bluuebunny/crossref_metadata_2025 huggingface dataset:
On the subsystems of topological Markov chains
Authors: Krieger Wolfgang
Published: September 2008

Let SA be an irreducible and aperiodic topological Markov chain. If SĀ is an irreducible and aperiodic topological Markov chain, whose topological entropy is less than that of SA, then there exists an irreducible and aperiodic topological Markov chain, whose topological entropy equals the topological entropy at SĀ, and that is a subsystem of SA. If S is an expansive homeomorphism of the Cantor discontinuum, whose topological entropy is less than that of SA, and such that for every j∈ℕ the number of periodic points of least period j of S is less than or equal to the number of periodic points of least period j of SA, then S is topological conjugate to a subsystem of SA.
Periodic points of maps of degree one of a circle
Authors: Misiurewicz MichaŁ
Published: September 2008

There is a theorem of Sarkovskii which characterizes the sets of periods of periodic points for continuous maps of an interval into itself. We give a similar characterization for continuous maps of degree one of a circle into itself.
Singularities of Legendre varieties, of evolvents and of fronts at an obstacle
Authors: Arnol'd V. I.
Published: September 2008

The Legendre manifolds are the maximal integral submanifolds of a contact manifold. The shortest path problem on a manifold with boundary leads to Legendre varieties. We find normal forms of their generic singularities in terms of binary forms invariants theory.
Entropy and closed geodesies
Authors: Katok A.
Published: September 2008

We study asymptotic growth of closed geodesies for various Riemannian metrics on a compact manifold which carries a metric of negative sectional curvature. Our approach makes use of both variational and dynamical description of geodesies and can be described as an asymptotic version of length-area method. We also obtain various inequalities between topological and measure-theoretic entropies of the geodesic flows for different metrics on the same manifold. Our method works especially well for any metric conformally equivalent to a metric of constant negative curvature. For a surface with negative Euler characteristics every Riemannian metric has this property due to a classical regularization theorem. This allows us to prove that every metric of non-constant curvature has strictly more close geodesies of length at most T for sufficiently large T then any metric of constant curvature of the same total area. In addition the common value of topological and measure-theoretic entropies for metrics of constant negative curvature with the fixed area separates the values of two entropies for other metrics with the same area.
Finitely-additive invariant measures on Euclidean spaces
Authors: Margulis G. A.
Published: September 2008

It is shown that for n ≥ 3 the Lebesgue measure is the unique finitely-additive isometry-invariant measure on the ring of bounded Lebesgue measurable subsets of the n-dimensional Euclidean space.
Glancing billiards
Authors: Mather John N.
Published: September 2008

Consider the billiard ball problem in an open, convex, bounded region of the plane whose boundary is C2 and has at least one point of zero curvature. Then there are trajectories which come arbitrarily close to being positively tangent to the boundary and also come arbitrarily close to being negatively tangent to the boundary.
Attracting Cantor set of positive measure for a C∞ map of an interval
Authors: Misiurewicz Michał
Published: September 2008

We give an example of a smooth map of an interval into itself, conjugate to the Feigenbaum map, for which the attracting Cantor set has positive Lebesgue measure.
Examples of conservative diffeomorphisms of the two-dimensional torus with coexistence of elliptic and stochastic behaviour
Authors: Przytycki Feliks
Published: September 2008

We find very simple examples of C∞-arcs of diffeomorphisms of the two-dimensional torus, preserving the Lebesgue measure and having the following properties: (1) the beginning of an arc is inside the set of Anosov diffeomorphisms; (2) after the bifurcation parameter every diffeomorphism has an elliptic fixed point with the first Birkhoff invariant non-zero (the KAM situation) and an invariant open area with almost everywhere non-zero Lyapunov characteristic exponents, moreover where the diffeomorphism has Bernoulli property; (3) the arc is real-analytic except on two circles (for each value of parameter) which are inside the Bernoulli property area.
Factors of horocycle flows
Authors: Ratner Marina
Published: September 2008

We classify up to an isomorphism all factors of the classical horocycle flow on the unit tangent bundle of a surface of constant negative curvature with finite volume.
Ergodic behaviour of Sullivan's geometric measure on a geometrically finite hyperbolic manifold
Authors: Rudolph Daniel J.
Published: September 2008

Sullivan's geometric measure on a geometrically finite hyperbolic manifold is shown to satisfy a mean ergodic theorem on horospheres and through this that the geodesic flow is Bernoulli.
On the ergodic properties of piecewise linear perturbations of the twist map
Authors: Wojtkowski M.
Published: September 2008

It is proved that for a sequence of arbitrarily small piecewise linear perturbations of the twist map, there is a domain with stochastic behaviour (almost hyperbolicity). The measure of this domain has the asymptotics

where A is the magnitude of the perturbation.
Bifurcation of an attracting invariant circle: a Denjoy attractor
Authors: Hall Glen R.
Published: September 2008

We construct an example of a C∞ diffeomorphism of an annulus into itself which has an attracting invariant circle such that the map restricted to this circle has no periodic points and no dense orbits. By studying two parameter families of maps of the plane which undergo Hopf bifurcation, particularly the set of parameter values for which the rotation number is irrational, we see that the above example can be considered as a ‘worst case’ of the loss of smoothness of an attracting invariant circle without periodic orbits.
On a construction of unitary cocycles and the representation theory of amenable groups
Authors: Sutherland Colin E.
Published: September 2008

If K is a countable amenable group acting freely and ergodically on a probability space (Γ, μ), and G is an arbitrary countable amenable group, we construct an injection of the space of unitary representations of G into the space of unitary 1-cocyles for K on (Γ, μ); this injection preserves intertwining operators. We apply this to show that for many of the standard non-type-I amenable groups H, the representation theory of H contains that of every countable amenable group.
Errata to ‘A proof of Pesin's formula’
Authors: Mañé R.
Published: September 2008

In the paper ‘A proof of Pesin's formula’ (R. Mañé, Ergod. Th. & Dynam. Sys. (1981), 1, 95–102) there are some minor but misleading mistakes. The following corrections are necessary:

(1) On page 99, inequality (11) states:

It should read:

The first inequality is obviously false in general. The second one is an immediate corollary of Oseledec's theorem. This misprint is of no consequence because in the rest of the paper we always used the correct inequality.
Which groups act distally?
Authors: Abels Herbert
Published: September 2008

The following question is discussed: which locally compact topological groups have an effective distal action on some compact metrizable space?
Bifurcation and chain recurrence
Authors: Hurley M.
Published: September 2008

We show that there is a residual subset of the set of C1 diffeomorphisms on any compact manifold at which the map

is continuous. As this number is apt to be infinite, we prove a localized version, which allows one to conclude that if f is in this residual set and X is an isolated chain component for f, then

(i) there is a neighbourhood U of X which isolates it from the rest of the chain recurrent set of f, and

(ii) all g sufficiently C1 close to f have precisely one chain component in U, and these chain components approach X as g approaches f.

(ii) is interpreted as a generic non-bifurcation result for this type of invariant set.
Hausdorff dimension for horseshoes
Authors: McCluskey Heather , Manning Anthony
Published: September 2008

We shall measure how thick a basic set of a C1 axiom A diffeomorphism of a surface is by the Hausdorff dimension of its intersection with an unstable manifold. This depends continuously on the diffeomorphism. Generically a C2 diffeomorphism has attractors whose Hausdorff dimension is not approximated by the dimension of its ergodic measures.
The C1 Closing Lemma, including Hamiltonians
Authors: Pugh Charles C. , Robinson Clark
Published: September 2008

An Axiom of Lift for classes of dynamical systems is formulated. It is shown to imply the Closing Lemma. The Lift Axiom is then verified for dynamical systems ranging from C1 diffeomorphisms to C1 Hamiltonian vector fields.
The orbit of a Hölder continuous path under a hyperbolic toral automorphism
Authors: Irwin M. C.
Published: September 2008

Let f:T3→T3 be a hyperbolic toral automorphism lifting to a linear automorphism with real eigenvalues. We prove that there is a Hölder continuous path in T3 whose orbit-closure is 1-dimensional. This strengthens results of Hancock and Przytycki concerning continuous paths, and contrasts with results of Franks and Mañé concerning rectifiable paths.
Constant-to-one and onto global maps of homomorphisms between strongly connected graphs
Authors: Nasu Masakazu
Published: September 2008

The global maps of homomorphisms of directed graphs are very closely related to homomorphisms of a class of symbolic dynamical systems called subshifts of finite type. In this paper, we introduce the concepts of ‘induced regular homomorphism’ and ‘induced backward regular homomorphism’ which are associated with every homomorphism between strongly connected graphs whose global map is finite-to-one and onto, and using them we study the structure of constant-to-one and onto global maps of homorphisms between strongly connected graphs and that of constant-to-one and onto homomorphisms of irreducible subshifts of finite type. We determine constructively, up to topological conjugacy, the subshifts of finite type which are constant-to-one extensions of a given irreducible subshift of finite type. We give an invariant for constant-to-one and onto homomorphisms of irreducible subshifts of finite type.
Croissance des boules et des géodésiques fermées dans les nilvariétés
Authors: Pansu Pierre
Published: September 2008

If (M, g) is a riemannian nilmanifold, the homothetic metrics εg˜ on the universal cover M converge in the sense of Gromov for small ε. In this convergence the volume of balls and the number of closed geodesies go to a limit, and precise asymptotic estimates are given for these numbers.
Strong shift equivalence of 2 × 2 matrices of non-negative integers
Authors: Baker Kirby A.
Published: September 2008

The concept of strong shift equivalence of square non-negative integral matrices has been used by R. F. Williams to characterize topological isomorphism of the associated topological Markov chains. However, not much has been known about sufficient conditions for strong shift equivalence even for 2×2 matrices (other than those of unit determinant). The main theorem of this paper is: If A and B are positive 2×2 integral matrices of non-negative determinant and are similar over the integers, then A and B are strongly shift equivalent.
Persistence in expansive systems
Authors: Lewowicz Jorge
Published: September 2008

We give some sufficient conditions for an expansive diffeomorphism ƒ of a compact manifold to be such that every neighbouring diffeomorphism shows, roughly, all the dynamical features of ƒ. These results are then applied to prove a structural stability theorem for pseudo-Anosov maps.
Topological minimal self-joinings
Authors: Markley Nelson G.
Published: September 2008

There is an interesting duality between some of the concepts of ergodic theory and those of topological dynamics. This paper is a first attempt at developing a topological analogue to the measure-theoretic notion of a transformation having minimal self-joinings. The main problem is to understand the dynamics of the composition of a cartesian product of powers of a transformation having topological minimal self-joinings with a compact permutation of the coordinates. Most of the results are about the minimal subsets of such a composition.
Embedding some transformation group C*-algebras into AF-algebras
Authors: Pimsner Mihai V.
Published: September 2008

For a homeomorphism of a compact metrizable space X, we show that the property that every point of X is pseudo-non-wandering (see definition 2) is equivalent to the possibility of embedding the corresponding transformation group C*-algebra into an AF-algebra.
Immediate conditional hyperbolicity in dynamical systems
Authors: Rosenblatt Joseph , Swanson Richard
Published: September 2008

For many diffeomorphisms of a compact manifold X, eventual conditional hyperbolicity implies immediate conditional hyperbolicity in some (possibly new) Finsler structures. That is, if A and B are vector bundle isomorphisms over the mapping ƒ of the base X, such that uniformly on X, then there exist new norms for A and B such that uniformly on X, whenever the mapping ƒ satisfies the condition that there exist infinitely many N ≥ 1 such that any ƒ-invariant. For example, this condition on ƒ holds if any one of the following conditions holds: (1) ƒ is periodic; (2) ƒ is periodic on its non-wandering set; (3) ƒ has a finite non-wandering set (for example, ƒ is a Morse-Smale diffeomorphism); (4) ƒ is an almost periodic mapping of a connected base X; (5) ƒ is a mapping of the circle with no periodic points; or (6) ƒ and all its powers are uniquely ergodic. We consider various types of eventually conditionally hyperbolic systems and describe sufficient conditions on ƒ to have immediate conditional hyperbolicity of these systems in some new Finsler structures. Thus, for a sizable class of dynamical systems, we settle, in the affirmative, a question raised by Hirsch, Pugh, and Shub.
Fat baker's transformations
Authors: Alexander J. C. , Yorke J. A.
Published: September 2008

We investigate a variant of the baker's transformation in which the mapping is onto but is not one-to-one. The Bowen-Ruelle measure for this map is investigated.
On orbits of unipotent flows on homogeneous spaces
Authors: Dani S. G.
Published: September 2008

Let G be a connected Lie group and let Γ be a lattice in G (not necessarily co-compact). We show that if (ut) is a unipotent one-parameter subgroup of G then every ergodic invariant (locally finite) measure of the action of (ut) on G/Γ is finite. For ‘arithmetic lattices’ this was proved in [2]. The present generalization is obtained by studying the ‘frequency of visiting compact subsets’ for unbounded orbits of such flows in the special case where G is a connected semi-simple Lie group of ℝ-rank 1 and Γ is any (not necessarily arithmetic) lattice in G.
Dynamics of exp (z)
Authors: Devaney Robert L. , Krych Michal
Published: September 2008

We describe the dynamical behaviour of the entire transcendental function exp(z). We use symbolic dynamics to describe the complicated orbit structure of this map whose Julia Set is the entire complex plane. Bifurcations occurring in the family c exp(z) are discussed in the final section.
Flow equivalence of subshifts of finite type
Authors: Franks John
Published: September 2008

A complete set of computable invariants is given for deciding whether two irreducible subshifts of finite type have topologically equivalent suspension flows.
Combined structural and topological stability are equivalent to Axiom A and the strong transversality condition
Authors: Hurley Mike
Published: September 2008

The purpose of this paper is to develop necessary conditions for a diffeomorphism to be topologically stable (lower semistable). Our results combine with a recent theorem of R. Mañé and with earlier results of J. Robbin, C. Robinson, and Z. Nitecki to give a complete characterization of diffeomorphisms of compact manifolds that are both topologically and structurally stable: they are precisely the Axiom A diffeomorphisms that satisfy the strong transversality condition.
Any flow is an orbit factor of any flow
Authors: Ornstein Donald , Weiss Benjamin
Published: September 2008

We prove that given any two ergodic non-singular flows , St, the first can be time changed to t so that St is a factor of t. A corresponding result for transformations is that if , T are any two ergodic non-singular transformations then there is a tower over that has T as a factor.
Lorenz knots are prime
Authors: Williams R. F.
Published: September 2008

Lorenz knots are the periodic orbits of a certain geometrically defined differential equation in ℝ3. This is called the ‘geometric Lorenz attractor’ as it is only conjecturally the real Lorenz attractor. These knots have been studied by the author and Joan Birman via a ‘knot-holder’, i.e. a certain branched two-manifold H. To show such knots are prime we suppose the contrary which implies the existence of a splitting sphere, S2. The technique of the proof is to study the intersection S2∩H. A novelty here is that S2∩H is likewise branched.
Simple permutations with order a power of two
Authors: Bernhardt Chris
Published: September 2008

Continuous maps from the real line to itself give, in a natural way, a partial ordering of permutations. This paper studies the structure of simple permutations which have order a power of two, where simple permutations are permutations corresponding to the simple orbits of Block.
One-parameter automorphism groups of operator algebras allowing spectral projections
Authors: Elliott George A. , Zsidó László
Published: September 2008

The property of a one-parameter C*- (or W*-) dynamical system that the spectral subspaces corresponding to the three subsets (−∞, 0), {0}, and (0, +∞) add up to the whole algebra is reformulated. If the C*-algebra is prime (or the W*-algebra is a factor), an equivalent property is that the spectrum is finite.
Uncountably many topological models for ergodic transformations
Authors: Glasner S. , Rudolph D.
Published: September 2008

Given a topological process (X, µ, T) where T is a homeomorphism of the compact metric space X which preserves the probability measure µ and is ergodic, we show that there exists an uncountable family {(Xi, µi, Ti)}i∈I of topological processes such that for every i, (Xi, µi, Ti) is measure-theoretically isomorphic to (X, µ, T) but for every i ≠ j, (Xi, µi, Ti) and (Xj, µj, Tj) are not almost topologically conjugate.
The Lyapunov dimension of a nowhere differentiable attracting torus
Authors: Kaplan James L. , Mallet-Paret John , Yorke James A.
Published: September 2008

The fractal dimension of an attracting torusTkin×Tkis shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjecture. The cases studied here can have several Lyapunov numbers greater than 1 and several less than 1
The entropies of topological Markov shifts and a related class of algebraic integers
Authors: Lind D. A.
Published: September 2008

We give an algebraic characterization of the class of spectral radii of aperiodic non-negative integral matrices, and describe a method of constructing all such matrices with given spectral radius. The logarithms of the numbers in are the entropies of mixing topological Markov shifts. There is an arithmetic structure to , including factorization into irreducibles in only finitely many ways. This arithmetic structure has dynamical consequences, such as the impossibility of factoring the p-shift into a direct product of nontrivial homeomorphisms for prime p.
Weak disjointness and the equicontinuous structure relation
Authors: Auslander Joe , McMahon Doug , van der Woude Jaap , Wu Ta Sun
Published: September 2008

We discuss weak disjointness of homomorphisms of minimal transformation groups and use the techniques involved to deepen our knowledge of the equicontinuous structure relation.
Shift equivalence and the Jordan form away from zero
Authors: Boyle Mike
Published: September 2008

Only finitely many shift equivalence classes of non-negative aperiodic integral matrices may share a given diagonal Jordan form away from zero. The diagonal assumption is necessary.
Twist sets for maps of the circle
Authors: Misiurewicz Michał
Published: September 2008

Let f be a continuous map of degree one of the circle onto itself. We prove that for every number a from the rotation interval of f there exists an invariant closed set A consisting of points with rotation number a and such that f restricted to A preserves the order. This result is analogous to the one in the case of a twist map of an annulus.
The backward continued fraction map and geodesic flow
Authors: Adler Roy L. , Flatto Leopold
Published: September 2008

The ‘backward continued fraction’ map studied by A. Reyni is defined by y = g(x) where g(x) equals the fractional part of 1/(1−x) for 0 < x < 1. We show that it is a factor map of a special cross-section map for the geodesic flow on the unit tangent bundle of the modular surface. This gives an alternative derivation of the fact that this map preserves the infinite measure dx/x on the unit interval.
Minimal sets on tori
Authors: Berend Daniel
Published: September 2008

Let Σ be a commutative semigroup of continuous endomorphisms of the r-dimensional torus. Generalizing a result of Furstenberg dealing with the circle group, necessary and sufficient conditions are given here for Σ to possess the following property: Any Σ-minimal set consists of torsion elements. Semigroups not having this property are shown to admit minimal sets of positive Hausdorff dimension.
Subordinate Šil'nikov bifurcations near some singularities of vector fields having low codimension
Authors: Broer H. W. , Vegter G.
Published: September 2008

A specific singularity of a vector field on is considered, of codimension 2 in the dissipative case and of codimension 1 in the conservative case. In both contexts in generic unfoldings the existence is proved of subordinate Šil'nikov bifurcations, which have codimension 1. Special attention is paid to the C∞-flatness of this subordinate phenomenon.
A criterion for the positivity of the Liapunov characteristic exponent
Authors: Cornelis Eric , Wojtkowski Maciej
Published: September 2008

We formulate sufficient conditions under which, for a finite subset of SL (2, ℝ), the maximal Liapunov exponent is positive. These conditions are based on the notion of compatible hyperbolicity. We then give an analytical formulation of such a condition and we apply this criterion to prove mixing properties of a particular transformation of the two-dimensional torus.
Transitivity and invariant measures for the geometric model of the Lorenz equations
Authors: Robinson Clark
Published: September 2008

This paper concerns perturbations of the geometric model of the Lorenz equations and their associated one-dimensional Poincaré maps. R. Williams has shown that if a map of the interval of the type arising from the Lorenz equations satisfies f′(x) > 2½ everywhere except at the discontinuity then f is locally eventually onto (l.e.o.) and so topologically transitive [14]. We show roughly that if are the end points of the interval, and their iterates stay on the same side of the point of discontinuity for 0 ≤ j ≤ k, and f′(x) > 21/(k+1) everywhere, then f is l.e.o. Secondly, we show that the one-dimensional Poincaré map of any Cr perturbation of the geometric model (for large enough r) has an ergodic measure which is equivalent to Lebesgue measure. This result follows by showing it is C1+α and satisfies a theorem of Keller, Wong, Lasota, Li & Yorke.
The Perron-Frobenius operator in spaces of smooth functions on an interval
Authors: Szewc B.
Published: September 2008

The densities of invariant measures for Misiurewicz maps and Lasota-Yorke maps of class Cr are of class Cr-1 on certain intervals (forming the partition of an interval in case of Misiurewicz maps). For these maps the Perron-Frobenius operator has an unambiguous decomposition into the sum of projections onto eigenspaces (multiplied by the eigenvalues) and a remainder operator. The remainder operator has spectral radius less than one in certain spaces of smooth functions.
On the Bernoulli property for rational maps
Authors: Mañé Ricardo
Published: September 2008

Every rational function ƒ with degree has a unique invariant probability ƒƒ that maximizes entropy. It has been conjectured that the system (ƒ, μƒ) is equivalent to the one sided Bernoulli shift . In this paper we prove that there exists m >0 such that (ƒm, ƒƒ) is equivalent to .
Characterization of MorseSmale isotopy classes on surfaces
Authors: Fernando Luiz , Da Rocha C.
Published: September 2008

In this paper we use Thurston's work on the dynamics of diffeomorphisms on surfaces to show that a diffeomorphism ƒ on a surface is isotopic to a Morse- Smale one if and only if the growth rate of the length of the words representing elements of the fundamental group under iteration by ƒ is one. Morse-Smale isotopy classes are also shown to be the same as Nielsen's algebraically finite type.
A Morse equation in Conley's index theory for semiflows on metric spaces
Authors: Rybakowski Krzysztof P. , Zehnder Eduard
Published: September 2008

Given a compact (two-sided) flow, an isolated invariant set S and a Morse-decomposition (M1, …, Mn) of S, there is a generalized Morse equation, proved by Conley and Zehnder, which relates the Alexander-Spanier cohomology groups of the Conley indices of the sets Mi and S with each other. Recently, Rybakowski developed the technique of isolating blocks and extended Conley's index theory to a class of one-sided semiflows on non-necessarily compact spaces, including e.g. semiflows generated by parabolic equations. Using these results, we discuss in this paper Morse decompositions and prove the above-mentioned Morse equation not only for arbitrary homology and cohomology groups, but also in this more general semiflow setting.
Invariant families of cones and Lyapunov exponents
Authors: Wojtkowski Maciej
Published: September 2008

We show that in several cases preservation of cones leads to non-vanishing of (some) Lyapunov exponents. It gives simple and effective criteria for nonvanishing of the exponents, which is demonstrated on the example of the billiards studied by Bunimovich. It is also shown that geodesic flows on manifolds of non-positive sectional curvature can be treated from this point of view.
Some entire functions with multiply-connected wandering domains
Authors: Baker I. N.
Published: September 2008

A component U of the complement of the Julia set of an entire function ƒ is a wandering domain if the sets ƒn(U) are mutually disjoint, where n ∈ℕ and ƒn is the n-th iterate of ƒ. Examples are given of entire ƒ of order , which have multiply-connected wandering domains. An example is given where the connectivity is infinite.
The characterization of differential operators by locality: abstract derivations
Authors: Batty C. J. K. , Robinson D. W.
Published: September 2008

Let δ be a closed *-derivation on a commutative C*-algebra , suppose that is dense in for some n = 1, 2, …, ∞, and let be a linear operator satisfying the locality condition

It is shown that , for some finite integer p≤n and functions lm on X. Estimates on the coefficients lm are obtained and applied to flows and local flows.
A local limit theorem for a sequence of interval transformations
Authors: Calderoni P. , Campanino M. , Capocaccia D.
Published: September 2008

Let λ > 1 be a real eigenvalue of an automorphism of the two dimensional torus. We prove that for a dense, open subset of intervals the sequence where {x} denotes the fractional part of x and χ[a, b] the characteristic function of [a, b], satisfies the local limit theorem with respect to Lebesgue measure on [0, 1].
Simplicial systems for interval exchange maps and measured foliations
Authors: Kerckhoff S. P.
Published: September 2008

The spaces of interval exchange maps and measured foliations are considered and an alternative proof that almost all interval exchange maps and measured foliations are uniquely ergodic is given. These spaces are endowed with a refinement process, called a simplicial system, which is studied abstractly and is shown to be normal under a simple assumption. The results follow and thus are a corollary of a more general theorem in a broader setting.
The entropy of C2 surface diffeomorphisms in terms of Hausdorff dimension and a Lyapunov exponent
Authors: Mendoza Leonardo
Published: September 2008

In this paper we prove that if the entropy of an ergodic measure preserved by a C2 surface diffeomorphism is positive then it is equal to the product of the Hausdorff dimension of the quotient measure defined by the family of stable manifolds and the positive Lyapunov exponent.
Errata to ‘Hausdorff dimension for horseshoes’
Authors: Manning Anthony
Published: September 2008

In our paper ‘Hausdorff dimension for horseshoes’ (H. McCluskey and A. Manning, Ergod. Th. & Dynam. Sys. (1983) 3, 251–260), § 3 entitled ‘Continuity across a bifurcation’ should be deleted since the proof of theorem 3 there is wrong.

The mistake is that does not in fact imply that δs → 1. This is because, for q = q0, the function is is only non-positive rather than negative and so is not strictly decreasing function of t.(Thus the pressure curve in figure 3 may, for q = q0, have become horizontal before t = l.) A similar mistake occurs in the second half of the same proof. Thus the claim of theorem 3 that the Hausdorff dimension of the basic set changes continuously in the bifurcation from Anosov to DA remains unproved. The rest of the paper is not affected.
Super persistent chaotic transients
Authors: Grebogi Celso , Ott Edward , Yorke James A.
Published: September 2008

The unstable-unstable pair bifurcation is a bifurcation in which two unstable fixed points or periodic orbits of the same period coalesce and disappear as a system paremeter is raised. For parameter values just above that at which unstable orbits are destroyed there can be chaotic transients. Then, as the bifurcation is approached from above, the average length of a chaotic transient diverges, and, below the bifurcation point, the chaotic transient may be regarded as having been converted into a chaotic attractor. It is argued that unstable-unstable pair bifurcations should be expected to occur commonly in dynamical systems. This bifurcation is an example of the crisis route to chaos. The most striking fact about unstable-unstable pair bifurcation crises is that long chaotic transients persist even for parameter values relatively far from the bifurcation point. These long-lived chaotic transients may prevent the time asymptotic state from being reached during experiments. An expression giving a lower bound for the average lifetime of a chaotic transient is derived and shown to agree well with numerical experiments. In particular, this bound on the average lifetime, (τ), satisfies

for α near α*, where k1 and k2 are constants and α* is the value of the parameter a at which the crisis occurs. Thus, as a approaches α* from above, (τ) increases more rapidly than any power of (α − α*)−1. Finally, we discuss the effect of adding bounded noise (small random perturbations) on these phenomena and argue that the chaotic transients should be lengthened by noise.
Global shadowing of pseudo-Anosov homeomorphisms
Authors: Handel Michael
Published: September 2008

We prove that if f: M2→ M2 is pseudo-Anosov and if g ⋍ f then there is a closed subset X ⊂ M2 and a continuous surjection π: X→ M that is homotopic to inclusion such that f ° π = π ° g|X.
A counterexample to a positive entropy skew product generalization of the Pinsker conjecture
Authors: King Jonathan
Published: September 2008

The class of k-automorphisms is not contained in a certain class of skew products over a Bernoulli base. The non-identity fibre transformation in the skew is allowed to have positive or even infinite entropy. A difficulty presented by positive entropy is handled via an apparently new property of independent processes (lemma 7.24).
Relative equilibria of the four-body problem
Authors: Moeckel Richard
Published: September 2008

By employing a regularizing transformation, the problem of bifurcation of relative equilibria in the Newtonian 4-body problem is reduced to a study of an algebraic correspondence between real algebraic varieties. The finiteness theorems of algebraic geometry are used to find an upper bound for the number of affine equivalence classes of relative equilibria which holds for all masses in the complement of a proper, algebraic subset of the space of all masses.
Invariant measures for the flow of a first order partial differential equation
Authors: Rudnicki R.
Published: September 2008

We prove that the dynamical systems generated by first order partial differential equations are K-flows and chaotic in the sense of Auslander & Yorke.
k-fold mixing lifts to weakly mixing isometric extensions
Authors: Rudolph Daniel J.
Published: September 2008

If is a weakly mixing isometric extension of a finite measure preserving, k-fold mixing map T, then must also be k-fold mixing.
Almost topological classification of finite-to-one factor maps between shifts of finite type
Authors: Adler Roy , Kitchens Bruce , Marcus Brian
Published: September 2008

We classify finite-to-one factor maps between shifts of finite type up to almost topological conjugacy.
Growth rate of surface homeomorphisms and flow equivalence
Authors: Fried David
Published: September 2008

We study which algebraic integers λ ≥ l arise as the growth rate of a mapping class of a surface and give conditions that are necessary and perhaps sufficient. Flow equivalence and twisted Lefschetz zeta functions are used to generate families of λ's. Examples and open problems are included
Dimension of invariant measures for maps with exponent zero
Authors: Ledrappier F. , Misiurewicz M.
Published: September 2008

We give examples of maps of the interval with zero entropy for which the continuous invariant measure has no dimension, and we prove a dimension property for maps lying in the stable manifold of Feigenbaum's fixed points.
Symmetric S-unimodal mappings and positive Liapunov exponents
Authors: Nowicki Tomasz
Published: September 2008

Symmetric S-unimodal functions with positive Liapunov exponent of the critical value have an invariant measure absolutely continuous with respect to Lebesgue measure.
Almost sure invariance principle for some maps of an interval
Authors: Ziemian Krystyna
Published: September 2008

We prove an almost sure invariance principle and a central limit theorem for the process , where f is a map of an interval with a non-positive Schwarzian derivative whose trajectories of critical points stay far from the critical points, and F is a measurable function with bounded p-variation (p ≥ 1).

The almost sure invariance principle implies the Log-log laws, integral tests and a distributional type of invariance principle for the process .
On the Cr-closing lemma for flows on the torus T2
Authors: Gutierrez C.
Published: September 2008

Vector fields of , 1 ≤ r ≤ ∞, with non-trivial recurrent points are classified in two types, one of which we call the constant type inspired by the terminology of continued fractions. Let have finitely many singularities and p ∈ T2 be a non-wandering point of X. With the exception of the case when X is of constant type and, simultaneously, p is non-trivial recurrent, we prove that there exists arbitrarily close to X (in the Cr-topology) having a periodic trajectory through p.
Deciding eventual positivity of polynomials
Authors: Handelman David
Published: September 2008

Let P and f be polynomials in several (real) variables, with P having no negative coefficients. We give necessary and sufficient conditions for there to exist a positive integer n with Pnf having no negative coefficients; roughly speaking, the conditions involve the behaviour of f as a function on the positive orthant, together with its behaviour on a boundary constructed from the supporting monomials of P. This completes a series of results due to Poincaré (1883), Meissner (1911), and Polyà (1927). The former discusses the one variable case, the latter two deal with the situation that the Newton polyhedra of both P and f be, respectively, standard hypercubes, standard simplices.
Measured foliations and handlebodies
Authors: Masur Howard
Published: September 2008

We consider the action of the subgroup of the mapping class group consisting of homeomorphisms that extend to the handlebody on Thurston's sphere of measured foliations. Properties of the limit set and domain of discontinuity are described and for genus two it is shown the limit set has measure zero.
Rotation intervals for a class of maps of the real line into itself
Authors: Misiurewicz Michał
Published: September 2008

We study a class of maps of the real line into itself which are degree one liftings of maps of the circle and have discontinuities only of a special type. This class contains liftings of continuous degree one maps of the circle, lifting of increasing mod 1 maps and some maps arising from Newton's method of solving equations. We generalize some results known for the continuous case.
The Chebotarov theorem for Galois coverings of Axiom A flows
Authors: Parry William , Pollicott Mark
Published: September 2008

We consider G (Galois) coverings of Axiom A flows (restricted to basic sets) and prove an analogue of Chebotarev's theorem. The theorem provides an asymptotic formula for the number of closed orbits whose Frobenius class is a given conjugacy class in G. An application answers a question raised by J. Plante. The basic method is then extended to compact group extensions and applied to frame bundle flows defined on manifolds of variable negative curvature.
Fixed points of Lie group actions on surfaces
Authors: Plante J. F.
Published: September 2008

LetGbe a connected finite-dimensional Lie group andMa compact surface. We investigate whether, for a givenGandM, every continuous action ofGonMmust have a fixed (stationary) point. It is shown that whenGis nilpotent andMhas non-zero Euler characteristic that every action ofGonMmust have a fixed point. On the other hand, it is shown that the non-abelian 2-dimensional Lie group (affine group of the line) acts without fixed points on every compact surface. These results make it possible to complete this investigation for Lie groups of dimension at most 3.
On orbits of unipotent flows on homogeneous spaces, II
Authors: Dani S. G.
Published: September 2008

We show that if (ut) is a one-parameter subgroup of SL (n, ℝ) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, ℝ)/SL(n, ℤ) such that the following holds: for any g ∈ SL(n, ℝ) either m({t ∈ [0, T] | utg SL (n, ℤ) ∈ K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations.

Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation.
A finiteness theorem for a dynamical class of entire functions
Authors: Goldberg Lisa R. , Keen Linda
Published: September 2008

We define a class Σ of entire functions whose covering properties are similar to those of rational maps. The set Σ is closed under composition of functions, and we show that when regarded as dynamical systems of the plane, the elements of Σ share many properties with rational maps. In particular, they have finite dimensional spaces of quasiconformal deformations, and they contain no wandering domains in their stable sets.
Vector fields with transverse foliations, II
Authors: Goodman Sue
Published: September 2008

When does a non-singular flow on a 3-manifold have a 2-dimensional foliation everywhere transverse to it? A complete answer is given for a large class of flows, those with 1-dimensional hyperbolic chain recurrent set. We find a simple necessary and sufficient condition on the linking of periodic orbits of the flow.
Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets
Authors: Hockett Kevin , Holmes Philip
Published: September 2008

We investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronsonet al.[3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].
On Denjoy's theorem for endomorphisms
Authors: Malta I.
Published: September 2008

In this paper we give an extension to C2 endomorphisms of the circle of the well known Denjoy's theorem on C2 diffeomorphisms. We also give a simple proof of an extension of Block and Franke's theorem on the existence of periodic points for maps of the circle.
On Hausdorff dimension of invariant sets for expanding maps of a circle
Authors: Urbański Mariusz
Published: September 2008

Given an orientation preserving C2 expanding mapping g: S1 → Sl of a circle we consider the family of closed invariant sets Kg(ε) defined as those points whose forward trajectory avoids the interval (0, ε). We prove that topological entropy of g|Kg(ε) is a Cantor function of ε. If we consider the map g(z) = zq then the Hausdorff dimension of the corresponding Cantor set around a parameter ε in the space of parameters is equal to the Hausdorff dimension of Kg(ε). In § 3 we establish some relationships between the mappings g|Kg(ε) and the theory of β-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(ε).
Stochastic stability of hyperbolic attractors
Authors: Young Lai-Sang
Published: September 2008

We study the effects of small random errors on the asymptotic distribution of points in the basin of a hyperbolic attractor.
Corrections to the paper ‘On orbits of unipotent flows on homogeneous spaces’
Authors: Dani S. G.
Published: September 2008

The author regrets that there are certain errors in [1] and would like to give the following corrections.
Generating special Markov partitions for hyperbolic toral automorphisms using fractals
Authors: Bedford Tim
Published: September 2008

We show that given some natural conditions on a 3 × 3 hyperbolic matrix of integers A(det A = 1) there exists a Markov partition for the induced map A(x + ℤ3) = A(x)+ℤ3 on T3 whose transition matrix is (A−1)t. For expanding endomorphisms of T2 we construct a Markov partition so that there is a semiconjugacy from a full (one-sided) shift.
ω-limit sets for maps of the interval
Authors: Block Louis , Coven Ethan M.
Published: September 2008

Let f denote a continuous map of a compact interval to itself, P(f) the set of periodic points of f and Λ(f) the set of ω-limit points of f. Sarkovskǐi has shown that Λ(f) is closed, and hence ⊆Λ(f), and Nitecki has shown that if f is piecewise monotone, then Λ(f)=. We prove that if x∈Λ(f)−, then the set of ω-limit points of x is an infinite minimal set. This result provides the inspiration for the construction of a map f for which Λ(f)≠.
From a differentiable to a real analytic perturbation theory, applications to the Kupka Smale theorems
Authors: Broer H. W. , Tangerman F. M.
Published: September 2008

Kupka-Smale like theorems are proven in the real analytic case, using existing perturbation schemes for the smooth case and the heat operator. As a consequence, a topological proof is obtained of Siegel's theorem on the generic divergence of normal form transformations.
Analytic models of pseudo-Anosov maps
Authors: Lewowicz Jorge , De Sá Eduardo Lima
Published: September 2008

We give a new proof of the existence of analytic models of pseudo-Anosov maps. The persistence properties of Thurston's maps ensure that any Co-perturbation of them presents all their dynamical features. Using Lyapunov functions of two variables we are able to choose certain analytic perturbations which do not add any new dynamical behaviour to the original pseudo-Anosov map.
Monotone twist mappings and the calculus of variations
Authors: Moser Jürgen
Published: September 2008

It is shown that a smooth area-preserving monotone twist mapping ϕ of an annulus A can be interpolated by a flow ϕt which is generated by a t-dependent Hamiltonian in ℝ × A having the period 1 in t and satisfying a Legendre condition. In other words, any such monotone twist mapping can be viewed as a section mapping for the extremals of variational problem on a torus:

where F has period 1 in t and x and satisfies the Legendre condition Fẋẋ>0.
Perron units which are not Mahler measures
Authors: Boyd David W.
Published: September 2008

The Mahler measure M(α) of an algebraic integer α is the product of the absolute value of the conjugates of α which lie outside the unit circle. The quantity log M(α) occurs in ergodic theory as the entropy of an endomorphism of the torus. Adler and Marcus showed that if β = M(α) then β is a Perron number which is a unit if α is a unit. They asked whether the Perron number β whose minimal polynomial is tm −t −1 is the measure of any algebraic integer. We show here that the answer is negative for all m > 3.
Restricted orbit changes of ergodic ℤd-actions to achieve mixing and completely positive entropy
Authors: Fieldsteel Adam , Friedman N. A.
Published: September 2008

We show that for every ergodic ℤd-action T, there is a mixing ℤd-action S which is orbit equivalent to T via an orbit equivalence that is a weak a-equivalence for all a ≥ 1 and a strong b-equivalence for all b ∈ (0, 1). If T has positive entropy, then S can be taken to have completely positive entropy. If the dimension d is greater than one, the orbit equivalence may be taken to be bounded and a strong b-equivalence for all b > 0.
Multiple attractors in Newton's method
Authors: Hurley Mike
Published: September 2008

For eachd≥ 2 there exists a polynomialpwith real coefficients such that the associated Newton functionz–[p(z)/p′(z)] has 2d–2 distinct attracting periodic orbits in the complex plane. According to a theorem ofG. Julia, this is the maximal number of attracting orbits that any rational function of degreedcan possess.
Geometrical Markov coding of geodesics on surfaces of constant negative curvature
Authors: Series Caroline
Published: September 2008

A natural geometrical representation of the geodesic flow on a surface M of constant negative curvature is given in which the base transformation is the shift on a (finite type) space of shortest words relative to a fixed generating set for π1(M) and the height function is the hyperbolic distance across a fundamental region for π1(M). This representation is obtained by comparing cutting sequences on M with generalised continued fraction expansions of endpoints on ℝ
On connection-preserving actions of discrete linear groups
Authors: Zimmer Robert J.
Published: September 2008

We study actions of arithmetic groups on compact manifolds which preserve a connection.
An extension of Ratner's rigidity theorem to n-dimensional hyperbolic space
Authors: Flaminio Livio
Published: September 2008

We prove that the horospherical foliations of two compact manifolds of constant negative curvature are measurably isomorphic if and only if the two manifolds are isometric.