is asymptotically stable, then it has finitely many connected components, which are cyclically permuted. If it is Liapunov stable, then A may have infinitely many connected components. Our main result states that these form a Cantor set on which f is topologically conjugate to an adding machine. A number of consequences are derived, including a complete classification of compact transitive sets for continuous maps of the interval and the Liapunov instability of the invariant Cantor set of Denjoy maps of the circle.
Minimal sets of homogeneous flows
Authors: Starkov A. N.
Published: September 2008

Generalizing Ratner's topological theorem for unipotent flows we obtain a sufficient condition for an orbit closure of a flow on a homogeneous space of finite volume to be algebraic. We give some applications of the result to studying birecurrent and minimal sets of homogeneous flows.
On topological entropy of maps
Authors: Hurley Mike
Published: September 2008

We introduce an ‘entropy’ hm(f) for a continuous mapping of a compact metric space to itself which is denned in terms of (n, ∈)-separated subsets of inverse images of individual points. This invariant is compared with the inverse-image entropy h_(f) introduced recently by Langevin and Walczak. The two main results are: (1) the inequality hm(f) ≤ h(f) ≤ hm(f) + h_(f) relating hm, h_ and the topological entropy h(f); (2) if pseudo-orbits are used in place of orbits in the definition of hm then the quantity that results is equal to the topological entropy. We actually establish an inequality that at least formally is slightly stronger than (1) by defining a variant of h_ which we call hi; it is trivial to show that hi ≤ h_, and we show that h ≤ hi + hm, from which (1) follows.
Pseudogroups of isometries of ℝ: reconstruction of free actions on ℝ-trees
Authors: Gaboriau Damien , Levitt Gilbert , Paulin Frédéric
Published: September 2008

Rips' theorem about free actions on ℝ-trees relies on a careful analysis of finite systems of partial isometries of ℝ. We associate a free action on an ℝ-tree to any finite system of isometries without reflection. Any free action may be approximated (strongly in the sense of Gillet-Shalen) by actions arising in this way. Proofs involve the use, in an essential way, of separation properties of systems of isometries. We also interpret these finite systems of isometries as generating sets of pseudogroups of partial isometries between closed intervals of ℝ.
Persistence of homoclinic tangencies in higher dimensions
Authors: Romero Neptali
Published: September 2008

In this paper we extend to a very general context Newhouse's phenomenon concerning the persistence of homoclinic tangencies and the coexistence of infinitely many sinks. This is done using the corresponding results in codimension one recently provedby J. Palis and M. Viana, and in a reduction of codimension in the unfolding of homoclinic tangencies developed in the present paper.
Conditional measure and flip invariance of Bowen-Margulis and harmonic measures on manifolds of negative curvature
Authors: Yue Chengbo
Published: September 2008

Kifer and Ledrappier have asked whether the harmonic measures {νx} on manifolds of negative curvature are equivalent to the conditional measures of the harmonic measure v of the geodesic flow associated with the fibration {SxM}x∈M. We settle this question with a rigidity result. We also clear up the same problem concerning the Patterson-Sullivan measure and the Bowen–Margulis measure.
Successive derivatives of a first return map, application to the study of quadratic vector fields
Authors: Francoise J. P.
Published: September 2008

We provide an algorithm to compute the first non-zero derivative of the return map r ↦ L(r, ε) of a planar vector field which is a polynomial perturbation of . It yields a new method for finding the centre conditions in the case of a homogeneous perturbative part.
Ergodic transformations conjugate to their inverses by involutions
Authors: Goodson Geoffrey R. , del Junco Andrés , Lemańczyk Mariusz , Rudolph Daniel J.
Published: September 2008

LetTbe an ergodic automorphism defined on a standard Borel probability space for whichTandT−1are isomorphic. We investigate the form of the conjugating automorphism. It is well known that ifTis ergodic having a discrete spectrum andSis the conjugation betweenTandT−1, i.e.SsatisfiesTS=ST−1thenS2=Ithe identity automorphism. We show that this result remains true under the weaker assumption thatThas a simple spectrum. IfThas the weak closure property and is isomorphic to its inverse, it is shown that the conjugationSsatisfiesS4=I. Finally, we construct an example to show that the conjugation need not be an involution in this case. The example we construct, in addition to having the weak closure property, is of rank two, rigid and simple for all orders with a singular spectrum of multiplicity equal to two.
Measure of minimal sets of polymodal maps
Authors: Vargas E.
Published: September 2008

We consider aC2non-renormalizable polymodal mapwith finitely many non-flat critical points of turning type and we prove that any minimal set offhas zero Lebesgue measure.
Topological Wiener–Wintner ergodic theorems and a randomL2ergodic theorem
Authors: Walters Peter
Published: September 2008

We give some topological ergodic theorems inspired by theWiener-Wintnerergodic theorem. These theorems are used to give results for uniquely ergodic transformations and to study unique equilibrium states for shift maps. These latter results give randomL2ergodic theorems for a finite set of commuting measure-preserving transformations.
The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters
Authors: Akcoglu Mustafa , Bellow Alexandra , Jones Roger L. , Losert Viktor , Reinhold-Larsson Karin , Wierdl Máté
Published: September 2008

In this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kronecker's lemma on sequences linearly independent over the rationals.
Lexicographic semigroupoids
Authors: Power S. C.
Published: September 2008

The natural lexicographic semigroupoids associated with Cantor product spaces indexed by countable linear orders are classified. Applications are given to the classification of triangular operator algebras which are direct limits of upper-triangular matrix algebras.
Systèmes dynamiques gaussiens d'entropie nulle, lâchement et non lâchement Bernoulli
Authors: De La Rue Thierry
Published: September 2008

We construct two real Gaussian dynamical systems of zero entropy; the first one is not loosely Bernoulli, and the second is a loosely Bernoulli Gaussian-Kronecker system. To get loose-Bernoullicity for the second system, we prove and use a property of planar Brownian motion on [0, 1]: we can recover the whole trajectory knowing only some angles formed by the motion.