bluuebunny/crossref_metadata_2025 huggingface dataset:
Quasi-transversal saddle-node bifurcation on surfaces
Authors: Beloqui J. , Pacifico M. J.
Published: September 2008

In this paper we give a complete set of invariants (moduli) for mild and strong semilocal equivalence for certain two parameter families of diffeomorphisms on surfaces. These families exhibit a quasi-transversal saddle-connection between a saddle-node and a hyperbolic periodic point.
Suspensions of topological transformation groups
Authors: Ellis David B.
Published: September 2008

Let S be a subgroup of a topological group T, and suppose that S acts on a space X. One can form a T-transformation group (X ×sT, T) called the suspension of the S-transformation group (X, S). In this paper we study the relationship between the dynamical properties of (X, S) and those of its suspension when S is syndetic in T. The main tool used in this study is a notion of the group of a minimal flow (X, T) which is sensitive to the topology on the group T. We are able, using this group and the enveloping semigroup to obtain results on which T-transformation groups can be realized as suspensions of S-transformation groups, and give conditions under which the suspension of an equicontinuous S-flow is an equicontinuous T-flow.
Invariant curves around a parabolic fixed point at infinity
Authors: Aharonov Dov , Elias Uri
Published: September 2008

The stability of a fixed point of an area-preserving transformation in the plane is characterized by the invariant curves which surround it. The existence of invariant curves had been extensively studied for elliptic fixed points. Here we study the similar problem for parabolic fixed points. In particular we are interested in the case where the fixed point is at infinity.
Homotopical stability of pseudo-Anosov diffeomorphisms
Authors: Fathi Albert
Published: September 2008

We show that the pseudo-Anosov diflEeomorphisms have a kind of stability even outside their own homotopy class, this generalizes some results of Lewowicz and Handel. As a corollary, we show that two pseudo-Anosov maps, with the same dilatation coefficient, which are semi-conjugate on the π1 level are also semiconjugate as dynamical systems by a map which is a ramified cover.
Approximations of continuous functions by squares
Authors: Humke Paul D. , Laczkovich Miklós
Published: September 2008

Let C denote the space of continuous functions mapping [0,1] into itself and endowed with the sup metric. It has been shown that C2 = {f ∘ f: ∈ C} is an analytic but non-Borel subset of C. This implies that there is no simple geometric characterization for a function being a square. In this paper we consider the problem of characterizing those functions which can be approximated by squares. In the first section we prove that any continuous function mapping a closed proper subset of [0,1 ] into [0,1 ] can be extended to a square. In particular this shows that C2 is Lp dense in C. On the other hand, C2 does not contain a ball when C is endowed with the sup metric. In the second section we prove that no strictly decreasing function can be uniformly approximated by squares, although the distance between the class of strictly decreasing functions and C2 is zero. In the last section we investigate the function f(x) = 1 − x and show that for every g ∈ C and that ¼ cannot be improved.
Approximate solutions of cohomological equations associated with some Anosov flows
Authors: Katok Svetlana
Published: September 2008

The Livshitz theorem reported in 1971 asserts that any C1 function having zero integrals over all periodic orbits of a topologically transitive Anosov flow is a derivative of another C1 function in the direction of the flow. Similar results for functions of higher differentiability have also appeared since. In this paper we prove a ‘finite version’ of the Livshitz theorem for a certain class of Anosov flows on 3-dimensional manifolds which include geodesic flows on negatively curved surfaces as a special case.
An algorithm for sofic shift equivalence
Authors: Kim K. H. , Roush F. W.
Published: September 2008

We prove the decidability of shift equivalence of sofic systems and discuss algebraic invariants.
Open index pairs, the fixed point index and rationality of zeta functions
Authors: Mrozek Marian
Published: September 2008

We define open index pairs of an isolated invariant set, prove their existence and compute the fixed point index of an isolating neighbourhood in terms of the Lefschetz number of a certain map associated with the open index pair. We use this to establish rationality of zeta functions and Lefschetz zeta functions.
Maximal absolutely continuous invariant measures for piecewise linear Markov transformations
Authors: Byers W. , Góra P. , Boyarsky A.
Published: September 2008

Let be an irreducible 0–1 matrix such that the non-zero entries in each row are consecutive. Let be the class of piecewise linear Markov transformations τ on [0, 1] into [0, 1] induced by for which the absolutely continuous invariant measure has maximal entropy. The main result presents necessary and sufficient slope conditions on τ which guarantee that τ ∈ .
Ergodic aspects of cellular automata
Authors: Hurley Mike
Published: September 2008

This paper contains a study of attractors in cellular automata, particularly the minimal attractors as defined by J. Milnor. Milnor's definition of attractor uses a measure on the state space; the measures that we consider are Bernoulli product measures. Given a Bernoulli measure it is shown that a cellular automaton has at most one minimal attractor; when there is one, it is the omega-limit set of almost all points. Examples are given to show that the minimal attractor can change as the Bernoulli measure is varied. Other examples illustrate the difference between this result and the corresponding result that is obtained by replacing Milnor's definition of attractor by the purely topological definition used by C. Conley. The examples also show that certain invariant sets of cellular automata are less well-behaved than one might hope: for instance the periodic points are not necessarily dense in the nonwandering set.
Correction to ‘Equivariant spectral decomposition for flows with a ℤ-action’
Authors: Mosher Lee
Published: September 2008

The statement and proof of the ℤ-spectral decomposition theorem for a pseudo-Anosov flow φ on a 3-manifold M are in error (see [1]). There are counter-examples which show that the theorem as stated is false. We remark that the results of § 9 of [1], concerning analogues of the ℤ-spectral decomposition theorem for basic sets of Axiom A flows, are unaffected by this error.
The entropy of polynomial diffeomorphisms of C2
Authors: Smillie John
Published: September 2008

In this note we answer a question raised by Friedland and Milnor in [FM] concerning the topological entropy of polynomial diffeomorphisms of C2.
Extensions à fibre constante
Authors: Blanchard F.
Published: September 2008

The aim of this paper is to prove that any transitive constant-to-one extension of a coded system is still coded. The corresponding statement for sofic systems is false, and it is an open question whether it is true for subshifts of finite type.
Eventual factor maps and compositions of closing maps
Authors: Kitchens Bruce , Marcus Brian , Trow Paul
Published: September 2008

We prove some results related to the question of the existence of factor maps and eventual factor maps between shifts of finite type. Our main result is that if A and B are integral eventually positive (IEP) matrices, and A eventually factors finite-to-one onto B, then there exists an IEP matrix C such that A eventually factors onto C by left closing maps and C eventually factors onto B by right closing maps. This answers the question of the existence of finite-to-one eventual factor maps when the spectrum of A is simple. As a corollary, if in addition to the above hypothesis, χ*A=χ*B, (where χ*A is the characteristic polynomial of A modulo x), then A is shift equivalent to B.
Rotation vectors and entropy for homeomorphisms of the torus isotopic to the identity
Authors: Llibre J. , Mackay R. S.
Published: September 2008

We show that if a homeomorphism f of the torus, isotopic to the identity, has three or more periodic orbits with non-collinear rotation vectors, then it has positive topological entropy. Furthermore, for each point ρ of the convex hull Δ of their rotation vectors, there is an orbit of rotation vector ρ, for each rational point p/q, p ∈ ℤ2, q ∈ ℕ, in the interior of Δ, there is a periodic orbit of rotation vector p / q, and for every compact connected subset C of Δ there is an orbit whose rotation set is C. Finally, we prove that f has ‘toroidal chaos’.
Systems of differential equations that are competitive or cooperative. VI: A localCrClosing Lemma for 3-dimensional systems
Authors: Hirsch Morris W.
Published: September 2008

For certainCr3-dimensional cooperative or competitive vector fieldsF, whereris any positive integer, it is shown that for any nonwandering pointp, every neighborhood ofFin theCrtopology contains a vector field for whichpis periodic, and which agrees withFoutside a given neighborhood ofp. The proof is based on the existence of invariant planar surfaces throughp.
Stability of Lyapunov exponents
Authors: Ledrappier F. , Young L.-S.
Published: September 2008

We consider small random perturbations of matrix cocycles over Lipschitz homeomorphisms of compact metric spaces. Lyapunov exponents are shown to be stable provided that our perturbations satisfy certain regularity conditions. These results are applicable to dynamical systems, particularly to volume-preserving diffeomorphisms.
Can one always lower topological entropy?
Authors: Shub M. , Weiss B.
Published: September 2008

We consider the problem of when does a positive entropy topological system have a continuous factor with strictly smaller entropy. In many cases it is shown that such small entropy factors exist. On the other hand, classes of examples are given where differentiable factors must preserve some of the original entropy.
-Convergence to equilibrium and space—time bernoullicity for spin systems in the M < ε case
Authors: Steif Jeffrey E.
Published: September 2008

Liggett has proved that for spin systems, Markov processes with state space {0,1}ℤn, there is a unique stationary distribution in the M < ε regime and all initial configurations uniformly approach this unique stationary distribution exponentially in the weak topology. Here, M and ε are two parameters of the system. We extend this result to discrete time but strengthen it by proving exponential convergence in the stronger - metric instead of the usual weak topology. This is then used to show that the unique stationary process with state space {0,1}ℤn and index set ℤ is isomorphic (in the sense of ergodic theory) to an independent process indexed by ℤ. In the translation invariant case, we prove the stronger fact that this stationary process viewed as a {0, l}-valued process with index set ℤn × ℤ (spacetime) is isomorphic to an independent process also indexed by ℤn × ℤ. This shows that this process is in some sense the most random possible. An application of this last result to approximating by an infinite number of finite systems concatenated independently together is also presented. Finally, we extend all of these results to continuous time.
Iterates of meromorphic functions III: Preperiodic domains
Authors: Baker I. N. , Kotus J. , Yinian Lü
Published: September 2008

The paper discusses the connectivity of periodic and preperiodic domains in the stable set in the iteration of a meromorphic function. The connectivity of an invariant component has one of the values 1, 2, ∞. Examples are constructed to show that the connectivity of a preperiodic component may take any value.
Rotation and periodicity in plane separating continua
Authors: Barge Marcy , Gillette Richard M.
Published: September 2008

We prove that ifFis an orientation-preserving homeomorphism of the plane that leaves invariant a continuum Λ which irreducibly separates the plane into exactly two domains, then the convex hull of the rotation set ofFrestricted to Λ is a closed interval and each reduced rational in this interval is the rotation number of a periodic orbit in Λ. We also show that the interior and exterior rotation numbers ofFassociated with Λ are contained in the convex hull of the rotation set ofFrestricted to Λ and that if this set is nondegenerate then Λ is an indecomposable continuum.
Geodesic flows on manifolds of negative curvature with smooth horospheric foliations
Authors: Feres Renato
Published: September 2008

We improve and extend a result due to M. Kanai about rigidity of geodesic flows on closed Riemannian manifolds of negative curvature whose stable or unstable horospheric foliation is smooth. More precisely, the main results proved here are: (1) Let M be a closed C∞ Riemannian manifold of negative sectional curvature. Assume the stable or unstable foliation of the geodesic flow φt: V → V on the unit tangent bundle V of M is C∞. Assume, moreover, that either (a) the sectional curvature of M satisfies −4 < K ≤ −1 or (b) the dimension of M is odd. Then the geodesic flow of M is C∞-isomorphic (i.e., conjugate under a C∞ diffeomorphism between the unit tangent bundles) to the geodesic flow on a closed Riemannian manifold of constant negative curvature. (2) For M as above, assume instead of (a) or (b) that dim M ≡ 2(mod 4). Then either the above conclusion holds or φ1, is C∞-isomorphic to the flow , on the quotient Γ\, where Γ is a subgroup of a real Lie group ⊂ Diffeo () with Lie algebra is the geodesic flow on the unit tangent bundle of the complex hyperbolic space ℂHm, m = ½ dim M.
Spectral properties of a class of operators associated with maps in one dimension
Authors: Ruelle David
Published: September 2008

Let f be a piecewise monotone map of the interval [0,1] to itself, and g a function of bounded variation on [0, 1]. Hofbauer, Keller and Rychlik have studied operators on functions of bounded variation, where

Among other things, they show that the essential spectral radius of is in many cases strictly smaller than the spectral radius; there exist therefore isolated eigenvalues of finite multiplicity. The purpose of the present paper is to prove similar results for a more general class of operators forming an algebra (and therefore containing sums of operators like ). An analogous extension was presented by Ruelle for operators associated with expanding maps.
The Hausdorff dimension of Julia sets of entire functions
Authors: Stallard Gwyneth M.
Published: September 2008

We construct a set of transcendental entire functions such that the Hausdorff dimensions of the Julia sets of these functions have greatest lower bound equal to one.
Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps
Authors: Benedicks Michael , Young Lai-Sang
Published: September 2008

We study the quadratic family and show that for a positive measure set of parameters the map has an absolutely continuous invariant measure that is stable under small random perturbations.
Random sets for the pointwise ergodic theorem
Authors: Huang Yenkun
Published: September 2008

We generalize a result of Bourgain and devise more general criteria which guarantee that the corresponding random set in Z+ almost surely satisfies a pointwise ergodic theorem on Lp for p > 1. Several large classes of examples are constructed. We also show that under a simple condition the corresponding random set in Z+ almost surely satisfies a pointwise ergodic theorem not only on Lp for p > 1 but also on L1. On the other hand, we establish a criterion to conclude that a certain class of random sets have Banach density zero. In particular, all of the examples mentioned have Banach density zero.
Dynamical properties of the shift maps on the inverse limit spaces
Authors: Li Shihai
Published: September 2008

In this paper, we prove the following results about the shift map on the inverse limit space of a compact metric space and a sole bonding map: (1) the chain recurrent set of the shift map equals the inverse limit space of the chain recurrent set of the sole bonding map. Similar results are proved for the nonwandering set, ω-limit set, recurrent set, and almost periodic set. (2) The shift map on the inverse limit space is chaotic in the sense of Devaney if and only if its sole bonding map is chaotic. (3) If the sole bonding map is ω-chaotic, then the shift map on the inverse limit space is ω-chaotic. With a modification of the definition of ω-chaos we show the converse is true. (4) We prove that a transitive map on a closed invariant set containing an interval must have sensitive dependence on initial conditions on the whole set. Then it is both ω-chaotic and chaotic in the sense of Devaney with chaotic set the whole set. At last we give a new method of constructing chaotic homeomorphisms on chainable continua, especially on pseudo-arcs.
The Bernoulli property of inner functions
Authors: Craizer Marcos
Published: September 2008

Let f: D → D be an inner function with a fixed point p ∈ D, and f*: S1 → S1 be its extension to the unit circle. We prove in this paper that the Rohlin invertible extension of the system (f*, λp) is equivalent to a generalized Bernoulli shift, where λp is the harmonic measure associated with p.
A classification of the isometric extensions of a multidimensional Bernoulli shift
Authors: Kammeyer Janet Whalen
Published: September 2008

The isometric extensions of a multidimensional Bernouli shift are classified completely, up to C-isomorphism, and up to isomorphism. If such an extension is weakly mixing then it must be Bernoulli; otherwise, it has a rotation factor, which has a Bernoulli complementary algebra. This result is extended to multidimensional Bernoulli flows and Bernoulli shifts of infinite entropy.
Countable sections for locally compact group actions
Authors: Kechris Alexander S.
Published: September 2008

It has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.
Positive Lyapunov exponents for a dense set of bounded measurable SL(2, ℝ)-cocycles
Authors: Knill Oliver
Published: September 2008

Let T be an aperiodic automorphism of a standard probability space (X, m). We prove, that the set

is dense in L∞(X, SL(2, ℝ)).
Rotation number and one-parameter families of circle diffeomorphisms
Authors: Tsujii Masato
Published: September 2008

We consider one-parameter families of circle diffeomorphisms, f1(x) = f(x) + t(t ∈ S1), where f: S1 is a Cr-diffeomorphism (r≥3). We show that, for Lebesgue almost every t ∈ S1 the rotation number of f1, is either a rational number or an irrational number of Roth type. In the former case, f1, has periodic orbits and, in the latter case, f1, is Cr − 2-conjugate to an irrational rigid rotation from well-known theorems of Herman and Yoccoz.
A new maximal inequality and its applications
Authors: Rosenblatt Joseph M. , Wierdl Mate
Published: September 2008

There is a maximal inequality on the integers which implies not only the classical ergodic maximal inequality and certain maximal inequalities for moving averages and differentiation theory, but it also has the following consequence: let P1 ≤ P2 ≤ … ≤ Pk+1 be positive integers. For a σ-finite measure-preserving system (Ω, β, μ, T) and an a.e. finite β-measurable f denote

Then for any λ > 0 and f ∈ L1(Ω)

We show how the multi-parametric and superadditive versions of the previous equation can be obtained from the corresponding inequality for reversed supermartingales. The possibility of similar theorems for martingales and other sequences is also discussed.
Dynamics of periodically forced parabolic equations on the circle
Authors: Sandstede Björn , Fiedler Bernold
Published: September 2008

We consider the dynamics of scalar reaction-diffusion equations:

We show that every prescribed vectorfield in the plane embeds into equation (1) for a suitable f = f (t, x, u, ux). In contrast, suppose f = f (t, u, ux) is independent of x and T-periodic in t. Then the ω-limit set of any bounded solution is a subset of some two-dimensional torus carrying a linear flow.
Pseudo-orbit tracing property and structural stability of expanding maps of the interval
Authors: Šlačkov S. V.
Published: September 2008

A version of pseudo-orbit tracing property for piecewise-continuous piecewise-expanding maps of the interval is proved. It is shown that the typical map of such a kind can be included in a 2m-parameter structurally stable family where m is the number of critical points.
Følner Independence and the amenable Ising model
Authors: Adams Scot
Published: September 2008

We define a criterion called Følner Independence for a stationary process over an amenable group. Intuitively, a process satisfies the criterion if, for sufficiently invariant Følner sets, the process in the Følner set is nearly independent of the process outside. We show that Følner Independence implies Finitely Determined. As an application, we show that, in its extreme Gibbs states, the amenable attractive Ising model is Følner Independent (hence Finitely Determined, hence Bernoulli).
An area preserving homeomorphism of T2 that is fixed point free but does not move any essential simple closed curve off itself
Authors: Bestvina Mladen , Handel Michael
Published: September 2008

We construct an area preserving homeomorphism ƒ: T2→T2 that is isotopic to the identity and fixed point free, but has the property that every essential simple closed curve C satisfies ƒ(C)∩C ≠ Ø. This answers a question of Guillou.
The equivalence theorem for ℤd-actions of positive entropy
Authors: Hasfura-Buenaga J. Roberto
Published: September 2008

First, the class of id-finitely fixed actions of ℤdon a Lebesgue space is defined. Then, it is demonstrated that this property is stable under id-Kakutani equivalence and that, conversely, any two id-finitely fixed ℤd-actions of the same (finite) positive entropy are id-Kakutani equivalent. By id-Kakutani equivalence we mean that element in A. del Junco and D. Rudolph's family of relations on ℤd-actions corresponding to the identityd×dmatrix.
On the classification of some two-dimensional Markov shifts with group structure
Authors: Shereshevsky Mark A.
Published: September 2008

For a finite Abelian group G define the two-dimensional Markov shift for all . Let μG be the Haar measure on the subgroup . The group ℤ2 acts on the measure space (XG, MG) by shifts. We prove that if G1, and G2 are p-groups and E(G1,) ≠ E(G2), where E(G) is the least common multiple of the orders of the elements of G, then the shift actions on and are not measure-theoretically isomorphic. For any finite Abelian groups G1 and G2 the shift actions on and are topologically conjugate if and only if G1 and G2 are isomorphic.
A characterization of ω-limit sets of maps of the interval with zero topological entropy
Authors: Bruckner A. M. , Smítal J.
Published: September 2008

We prove that an infiniteW⊂ (0, 1) is an ω-limit set for a continuous map ƒ of [0,1] with zero topological entropy iffW=Q∪PwhereQis a Cantor set, andPis countable, disjoint fromQ, dense inWif non-empty, and such that for any intervalJcontiguous toQ, card (J∩P) ≤ 1 if 0 or 1 is inJ, and card (J∩P) ≤ 2 otherwise. Moreover, we prove a conjecture by A. N. Šarkovskii from 1967 thatPcan contain points from infinitely many orbits, and consequently, that the system of ω-limit sets containingQand contained inW, can be uncountable.
Regularity of time-preserving conjugacies for contact Anosov flows withC1-Anosov splitting
Authors: Hamenstädt Ursula
Published: September 2008

Let(resp.) be a smooth contact flow on a compact manifoldV1(resp.)V2with Anosov splitting of classC1. We show that every time-preserving conjugacy Λ:(V1,)→(V2,) is necessarily of classC2.
Uniformly quasi-isometric foliations
Authors: Kellum Mark
Published: September 2008

Consider the natural action of 1-jets of the holonomy pseudogroup H on the transverse tangent bundle of a C1 compact foliated manifold (M, ℱ). If these 1-jets act in an equicontinuous way then it is possible to use a C1 ‘Ellis semi-group’ technique, as applied to a neighborhood of the identity in H, to produce a sheaf of compact local transformation groups whose orbits are the topological closures of leaves of ℱ. This action on the transverse manifold to ℱ then decomposes M into a union of minimal sets. These minimal sets we show to be C1 embedded submanifolds of M and the action of H on them is locally transitive.
Algebraic torsion, zeta function and Dirichlet series for graph links in homology 3-spheres
Authors: Némethi András
Published: September 2008

In this paper, we define the algebraic torsion τ associated with an -constructible sheaf ℱ on a topological space X and a ring homomorphism π1(X) → Z (with an acyclicity condition). If (X, ) has a fiber structure over S1, then τ is equal to the zeta function of the monodromy acting on the hypercohomology of the fiber. If X is the complement of a link in a homology 3-sphere and is given by a link such that admits a fiberable splice diagram, then this zeta function has a product decomposition (corresponding to the Jaco-Shalen-Johansson decomposition). This can be interpreted as a ‘Lefschetz type formula’ for a dynamical system suitably chosen.
Multiplicity two actions and loop space homology
Authors: Paternain Gabriel P.
Published: September 2008

We study Hamiltonian actions of compact Lie groups with low dimensional Marsden-Weinstein reduced spaces. We show that Hamiltonian systems with such symmetry groups have zero topological entropy and easily described dynamics. As a result we show that if the geodesic flow of a compact simply connected Riemannian manifold admits such a symmetry group, then the loop space homology of the manifold grows sub-exponentially with any field coefficient. Topological obstructions for collective complete integrability are thus obtained.
Expansive geodesic flows on surfaces
Authors: Paternain Miguel
Published: September 2008

We prove the following result: if M is a compact Riemannian surface whose geodesic flow is expansive, then M has no conjugate points. This result and the techniques of E. Ghys imply that all expansive geodesic flows of a compact surface are topologically equivalent.
Hausdorff dimension for non-invertible maps
Authors: Simon Károly
Published: September 2008

Some conditions are given under which the Hausdorff dimension of an attractor of an endomorphism of a surface can be determined using the pressure formula.
Core dimension group constraints for factors of sofic shifts
Authors: Trow Paul , Williams Susan
Published: September 2008

We give constraints on the existence of factor maps between sofic shifts. These constraints yield examples of sofic shifts of entropy lognwhich do not factor onto the fulln-shift. We also show that any prime which divides the degree of an endomorphism of a sofic shift must divide the non-leading coefficients of the characteristic polynomial of the core matrix of the shift.
Absolutely continuous conjugacy of expanding endomorphisms
Authors: Arteaga Carlos
Published: September 2008

In this article we consider conjugacies between differentiable expanding maps. We give a necessary and sufficient condition for a conjugating map to be absolutely continuous, in terms of the Jacobian around periodic orbits.
On the asymptotic range of cocyles for shifts of finite type
Authors: Coelho Zaqueu
Published: September 2008

We study the problem of lifting ergodicity to certain skew-product extensions over shifts of finite type. This can be done by computing the asymptotic range of the defining cocycle. For a class of ergodic shift-invariant measures and a class of functions (defining conservative extensions) we give a characterization of ergodicity in terms of weights of periodic orbits in the shift space.
Homoclinic tangencies: moduli and topology of separatrices
Authors: Posthumus Rense A. , Takens Floris
Published: September 2008

We consider two-dimensional diffeomorphisms φ:M→M. For a fixed pointp, i.e.,p∈Mand φ(p) =p, we say thatqis homoclinic topifp≠qand if both limi→+∞φi(q) and limi→−∞φi(q) are equal top.
Closed orbits in homology classes for Anosov flows
Authors: Sharp Richard
Published: September 2008

We consider transitive Anosov flows φ:M→Mand give necessary and sufficient conditions for every homology class inH1(M,ℤ) to contain a closed φ-orbit. Under these conditions, we derive an asymptotic formula for the number of closed φ-orbits in a fixed homology class, generalizing a result of Katsuda and Sunada.
A stochastic analogue of a theorem of Boyle's on almost flow equivalence
Authors: Araújo Paulo Ventura
Published: September 2008

We study a new topological classification of suspension flows on subshifts of finite type, and obtain a new proof of a theorem of Boyle's which states that, in an appropriate sense, all such flows are alike. We prove that the stochastic version of this classification is non-trivial by exhibiting a certain invariant, and show that this invariant is complete in a particular case, although not in general. Symbolic flows are important as models of basic sets of Axiom A flows, and so we discuss the significance of our results for this latter type of flow.
Crossed products of totally disconnected spaces by
Authors: Bratteli Ola , Evans David E. , Kishimoto Akitaka
Published: September 2008

Let Ω be a totally disconnected compact metrizable space, and let α be a minimal homeomorphism of Ω. Let σ be a homeomorphism of order 2 on Ω such that ασ = σα−1, and assume that σ or ασ has a fixed point. We prove (Theorem 3.5) that the crossed productis an AF-algebra.
Gradient-like flows on 3-manifolds
Authors: de Rezende K. A.
Published: September 2008

In this paper, we determine properties that a Lyapunov graph must satisfy for it to be associated with a gradient-like flow on a closed orientable three-manifold. We also address the question of the realization of abstract Lyapunov graphs as gradient-like flows on three-manifolds and as a byproduct we prove a partial converse to the theorem which states the Morse inequalities for closed orientable three-manifolds. We also present cancellation theorems of non-degenerate critical points for flows which arise as realizations of canonical abstract Lyapunov graphs.
Differentiable conjugacy for expanding maps on the circle
Authors: Arteaga Carlos
Published: September 2008

In this article we give a necessary and sufficient condition for the conjugacy of the expanding maps on the circle S1 to be differentiable.
α-equivalence: a refinement of Kakutani equivalence
Authors: Fieldsteel Adam , Del Junco Andrés , Rudolph Daniel J.
Published: September 2008

For a fixed irrational α > 0 we say that probability measure-preserving transformationsSandTare α-equivalent if they can be realized as cross-sections in a common flow such that the return time functions on the cross-sections both take values in {1, 1 +α} and have equal integrals. Similarly we call two flowsFandGα-equivalent ifFhas a cross-sectionSandGhas a cross-sectionTisomorphic toSand again both the return time functions take values in {1, 1 + α} and have equal integrals. The integer kα(S), equal to the least positivesuchsuch that exp2πikα-1belongs to the point spectrum ofS, is an invariant of α-equivalence.

We obtain a characterization of a-equivalence as a particular type of restricted orbit equivalence and use this to prove that within the class of loosely Bernoulli mapska(S) together with the entropyh(S) are complete invariants of α-equivalence. There is a corresponding a-equivalence theorem for flows which has as a consequence, for example, that up to an obvious entropy restriction, any weakly mixing cross-section of a loosely Bernoulli flow can also be realized as a cross-section with a {1,1 + α}-valued return time function.

For the proof of the α-equivalence theorem we develop a relative Kakutani equivalence theorem for compact group extensions which is of interest in its own right. Finally, an example of Fieldsteel and Rudolph is used to show that in generalkα(S) is not a complete invariant of α-equivalence within a given even Kakutani equivalence class.
Un théorème de préparation pour fonctions à développement Tchébychévien
Authors: Joyal Pierre
Published: September 2008

In this article, we prove a preparation theorem for functions which admit a certain type of expansion called Chebychev expansion. Taylor expansions are particular cases of Chebychev expansions. The result is based on an approach essentially different from those used for the classical preparation theorems. It has applications in bifurcation theory of vector fields.
Rational functions with no recurrent critical points
Authors: Urbański Mariusz
Published: September 2008

Let h be the Hausdorff dimension of the Julia set of a rational map with no nonperiodic recurrent critical points. We give necessary and sufficient conditions for h-dimensional Hausdorff measure and h-dimensional packing measure of the Julia set to be positive and finite. We also show that either the Julia set is the whole Riemann sphere or h < 2 and that if a rational map (not necessarily with no nonperiodic recurrent critical points!) has a rationally indifferent periodic point, then h > 1/2.
Polynomial beta functions
Authors: Angelis Valerio De
Published: September 2008

The pointwise spectral radii of irreducible matrices whose entries are polynomials with positive, integral coefficients are studied in this paper. Most results are derived in the case that the resulting algebraic function, the beta function of S. Tuncel, is in fact a polynomial. We show that the set of beta functions forms a semiring, and the spectral radius of a matrix of beta functions is again a beta function. We also show that the coefficients of a polynomial beta function p must be real algebraic integers, and p satisfies (after a change of variables if necessary) the inequality for non-zero (and not all positive) complex numbers z1,…,zd. If and the ordered sequence of exponents appearing in p is of the form (m,m+1,…,M−,1,M) for some integers m and M, the same inequality is necessary and sufficient for p to be a beta function.
Regularity at infinity of compact negatively curved manifolds
Authors: Hamenstädt Ursula
Published: September 2008

It is shown that three different notions of regularity for the stable foliation on the unit tangent bundle of a compact manifold of negative curvature are equivalent. Moreover if is a time-preserving conjugacy of geodesic flows of such manifolds M, N then the Lyapunov exponents at corresponding periodic points of the flows coincide. In particular Δ also preserves the Lebesgue measure class.
Entropy, transverse entropy and partitions of unity
Authors: Langevin Rémi , Walczak Pawel G.
Published: September 2008

The topological entropy of a transformation is expressed in terms of partitions of unity. The transverse entropy of a flow tangential to a foliation is defined and expresed in a similar way. The geometric entropy of a foliation of a Riemannian manifold is compared with the transverse entropy of its geodesic flow.
Equivalence of boundary measures on covering trees of finite graphs
Authors: Lyons Russell
Published: September 2008

Let T be the universal covering tree of a finite graph, G. By analogy with an open problem concerning negatively curved covering manifolds, Kaimanovich asked when two of the three natural measure classes on ∂T can coincide, the three measures being harmonic measure, the Patterson measure, and visibility measure. We provide an almost complete answer and discuss related issues. The answer is quite surprising in some cases.
An extension of the theorem of Milnor and Thurston on the zeta functions of interval maps
Authors: Baladi V. , Ruelle D.
Published: September 2008

We consider a piecewise continuous, piecewise monotone interval map and a piecewise constant weight. With these data we associate a weighted kneading matrix which generalizes the Milnor—Thurston matrix. We show that the determinant of this matrix is related to a natural weighted zeta function.
Un exemple de flot d'Anosov transitif transverse à un tore et non conjugué à une suspension
Authors: Bonatti Christian , Langevin Remi
Published: September 2008

We construct an example of transitive Anosov flow on a compact 3-manifold, which admits a transversal torus and is not the suspension of an Anosov diffeomorphism.
Rotation numbers, twists and a Sharkovskii–Misiurewicz-type ordering for patterns on the interval
Authors: Blokh A. M.
Published: September 2008

We introducerotation numbersandpairscharacterizing cyclic patterns on an interval and a special order among them; then we prove the theorem which specializes the Sharkovskii theorem in this setting.
On normal forms in Hamiltonian dynamics, a new approach to some convergence questions
Authors: Delatte David
Published: September 2008

We consider the convergence of solutions to various conjugacy problems occurring in Hamiltonian dynamics. These give rise to nonlinear functional equations which we solve by using a norm well adapted to the problem. The approach provides an alternative to the classical majorization method.
Falconer's formula for the Hausdorff dimension of a self-affine set in R2
Authors: Hueter Irene , Lalley Steven P.
Published: September 2008

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak if

It is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.
On the integrability of intermediate distributions for Anosov diffeomorphisms
Authors: Jiang M. , Pesin Ya B. , de la Llave R.
Published: September 2008

We study the integrability of intermediate distributions for Anosov diffeomorphisms and provide an example of a C∞-Anosov diffeomorphism on a three-dimensional torus whose intermediate stable foliation has leaves that admit only a finite number of derivatives. We also show that this phenomenon is quite abundant. In dimension four or higher this can happen even if the Lyapunov exponents at periodic orbits are constant.
Anosov automorphisms for nilmanifolds and rigidity of group actions
Authors: Qian Nantian
Published: September 2008

We obtain the density of Lyapunov exponents for maximal abelian ℝ-split group in kth tensor product representation of a subgroup Γ ⊂ SL(n, ℤ) of finite index under certain conditions. Anosov and Cartan actions of such groups associated with irreducible representations of SL(n, ℝ) are also classified. Examples of rigidity of actions on nilmanifolds are discussed.
Ensembles invariants des flots géodésiques des variétés localement symétriques
Authors: Zeghib A.
Published: September 2008

We study the rectifiable invariant subsets of algebraic dynamical systems determined by ℝ-semisimple one parameter groups. We show that their ergodic components are algebraic. A more precise geometric description of these components is possible in some cases of geodesic flows of locally symmetric spaces with non-positive curvature.
A topological invariant for volume preserving diffeomorphisms
Authors: Gambaudo Jean-Marc , Pécou Elisabeth
Published: September 2008

For a smooth diffeomorphism f in ℝn+2, which possesses an invariant n-torus , such that the restriction f is topologically conjugate to an irrational rotation, we define a number which represents the way the normal bundle to the torus asymptotically wraps around . We prove that this number is a topological invariant among volume-preserving maps. This result can be seen as a generalization of a theorem by Naishul, for which we give a simple proof.
Ergodic theory on Galton—Watson trees: speed of random walk and dimension of harmonic measure
Authors: Lyons Russell , Pemantle Robin , Peres Yuval
Published: September 2008

We consider simple random walk on the family treeTof a nondegenerate supercritical Galton—Watson branching process and show that the resulting harmonic measure has a.s. strictly smaller Hausdorff dimension than that of the whole boundary ofT. Concretely, this implies that an exponentially small fraction of thenth level ofTcarries most of the harmonic measure. First-order asymptotics for the rate of escape, Green function and the Avez entropy of the random walk are also determined. Ergodic theory of the shift on the space of random walk paths on trees is the main tool; the key observation is that iterating the transformation induced from this shift to the subset of ‘exit points’ yields a nonintersecting path sampled from harmonic measure.
Ergodic properties of a dynamical system arising from percolation theory
Authors: Kraaikamp Cor , Meester Ronald
Published: September 2008

We consider the following dynamical system: take a d-dimensional real vectorwith positive coordinates. Now keep the smallest coordinate and subtract this one from the others, and iterate this process. When the starting vector is x we denote by xn the result after n iterations. It is shown that for almost all x, limn→∞xn ≠ 0 (the null vector). This is shown to be equivalent to the conjectured finiteness of an algorithm which produces the critical probability in a certain dependent percolation model.
On the sofic limit sets of cellular automata
Authors: Maass Alejandro
Published: September 2008

It is not known in general whether any mixing sofic system is the limit set of some one-dimensional cellular automaton. We address two aspects of this question. We prove first that any mixing almost of finite type (AFT) sofic system with a receptive fixed point is the limit set of a cellular automaton, under which it is attained in finite time. The AFT condition is not necessary: we also give examples of non-AFT sofic systems having the same properties. Finally, we show that near Markov sofic systems (a subclass of AFT sofic systems) cannot be obtained as limit sets of cellular automata at infinity.
Phase transition in subhyperbolic Julia sets
Authors: Makarov N. , Smirnov S.
Published: September 2008

We study the pressure function for critically finite polynomials and analyze the case when this function fails to be real analytic.
Square functions in ergodic theory
Authors: Jones Roger L. , Ostrovskii Iosif V. , Rosenblatt Joseph M.
Published: September 2008

Given the usual averages in ergodic theory, let n1 ≤ n2 ≤ … and . There is a strong inequality ‖Sf‖2 ≤ 25‖f‖2 and there is a weak inequality m{Sf > λ} ≤ (7000/λ)‖f‖1. Related results and questions for other variants of this square function are also discussed.
Measures of full dimension on affine-invariant sets
Authors: Kenyon R. , Peres Y.
Published: September 2008

We determine the Hausdorff and Minkowski dimensions of some self-affine Sierpinski sponges, extending results of McMullen and Bedford. This result is used to show that every compact set invariant under an expanding toral endomorphism which is a direct sum of conformal endomorphisms supports an invariant measure of full dimension.
Regularity results for the solutions of the Livsic cohomology equation with values in diffeomorphism groups
Authors: Niţcă Viorel , Török Andrei
Published: September 2008

In this paper we obtain regularity results for the Livsic cohomology equation for cocycles with values in diffeomorphism groups and Lie groups.
Reidemeister torsion and Morse–Smale flows
Authors: Sánchez-Morgado Héctor
Published: September 2008

For several types of stable flows φ and representations ρ of the fundamental group of the underlying manifold, R-torsion for ρ can be computed from the periodic orbits of φ. However, there are counterexamples and the purpose of this paper is to describe the role of heteroclinic orbits in such counterexamples.
Expansive homeomorphisms and hyperbolic diffeomorphisms on 3-manifolds
Authors: Vieitez José L.
Published: September 2008

This paper is a contribution to the classification problem of expansive homeomorphisms. Let M be a compact connected oriented three dimensional topological manifold without boundary and f: M → M an expansive homeomorphism.We show that if the topologically hyperbolic period points of f are dense in M then M = , and f is conjugate to an Anosov diffeomorphism. This follows from our basic result: for such a homeomorphism, all stable and unstable sets are (tamely embedded) topological manifolds.
Minimal entropy and Mostow's rigidity theorems
Authors: Besson Gérard , Courtois Gilles , Gallot Sylvestre
Published: September 2008

Let (Y, g) be a compact connectedn-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. Ify∈, we define

whereB(y,R) denotes the ball of radiusRaroundyin. It is a well known fact that this limit exists and does not depend ony([Man]). The invarianth(g) is called the volume entropy of the metricgbut, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sensitive to changes of scale which makes it a bad invariant: however, this is a minor drawback that can be circumvented by looking at the behaviour of the entropy functional on the space of metrics with fixed volume (equal to one for example). Nevertheless, it seems very unlikely that two numbers, the entropy and the volume, might characterize any metric. The very first person to consider such a possibility was Katok ([Kat1]). In this article the entropy is thought of as a dynamical invariant which actually is suggested by its name. More precisely, let us define this dynamical invariant, which is called the topological entropy: let (M,d) be a compact metric space and ψt, a flow on it, we define

.
Fixed points of elliptic reversible transformations with integrals
Authors: Gong Xianghong
Published: September 2008

We show that for a certain family of integrable reversible transformations, the curves of periodic points of a general transformation cross the level curves of its integrals. This leads to the divergence of the normal form for a general reversible transformation with integrals. We also study the integrable holomorphic reversible transformations coming from real analytic surfaces in ℂ2 with non-degenerate complex tangents. We show the existence of real analytic surfaces with hyperbolic complex tangents, which are contained in a real hyperplane, but cannot be transformed into the Moser—Webster normal form through any holomorphic transformation.
Estimating invariant measures and Lyapunov exponents
Authors: Hunt Brian R.
Published: September 2008

This paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius—Perron operator for such maps. The method is applied to estimate the Lyapunov exponents for a planar map of recent interest.
Invariant measures for higher-rank hyperbolic abelian actions
Authors: Katok A. , Spatzier R. J.
Published: September 2008

We investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of ℝk, ℤkandWe show that they are either Haar measures or that every element of the action has zero metric entropy.