bluuebunny/crossref_metadata_2025 huggingface dataset:
Flows associated with product type odometers
Authors: Osikawa M.
Published: September 2008

An AC-flow is the associated flow of a product type odometer (PTO). We give examples of AC-flows and compute their L∞-point-spectra. We also introduce an invariant for isomorphism of aperiodic conservative ergodic nonsingular flows which is a closed subset of the unit interval and contains 0 and 1. We give a necessary condition for the associated flow of an approximately finite ergodic group to be finite measure preserving.
Rational ergodicity of geodesic flows
Authors: Aaronson Jon , Sullivan Dennis
Published: September 2008

We prove the rational egodicity of geodesic flows on divergence type surfaces of constant negative curvature, and identify their asymptotic types.
Twofold mixing implies threefold mixing for rank one transformations
Authors: Kalikow Steven Arthur
Published: September 2008

This paper addresses the following long-standing open question: If a stationary transformation on a probability space obeys the property

for all measurable sets A1, A2, does it follow that

for all measurable sets A1, A2, A3? Here we answer the question affirmatively for a certain class of transformations.
Ergodic rational maps with dense critical point forward orbit
Authors: Rees Mary
Published: September 2008

Examples are constructed of ergodic rational maps with dense critical orbit.
On the notion of the dimension with respect to a dynamical system
Authors: Pesin Ya. B.
Published: September 2008

For the invariant sets of dynamical systems a new notion of dimension-the so-called dimension with respect to a dynamical system-is introduced. It has some common features with the general topological notion of the dimension, but it also reflects the dynamical properties of the system. In the one-dimensional case it coincides with the Hausdorff dimension. For multi-dimensional hyperbolic sets formulae for the calculation of our dimension are obtained. These results are generalizations of Manning's results obtained by him for the Hausdorff dimension in the two-dimensional case.
Approximation by Brownian motion for Gibbs measures and flows under a function
Authors: Denker Manfred , Philipp Walter
Published: September 2008

Let denote a flow built under a Hölder-continuous function l over the base (Σ, μ) where Σ is a topological Markov chain and μ some (ψ-mining) Gibbs measure. For a certain class of functions f with finite 2 + δ-moments it is shown that there exists a Brownian motion B(t) with respect to μ and σ2 > 0 such that μ-a.e.

for some 0 < λ < 5δ/588. One can also approximate in the same way by a Brownian motion B*(t) with respect to the probability . From this, the central limit theorem, the weak invariance principle, the law of the iterated logarithm and related probabilistic results follow immediately. In particular, the result of Ratner ([6]) is extended.
On the structure of the family of Cherry fields on the torus
Authors: Boyd Colin
Published: September 2008

A class of vector fields on the 2-torus, which includes Cherry fields, is studied. Natural paths through this class are defined and it is shown that the parameters for which the vector field is unstable is the closure ofhas irrational rotation number}, where ƒ is a certain map of the circle andRtis rotation throught. This is shown to be a Cantor set of zero Hausdorff dimension. The Cherry fields are shown to form a family of codimension one submanifolds of the set of vector fields. The natural paths are shown to be stable paths.
Locally compact groups appearing as ranges of cocycles of ergodic ℤ-actions
Authors: Golodets V. Ya. , Sinelshchikov S. D.
Published: September 2008

The paper contains the proof of the fact that every solvable locally compact separable group is the range of a cocycle of an ergodic automorphism. The proof is based on the theory of representations of canonical anticommutation relations and the orbit theory of dynamical systems. The slight generalization of reasoning shows further that this result holds for amenable Lie groups as well and can be also extended to almost connected amenable locally compact separable groups.
Approximately transitive flows and ITPFI factors
Authors: Connes A. , Woods E. J.
Published: September 2008

We define a new property of a Borel group action on a Lebesgue measure space, which we call approximate transitivity. Our main results are (i) a type III0 hyperfinite factor is ITPFI if and only if its flow of weights is approximately transitive, and (ii) for ergodic transformations preserving a finite measure, approximate transitivity implies zero entropy.
Periodic points for piecewise monotonic transformations
Authors: Hofbauer Franz
Published: September 2008

Periodic points of piecewise monotonic transformations are investigated using an oriented graph, whose closed paths represent the periodic orbits of the transformation. In the first part it is shown that the inverse of the ζ-function is a kind of characteristic polynomial of this graph, which is a power series if the graph is infinite. In the second part, the sets are determined, which can occur as{withandwhereTis a monotonic mod one transformation.
Manifolds with non-positive curvature
Authors: Burns K. , Katok A. , Ballman W. , Brin M. , Eberlein P. , Osserman R.
Published: September 2008

This is an extended report on a problem session about geodesic flows held on May 30, 1984 at MSRI, Berkeley. The list of six authors reflects the principal participants in the discussion. Contributions of other participants are mentioned in the text. The session was chaired by A. Katok; this report was prepared by K. Burns in collaboration with A. Katok.
The chain recurrent set, attractors, and explosions
Authors: Block Louis , Franke John E.
Published: September 2008

Charles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x. In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.
On the measurable dynamics of z → ez
Authors: Ghys Etienne , Goldberg Lisa R. , Sullivan Dennis P.
Published: September 2008

We study the measure theoretic properties of the complex exponential map E(z) = ez.An particular, we show that the equivalence relation generated by E is recurrent and that E has no quasi-conformal deformations. This enables us to give some information concerning Devaney's semi-conjugacy between E and the shift map on sequences of integers.
Hyperbolic sets for twist maps
Authors: Goroff Daniel L.
Published: September 2008

An example is given of an area-preserving monotone twist map such that a uniformly hyperbolic structure exists on the closure of its Birkhoff maximizing orbits.
On upcrossing inequalities for subadditive superstationary processes
Authors: Krawczak Michael
Published: September 2008

Bishop [2]has given a proof of Birkhoff's ergodic theorem by establishing upcrossing inequalities similar to those of Doob. Such inequalities can be considered as quantitative improvements of convergence theorems: while convergence a.e. means that the number of upcrossings of any interval is a.e. finite, they assert integrability and prove bounds for the integrals. The main point of this paper is to prove upcrossing inequalities for the class of subadditive superstationary processes introduced by Abid [1] as a common generalization of Kingman's [5] subadditive stationary processes and Krengel's [6] superstationary processes. We make use of ideas of Smeltzer [7] who handled the subadditive stationary discrete parameter case in his unpublished thesis. In the continuous parameter case our upcrossing inequality requires more restrictive conditions than the corresponding convergence theorem, due to Hachem [3]. We actually show by example that the number of upcrossings need not be integrable under the assumptions of Hachem even for additive stationary processes.
Problems on rigidity of group actions and cocycles
Authors: Hurder S.
Published: September 2008

A conference on the interaction of ergodic theory, differential geometry and the theory of Lie Groups was held at the Mathematical Sciences Research Institute from May 24 to June 1, 1984. This is a report of the problem session organized by A. Katok and R. Zimmer and held on May 25, 1984 dealing with the topics in the title. Another problem session was centred on the rigidity of manifolds of non-positive curvature and related topics concerning their geodesic flows. This is reported on by K. Burns and A. Karok separately [2].
Twist periodic orbits and topological entropy for continuous maps of the circle of degree one which have a fixed point
Authors: Alsedà Lluís , Llibre Jaume , Misiurewicz Michał , Simó Carles
Published: September 2008

Let f be a continuous map from the circle into itself of degree one, having a periodic orbit of rotation number p/q ≠ 0. If (p, q) = 1 then we prove that f has a twist periodic orbit of period q and rotation number p/q (i.e. a periodic orbit which behaves as a rotation of the circle with angle 2πp/q). Also, for this map we give the best lower bound of the topological entropy as a function of the rotation interval if one of the endpoints of the interval is an integer.
Smale diffeomorphisms and surface topology
Authors: Batterson Steve , Smillie John
Published: September 2008

This paper concerns Smale diffeomorphisms of compact oriented surfaces. Relationships are found between the isotopy class of the map and the dynamics of its basic sets. The form of the dynamical properties involves restrictions on periods and reduced zeta functions.
Birkhoff periodic orbits for twist maps with the graph intersection property
Authors: Bernstein David
Published: September 2008

In this paper we show that Birkhoff periodic orbits actually exist for arbitrary monotone twist maps satisfying the graph intersection property.
Mixing and spectral multiplicity
Authors: Robinson E. Arthur
Published: September 2008

A mixing transformation is constructed which has non-simple spectrum of finite multiplicity. This example is based on a rank 1 mixing transformation and is constructed by cutting and stacking. It can be made to be mixing of all orders.
Transitivity and the centre for maps of the circle
Authors: Coven Ethan M. , Mulvey Irene
Published: September 2008

We study the dynamics of continuous maps of the circle with periodic points. We show that the centre is the closure of the periodic points and that the depth of the centre is at most two. We also characterize the property that every power is transitive in terms of transitivity of a single power and some periodic data.
Smoothing continuous flows on two-manifolds and recurrences
Authors: Gutierrez Carlos
Published: September 2008

Let φ: ℝ × M → M be a continuous flow on a compact C∞ two-manifold M. It is proved that there exists a C1 flow ψ on M which is topologically equivalent to φ, and that the following conditions are equivalent:

(a) any minimal set of φ is trivial;

(b) φ is topologically equivalent to a C2 flow;

(c) φ is topologically equivalent to a C∞ flow.

Also proved is a structure and an existence theorem for continuous flows with non-trivial recurrence.
On strong shift equivalence over a Boolean semiring
Authors: Kim Ki Hang , Roush Fred W.
Published: September 2008

Shift equivalence is the relation between A, B that there exists S, R, n > 0 with RA = BR, AS = SB, SR = An, RS = Bn. Strong shift equivalence is the equivalence relation generated by these equations with n = 1. We prove that for many Boolean matrices strong shift equivalence is characterized by shift equivalence and a trace condition. However, we also show that if A is strongly shift equivalent to B, then there exists a homomorphism from an iterated directed edge graph of A to the graph of B preserving the traces of powers. This yields results on colourings of iterated directed edge graphs and might distinguish new strong equivalence classes.
Hölder continuous paths and hyperbolic toral automorphisms
Authors: Irwin M. C.
Published: September 2008

Let f:Tn→Tn (n ≥ 3) be a hyperbolic toral automorphism. Let A be the set of α > 0 such that there is a Hölder continuous path of index α in Tn with 1-dimensional orbit-closure under f We prove that α0 = sup A can be expressed in terms of the eigenvalues of f and that α0 ∈ A if and only if α0 < 1.
Topological conjugacy for sofic systems
Authors: Nasu Masakazu
Published: September 2008

We prove that any topological conjugacy between subshifts is decomposed into the product of‘bipartite codes’, and obtain a natural generalization of Williams' theorem to sofic systems: two sofic systems are topologically conjugate iff the ‘representation matrices’ of the right [left] Krieger covers for them are ‘strong shift equivalent’ within right [left] Krieger covers; a similar result with respect to Fischer covers holds for transitive sofic systems.
The commutant is the weak closure of the powers, for rank-1 transformations
Authors: King Jonathan
Published: September 2008

In the class of rank-1 transformations, there is a strong dichotomy. For such a T, the commutant is either irivial, consisting only of the powers of T, or is uncountable. In addition, the commutant semigroup, C(T), is in fact a group. As a consequence, the notion of weak isomorphism between two transformations is equivalent to isomorphism, if at least one of the transformations is rank-1. In § 2, we show that any proper factor of a rank-1 must be rigid. Hence, neither Ornstein's rank-1 mixing nor Chacón's transformation, can be a factor of a rank-1.
Equilibrium measures for rational maps
Authors: Lopes Artur Oscar
Published: September 2008

For a polynomial map the measure of maximal entropy is the equilibrium measure for the logarithm potential in the Julia set [1], [4].

Here we will show that in the case where f is a rational map such that f(∞) = ∞ and the Julia set is bounded, then the two measures mentioned above are equal if and only if f is a polynomial.
Periodic points and invariant pseudomeasures for toral endomorphisms
Authors: Veech William A.
Published: September 2008

Extending a result of Livsic [10] it is proved that the coboundary equation f(Tx)−f(x) = g(x) admits a C∞ solution f for C∞g when T is an ergodic toral endomorphism and g sums to zero over every periodic orbit. The same statement is false with C1 in place of C∞, in contrast to the Livsic (hyperbolic) theorem. In one dimension the ‘Lip α’ case leads to questions relating to the generalized Riemann hypothesis.
Almost inductive limit automorphisms and embeddings into AF-algebras
Authors: Voiculescu Dan
Published: September 2008

The crossed product of an AF-algebra by an automorphism, a power of which is approximately inner, is shown to be embeddable into an AF-algebra. The proof uses almost inductive limit automorphisms, i.e. automorphisms possessing a sequence of almost invariant finite-dimensional C*-subalgebras converging to the given AF-algebra.
Dynamics of entire functions near the essential singularity
Authors: Devaney Robert L. , Tangerman Folkert
Published: September 2008

We show that entire functions which are critically finite and which meet certain growth conditions admit ‘Cantor bouquets’ in their Julia sets. These are invariant subsets of the Julia set which are homeomorphic to the product of a Cantor set and the line [0, ∞). All of the curves in the bouquet tend to ∞ in the same direction, and the map behaves like the shift automorphism on the Cantor set. Hence the dynamics near ∞ for these types of maps may be analyzed completely. Among the entire maps to which our methods apply are exp (z), sin (z), and cos (z).
A refinement of the Conley index and an application to the stability of hyperbolic invariant sets
Authors: Floer Andreas
Published: September 2008

A compact and isolated invariant set of a continuous flow possesses a so called Conley index, which is the homotopy type of a pointed compact space. For this index a well known continuation property holds true. Our aim is to prove in this context a continuation theorem for the invariant set itself, using an additional structure. This refinement of Conley's index theory will then be used to prove a global and topological continuation-theorem for normally hyperbolic invariant sets.
Classes of linear automata
Authors: Gilman Robert H.
Published: September 2008

This paper discusses a division of one-dimensional cellular automata into classes which seem to correspond to different types of behaviour.
Boshernitzan's criterion for unique ergodicity of an interval exchange transformation
Authors: Veech William A.
Published: September 2008

Confirming a conjecture by Boshernitzan, it is proved that ifTis a minimal non-uniquely ergodic interval exchange, the minimum spacing of the partition determined byTnis O(1/n).
Periodic orbits for dissipative twist maps
Authors: Casdagli Martin
Published: September 2008

We develop simple topological criteria for the existence of periodic orbits in maps of the annulus. These are applied to one-parameter families of dissipative twist maps of the annulus and their attractors. It follows that many of the motions found by variational methods in area preserving twist maps also occur in the dissipative case.
Time-like geodesic flows on Lorentz manifolds
Authors: Emch Gérard G. , Hong Sungpyo
Published: September 2008

An extension from Riemann to Lorentz manifolds is proved for Anosov's theorem on the hyperbolicity (or exponential sensitivity to initial conditions) of the geodesic flow.
Loose Bernoullicity is preserved under exponentiation by integrable functions
Authors: Rahe Maurice H. , Rudolph Daniel J.
Published: September 2008

It is known that if Ω is a Lebesgue space, T:Ω→Ω is a loosely Bernoulli transformation, and L is a fixed non-zero integer, then the transformation S = TL will again be loosely Bernoulli on each ergodic component. In this note, the above stated result is extended to include the case where L is an arbitrary integrable integer-valued function on Ω.
Dynamics of Markov chains and stable manifolds for random diffeomorphisms
Authors: Brin M. , Kifer Yu.
Published: September 2008

We consider the Markov chain on a compact manifold M generated by a sequence of random diffeomorphisms, i.e. a sequence of independent Diff2(M)-valued random variables with common distribution. Random diffeomorphisms appear for instance when diffusion processes are considered as solutions of stochastic differential equations. We discuss the global dynamics of Markov chains with continuous transition densities and construct non-random stable foliations for random diffeomorphisms.
A counter-example to a C2 closing lemma
Authors: Gutierrez C.
Published: September 2008

Let M be a compact manifold that contains a two-dimensional punctured torus. Given p ∈ M and an integer r ≥ 2, there exists X ∈ ∞(M) having non-trivial recurrent trajectories and such that, for some neighbourhood of X|(M−{p}) in r(M−{p}), no Y ∈ has closed orbits.
On ergodic actions whose self-joinings are graphs
Authors: del Junco A. , Rudolph D.
Published: September 2008

We call an ergodic measure-preserving action of a locally compact group G on a probability space simple if every ergodic joining of it to itself is either product measure or is supported on a graph, and a similar condition holds for multiple self-joinings. This generalizes Rudolph's notion of minimal self-joinings and Veech's property S.

Main results The joinings of a simple action with an arbitrary ergodic action can be explicitly descnbed. A weakly mixing group extension of an action with minimal self-joinings is simple. The action of a closed, normal, co-compact subgroup in a weakly-mixing simple action is again simple. Some corollaries. Two simple actions with no common factors are disjoint. The time-one map of a weakly mixing flow with minimal self-joinings is prime Distinct positive times in a -action with minimal self-joinings are disjoint.
Recurrence without uniform recurrence
Authors: Korner T. W.
Published: September 2008

We construct a minimal homeomorphism of a compact space such that a sequence of its iterates converges pointwise to the identity but no sequence of its iterates converges uniformly.
Points génériques de Champernowne sur certains systèmes codes; application aux θ-shifts
Authors: Bertrand-Mathis Anne
Published: September 2008

We show that if messages of length one, two, three…of a code X which verifies ρx* < ρx are concatenated, we obtain a point generic for an invariant measure on the dynamical system associated to the code; this measure is induced by the maximal measure on the tower studied by Hansel and Blanchard.

Résumé. Nous montrons que si nous alignons les messages de longueur 1, puis 2, et ainsi de suite, d'un code X vérifiant ρx* < ρx, nous obtenons un point générique pour une mesure invariante sur le système dynamique associé au code en question; nous montrons que cette mesure se trouve être la mesure induite par la mesure d'entropie maximale sur la tour associée au code introduite par Hansel et Blanchard. Nous en déduisons des points génériques pour les systèmes sofiques et les θ-shifts munis de leur mesure maximale.
On the number of periodic reflecting rays in generic domains
Authors: Petkov Vesselin M. , Stojanov Luchezar N.
Published: September 2008

We prove that for generic domains Ω ⊂ ℝn with smooth boundary X for every integer s ≥ 2 there is at most a finite number of periodic reflecting rays with just s reflections on X.
The attracting centre of a continuous self-map of the interval
Authors: Jincheng Xiong
Published: September 2008

Let ƒ denote a continuous map of a compact interval I to itself. A point x ∈ I is called a γ-limit point of ƒ if it is both an ω-limit point and an α-limit point of some point y ∈ I. Let Γ denote the set of γ-limit points. In the present paper, we show that (1) −Γ is either empty or countably infinite, where denotes the closure of the set P of periodic points, (2) x ∈ I is a γ-limit point if and only if there exist y1 and y2 in I such that x is an ω-limit point of y1, and y1 is an ω-limit point of y2, and if and only if there exists a sequence y1, y2,…of points in I such that x is an ω-limit point of y1, and yi is an ω-limit point of yi+1 for every i ≥ 1, and (3) the period of each periodic point of ƒ is a power of 2 if and only if every γ-limit point is recurrent.
On the construction of smooth ergodic skew-products
Authors: Nerurkar Mahesh G.
Published: September 2008

In this paper we prove results about lifting dynamical and ergodic properties of a given smooth dynamical system to its skew-product extensions by smooth cocycles. The classical small divisor argument shows that in general such results are not possible. However, using the notion of the ‘fast periodic approximation’ introduced by A. Katok, we will show that if the dynamical system admits such a ‘fast periodic approximation’ then indeed a certain qualitative behaviour which is prohibited by small divisor type conditions is now in fact generic. The techniques are also applied to show that ‘recurrent-proximal’ behaviour of solutions of linear differential equations with almost periodic coefficients is generic under suitable conditions on the coefficient matrix.
A method for constructing attractors
Authors: Barge Marcy
Published: September 2008

A procedure is developed for constructing C1 diffeomorphisms of the two sphere having inverse limits of certain interval maps as attractors. The method is carried out for a particular interval map yielding a diffeomorphism with a transitive non-hyperbolic attractor.
Joint ergodicity for group actions
Authors: Bergelson Vitaly , Rosenblatt Joseph
Published: September 2008

Let T1,…,Tn be continuous representations of a σ-compact separable locally compact amenable group G as measure-preserving transformations of a non-atomic separable probability space (X, β, m). Let (Kn) be a right Følner sequence of compact sets in G. If T1,…,Tn are pairwise commuting in the sense that Ti(g)Tj(h) = Tj(h)Ti(g) for i ≠ j and g, h ∈ G, then necessary and sufficient conditions can be given, in terms of the ergodicity of certain tensor products, for the following to hold: for all F1,…,Fn∈L∞, the sequence AN(x) where

converges in L2(X) to . The necessary and sufficient conditions are that each of the following representations are ergodic: Tn, Tn−1⊗Tn−1Tn,…,T2⊗T2T3⊗…⊗T2…Tn, T1⊗T1T2⊗…⊗T1…Tn.

In order to prove this theorem, specific properties of the decomposition of L2(X) into its weakly mixing and compact subspaces with respect to a representation Ti are needed. These properties are also used to prove some generalizations of wellknown facts from ergodic theory in the case where G is the integer group Z.
Automorphisms of solenoids and p-adic entropy
Authors: Lind D. A. , Ward T.
Published: September 2008

We show that a full solenoid is locally the product of a euclidean component and p-adic components for each rational prime p. An automorphism of a solenoid preserves these components, and its topological entropy is shown to be the sum of the euclidean and p-adic contributions. The p-adic entropy of the corresponding rational matrix is computed using its p-adic eigenvalues, and this is used to recover Yuzvinskii's calculation of entropy for solenoidal automorphisms. The proofs apply Bowen's investigation of entropy for uniformly continuous transformations to linear maps over the adele ring of the rationals.
A positive Liapunov exponent for the critical value of an S-unimodal mapping implies uniform hyperbolicity
Authors: Nowicki Tomasz
Published: September 2008

A positive Liapunov exponent for the critical value of an S-unimodal mapping implies a positive Liapunov exponent of the backward orbit of the critical point, uniform hyperbolic structure on the set of periodic points and an exponential diminution of the length of the intervals of monotonicity. This is the proof of the Collet-Eckmann conjecture from 1981 in the general case.
A sofic system which is not spectrally of finite type
Authors: Williams Susan
Published: September 2008

We exhibit a transitive sofic system for which the core matrix has negative trace, and hence cannot share the nonzero spectrum of any subshift of finite type cover. We also show that every transitive sofic system has an integral core matrix.
On orbits of endomorphisms of tori and the Schmidt game
Authors: Dani S. G.
Published: September 2008

We show that there exists a subset F of the n-dimensional torus n such that F has Hausdorff dimension n and for any x∈F and any semisimple automorphism σ of n the closure of the σ-orbit of x contains no periodic points.
Entropy and semi-conjugacy in dimension two
Authors: Handel Michael
Published: September 2008

We prove that if a diffeomorphism f of a closed surface is homotopic to and has the same topological entropy as a pseudo-Anosov homeomorphism g, then f is semi-conjugate to g. As part of the proof, a necessary and sufficient condition is given for a pseudo-orbit of a pseudo-Anosov homeomorphism g to be shadowed by an actual orbit of g.
Mixing and asymptotic distribution modulo 1
Authors: Lyons Russell
Published: September 2008

If μ is a probability measure which is invariant and ergodic with respect to the transformationx↦qxon the circle ℝ/ℤ, then according to the ergodic theorem, {qnx} has the asymptotic distribution μ for μ-a.e.x. On the other hand, Weyl showed that when μ is Lebesgue measure, λ, and {mj} is an arbitrary sequence of integers increasing strictly to ∞, the asymptotic distribution of {mjx} is λ for λ-a.e.x. Here, we investigate the asymptotic distributions of {mjx} μ-a.e. for fairly arbitrary {mj} under some strong mixing conditions on μ. The result is a kind of stable ergodicity: the distributions are obtained from simple operations applied to μ. The ideas extend to the situation of a sequence of transformationsx↦qnxwhere invariance is not present. This gives us information about many Riesz products and Bernoulli convolutions. Finally, we apply the theory to resolve some questions aboutH-sets.
Ω-Stability is not dense in Axiom A
Authors: Patterson S. E.
Published: September 2008

An example of a diffeomorphism f on a three dimensional manifold M3 is constructed so that f satisfies Axiom A, has a cycle and f has a neighborhood N in Diffr (M3) so that each g in N is not Ω-stable. Existing techniques allow one to extend this example to any compact manifold of dimension greater than two.
Continuous invariant cone families and ergodicity of flows in dimension three
Authors: Burns Keith , Gerber Marlies
Published: September 2008

It is shown that a C2 flow on a compact three-dimensional manifold that preserves a smooth measure and has a continuous family of cones satisfying a certain invariance property must be ergodic.
Existence de cycles pour des multi-applications du cercle
Authors: Dufour Jean-Paul
Published: September 2008

We consider multi-applications Γ of the circleS1, the graphs of which are ‘degree (1, 1)’, continuous piece-wise monotonic curves ofS1×S1In general Γpis not a connected curve but it is a union of a degree (1, 1) continuous curve ΓpofS1×S1and of some other curves homotopic to a point. Using these Γpwe are able to study dynamics of Γ. We focus on the case where Γ has no periodic points and we see, for instance, that all ‘regular’ orbits have, onS1the same order as orbits of an irrational rotation. Using this we prove that such F without ‘cycles’ are obtained from a Denjoy's counter-example, perturbing it in the holes of the invariant set. Finally we generalize the classical result of Block and Franke showing that if Γ is aC2curve with no degenerate critical points, or if Γ is aC∞ curve with no ‘flat’ points, there are always ‘cycles’, unless Γ is an homeomorphism.
The number of periodic points of smooth maps
Authors: Matsuoka Takashi
Published: September 2008

Letf: M → M be a C1 map on a compact manifold. We give a topological condition under which f has an even number of periodic points with a given period. We also obtain a sufficient condition, in terms of homology, for ƒ to have infinitely many periodic points.
Singularities of vector fields on ℝ3 determined by their first non-vanishing jet
Authors: Bonckaert Patrick , Dumortier Freddy , Strien Sebastian Van
Published: September 2008

In this paper we will present a result which gives a sufficient condition for a vector field X on ℝ3 to be equivalent at a singularity to the first non-vanishing jet jkX(p) of X at p. This condition - which only depends on the homogeneous vector field defined by jkX(p) - is stated in terms of the blown-up vector field (which is defined on S2xℝ), and essentially means that there are no saddleconnections for |S2×{0}.

The key tool in the proof will be a result of local normal linearization along a codimension 1 submanifold M providing a C0 conjugacy having a normal derivative along M equal to 1.
Rigidity in topological dynamics
Authors: Glasner S. , Maon D.
Published: September 2008

By analogy with the ergodic theoretical notion, we introduce notions of rigidity for a minimal flow (X, T) according to the various ways a sequence Tni can tend to the identity transformation. The main results obtained are:

(i) On a rigid flow there exists a T-invariant, symmetric, closed relation Ñ such that (X, T) is uniformly rigid iff Ñ = Δ, the diagonal relation.

(ii) For syndetically distal (hence distal) flows rigidity is equivalent to uniform rigidity.

(iii) We construct a family of rigid flows which includes Körner's example, in which Ñ exhibits various kinds of behaviour, e.g. Ñ need not be an equivalence relation.

(iv) The structure of flows in the above mentioned family is investigated. It is shown that these flows are almost automorphic.
Equivariant spectral decomposition for flows with a ℤ-action
Authors: Mosher Lee
Published: September 2008

Given a manifold equipped with a free, properly discontinuous, cocompact ℤ-action, and a flow on which is ℤ-equivariant, we study the qualitative dynamics of . Under certain hypotheses on , we show that the chain recurrent set of has a decomposition which is the analogue, in the category of ℤ-equivariant flows, of Smale's spectral decomposition for recurrent sets of Axiom A flows.
Homomorphismes entre systèmes dynamiques définis par substitutions
Authors: Host B. , Parreau F.
Published: September 2008

On étudie ici les homomorphismes métriques entre systèmes dynamiques definis par des substitutions de même longueur. En général, ces homomorphismes sont continus, et ont une forme simple. Le commutant essentiel d'un système dynamique defini par une substitution est fini, et peut être déterminé explicitement.
An algorithm for calculating critical probabilities and percolation functions in percolation models defined by rotations
Authors: Meester Ronald W. J.
Published: September 2008

A class of percolation models on ℤd is developed in which the stochastic structure is provided by means of a d-parameter dynamical system. Of particular interest are those models generated by circle rotations. Unlike for independent models, the critical value and the percolation function can be explicitly calculated. These calculations lead to a conjecture concerning the behaviour of a related dynamical system.
Almost all interval exchange transformations with flips are nonergodic
Authors: Nogueira Arnaldo
Published: September 2008

Here we prove that almost all interval exchange transformations which reverse orientation, in at least one interval, have a periodic point where the derivative is − 1. Therefore they are periodic in an open neighborhood of the periodic point.
Trees associated with the configuration of Herman rings
Authors: Shishikura Mitshuhiro
Published: September 2008

For a rational function with Herman rings, we define a tree and a piece-wise linear map on it, which reflect the configuration of the Herman rings. Their properties are investigated and some examples are given. Moreover, it is possible to define a similar tree associated with (super) attractive basins or Siegel disks.
Subsystems, Perron numbers, and continuous homomorphisms of Bernoulli shifts
Authors: Tuncel Selim
Published: September 2008

Let S, T be subshifts of finite type, with Markov measures p, q on them, and let φ: (S, p) → (T, q) be a block code. Let Ip, Iq denote the information cocycles of p, q. For a subshift of finite type ⊂T, the pressure of equals that of . Applying this to Bernoulli shifts and using finiteness conditions on Perron numbers, we have the following. If the probability vector p = (p1…, pk+1) is such that the distinct transcendental elements of {p1/pk+1…pk/pk+1) are algebraically independent then the Bernoulli shift B(p) has finitely many Bernoulli images by block codes.
Non-smooth geodesic flows and the earthquake flow on Teichmüller space
Authors: Weiss Howard
Published: September 2008

Thurston generalized the notion of a twist deformation about a simple closed geodesic on a hyperbolic Riemann surface to a twisting or shearing along a much more complicated object called a measure geodesic lamination. This new deformation is called an earthquake and it generates a flow on the tangent bundle of Teichmüller space.

In this paper we study the earthquake flow. We show that the flow is not smooth and that it is not the geodesic flow for an affine connection. We also derive the explicit form of the system of differential equations which earthquake trajectories satisfy.
The geometry of measured geodesic laminations and measured train tracks
Authors: Weiss Howard
Published: September 2008

Thurston and Kerckhoff have shown that the space of measured geodesic laminations on a hyperbolic Riemann surface serves as a non-linear model of the tangent space to Teichmüller space at the surface. In this paper we show that the natural map between these manifolds has stronger than Hölder continuous regularity.
On the work of V. A. Rokhlin in ergodic theory
Authors: Weiss Benjamin
Published: September 2008

The impact of V. A. Rokhlin's work in ergodic theory is discussed with particular emphasis on his famous lemma and its generalizations and his foundational work on measurable partitions.
The differentiability of the conjugation of certain diffeomorphisms of the circle
Authors: Katznelson Y. , Ornstein D.
Published: September 2008

Our purpose in this paper is to present a more or less complete solution to the problem of the smoothness of the conjugation of aperiodic diffeomorphisms of the circle. We show that the rotation number and the smoothness of the diffeomorphism guarantee a certain smoothness for the homeomorphism which conjugates it with a rigid rotation, and obtain the best smoothness that can be guaranteed.
Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems 2. The smooth case
Authors: Blokh A. M. , Lyubich M. Yu.
Published: September 2008

We prove that an arbitrary one dimensional smooth dynamical system with non-degenerate critical points has no wandering intervals.
Attractors in cellular automata
Authors: Hurley Mike
Published: September 2008

We give a classification theorem for cellular automata, showing that either there is a minimal quasi-attractor whose basin has full measure, or else no chain component has a basin with positive measure.
On the almost everywhere convergence of the ergodic averages
Authors: Martín-Reyes F. J. , De La Torre A.
Published: September 2008

Let (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).
An explosion point for the set of endpoints of the Julia set of λ exp (z)
Authors: Mayer John C.
Published: September 2008

The Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to ∞. Since Aλ contains all the repelling periodic points of Eλ, it follows that Jλ = Cl (Aλ). We show that Aλ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, Aλ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.
Marker automorphisms of the one-sided d-shift
Authors: Ashley Jonathan
Published: September 2008

We identify a set of generators for the automorphism group of the one-sided d-shift. For the 3-shift, this set of generators has an application to the dynamics of cubic polynomials.
The splitting of separatrices for analytic diffeomorphisms
Authors: Fontich E. , Simó C.
Published: September 2008

We study families of diffeomorphisms close to the identity, which tend to it when the parameter goes to zero, and having homoclinic points. We consider the analytical case and we find that the maximum separation between the invariant manifolds, in a given region, is exponentially small with respect to the parameter. The exponent is related to the complex singularities of a flow which is taken as an unperturbed problem. Finally several examples are given.
Meromorphic extension of the zeta function for Axiom A flows
Authors: Haydn Nicolai T. A.
Published: September 2008

We prove the meromorphicity of the zeta function on shifts of finite type for Hölder continuous functions assuming that the essential spectrum of the associated Ruelle operator is contained in the open unit disc. This result allows to extend the region of meromorphicity of the zeta function for Axiom A flows by a strip whose width is determined by the contraction rate of the flow.
Minimal convergence on Lp spaces
Authors: Assani I.
Published: September 2008

Let (X, F, μ) be a probability measure space, p and β real numbers such that 1≤p<+∞ and 0<β<p. For any linear positive operator T satisfying T1, T*1 = 1 we prove the norm and pointwise convergence of the sequence We get then the pointwise and norm convergence in Lp, 0 < β ≥ 1 < p < 2, of the sequence sgn Sif for any positive linear operator on Lp(Ω, A, μ) (μ-σ-finite) verifying ∥(1 − α)I + αS∥p ≤ 1 for a real number 0 < α < 1. In the particular case α = 1, (S is a contraction), β = p−l, this result gives the pointwise and norm convergence of the sequences introduced by Beauzamy and Enflo in 1985 to the asymptotic center of the sequence .
Automorphisms of one-sided subshifts of finite type
Authors: Boyle Mike , Franks John , Kitchens Bruce
Published: September 2008

We prove that the automorphism group of a one-sided subshift of finite type is generated by elements of finite order. For one-sided full shifts we characterize the finite subgroups of the automorphism group. For one-sided subshifts of finite type we show that there are strong restrictions on the finite subgroups of the automorphism group.
Produits de matrices aléatoires et applications aux propriétés géometriques des sous-groupes du groupe linéaire
Authors: Guivarc'h Yves
Published: September 2008

Using the asymptotic properties of products of random matrices we study some properties of the subgroups of the linear group. These properties are centered around the theorem of J. Tits giving the existence of free subgroups in linear groups.
The C*-algebras of Morse—Smale flows on two-manifolds
Authors: Wang Xiaolu
Published: September 2008

We give a classification of all the C*-algebras of Morse-Smale flows on closed two-manifolds, and determine the relation between the invariants of dynamical systems and the topological invariants of the C*-algebras.
The box and Hausdorff dimension of self-affine sets
Authors: Bedford Tim , Urbański Mariusz
Published: September 2008

Under a natural assumption the Hausdorff dimension of a measure μ canonically associated with a given self-affine set is computed. A simplified proof of Bowen's formula for the box dimension of self-affine sets proved earlier is given. A condition for the box dimension and Hausdorff dimension to be equal is proven, and a collection of examples in which this condition can be checked is discussed.
Bernoulli shifts of the same entropy are finitarily and unilaterally isomorphic
Authors: Del Junco Andrés
Published: September 2008

Letp= {p1,  ,pn} andq= {qi,  ,qm} be finite probability vectors, each having at least three non-zero components, such that. LetC= {1,  ,n},D= {1,  ,m} and let (Cℤ,pℤ, σ) and (Dℤ,qℤ, σ) be the corresponding Bernoulli shifts. Then there exists an isomorphism φ between these shifts such that for a.a.x∈ Cℤσ(x)(0) is determined by finitely many of the future co-ordinatesx(0),x(1),   and for a. a.y∈Dℤφ−1(y)(0) is determined by finitely many of the co-ordinates  y(−1),y(0),y(1),  .
Exponents, attractors and Hopf decompositions for interval maps
Authors: Keller Gerhard
Published: September 2008

Our main results, specialized to unimodal interval maps T with negative Schwarzian derivative, are the following:

(1) There is a set CT such that the ω-limit of Lebesgue-a.e. point equals CT. CT is a finite union of closed intervals or it coincides with the closure of the critical orbit.

(2) There is a constant λT such that for Lebesgue-a.e. x.

(3) λT > 0 if and only if T has an absolutely continuous invariant measure of positive entropy.

(4) , i.e. uniform hyperbolicity on periodic points implies λT > 0, and λT < 0 implies the existence of a stable periodic orbit.
An area approach to wandering domains for smooth surface endomorphisms
Authors: Norton Alec
Published: September 2008

We prove an infinite-area lemma for maps which are area non-contracting on the boundaries of certain domains; the maps are required to be smooth to the extent that their Jacobians are twice differentiable (see Main Lemma below). It will follow that a hyperbolic rational map has no wandering simply connected domains. As a more direct corollary, a C3 diffeomorphism f of a compact smooth 2-manifold cannot have a wandering domain Δ if f is area non-contracting on the boundary of each forward image of Δ.
Analysis of a procedure for finding numerical trajectories close to chaotic saddle hyperbolic sets
Authors: Nusse Helena E. , Yorke James A.
Published: September 2008

In dynamical systems examples are common in which there are regions containing chaotic sets that are not attractors, e.g. systems with horseshoes have such regions. In such dynamical systems one will observe chaotic transients. An important problem is the ‘Dynamical Restraint Problem’: given a region that contains a chaotic set but contains no attractor, find a chaotic trajectory numerically that remains in the region for an arbitrarily long period of time.

We present two procedures (‘PIM triple procedures’) for finding trajectories which stay extremely close to such chaotic sets for arbitrarily long periods of time.
Local product structure for group actions
Authors: Ramsay Arlan
Published: September 2008

A differentiable G-space is introduced, for a Lie group G, into which every countably separated Borel G-space can be imbedded. The imbedding can be a continuous map if the space is a separable metric space. Such a G-space is called a universal G-space. This universal G-space has a local product structure for the action of G. That structure is inherited by invariant subspaces, giving a local product structure on general G-spaces. This information is used to prove that G-spaces are stratified by the subsets consisting of points whose orbits have the same dimension, to prove that G-spaces with stabilizers of constant dimension are foliated, to give a short proof that closed subgroups of Lie groups are Lie groups, to give a new proof and a stronger version of the Ambrose—Kakutani Theorem and to give a new proof of the existence of near-slices at points having compact stabilizers and hence of the existence of slices for Cartan G-spaces.
On the number of caustics for invariant tori of Hamiltonian systems with two degrees of freedom
Authors: Bialy Misha
Published: September 2008

Let X be a two-dimensional orientable connected manifold without boundary, H: T*X → ℝ a smooth hamiltonian function denned on the cotangent bundle. We will assume that H is of a ‘classical type’ that is convex and even on each fibre Tx*X. The goal of this paper is to describe the set Γ of all singular points of the projection Θ|L where ι: L → T*X is a smooth embedded 2-torus invariant under the hamiltonian flow h1, Θ: T*X → X is the canonical projection.
Alpha-congruence for dispersive billiards
Authors: Eloranta Kari
Published: September 2008

We show the stability in the sense of α-congruence of dispersive (Sinai) planar billiards that are Bernoulli flows. The perturbations are either billiards on slightly altered tables or geodesic flows on nearby manifolds.
Invariant curves and time-dependent potentials
Authors: Laederich Stephane , Lev Mark
Published: September 2008

In this paper we prove the existence of invariant curves and thus stability for all time for a class of Hamiltonian systems with time-dependent potentials, namely, for systems of the form

where

is a superquadratic polynomial potential with periodic coefficients. As a limiting case, a proof of the stability of Ulam's problem of a particle bouncing between two periodicially moving walls is given.