Showing papers in "Ergodic Theory and Dynamical Systems in 1983"
TL;DR: In this article, the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established and the equidistribution of pre-images and periodic points with respect to this measure is proved.
Abstract: In this paper the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established. The equidistribution of pre-images and periodic points with respect to this measure is proved.
430 citations
TL;DR: In this article, an Axiom of Lift for classes of dynamical systems is formulated and verified for C 1 diffeomorphisms and C 1 Hamiltonian vector fields, and the lift axiom is shown to imply the closing lemma.
Abstract: 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 C 1 diffeomorphisms to C 1 Hamiltonian vector fields.
256 citations
TL;DR: In this article, the homothetic metrics on the universal cover M converged in the sense of Gromov for small e.g., the volume of balls and the number of closed geodesies.
Abstract: If (M, g) is a riemannian nilmanifold, the homothetic metrics eg˜ on the universal cover M converge in the sense of Gromov for small e. 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.
231 citations
TL;DR: In this paper, the authors 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.
Abstract: 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.
220 citations
TL;DR: In this article, a sufficient condition for a unimodal map of the interval to have an invariant measure absolutely continuous with respect to the Lebesgue measure was given, and the condition requires positivity of the forward and backward Liapunov exponent of the critical point.
Abstract: We give a sufficient condition for a unimodal map of the interval to have an invariant measure absolutely continuous with respect to the Lebesgue measure. Apart from some weak regularity assumptions, the condition requires positivity of the forward and backward Liapunov exponent of the critical point.
161 citations
TL;DR: In this article, the authors consider examples of Finsler metrics symmetric or not) on Sn, Pnℂ and Pn ℂ with only finitely many closed geodesies.
Abstract: We consider examples of Finsler metrics symmetric or not) on Sn, Pnℂ, Pnℍ, and P2Ca with only finitely many closed geodesies or with only few short closed geodesies. The number of closed geodesies in these examples and properties of the closed geodesies are considered.
150 citations
TL;DR: A mixing subshift of finite type T is a factor of a sofic shift S of greater entropy if and only if the period of any periodic point of S is divisible by the periods of some periodic point in T as discussed by the authors.
Abstract: A mixing subshift of finite type T is a factor of a sofic shift S of greater entropy if and only if the period of any periodic point of S is divisible by the period of some periodic point of T. Mixing sofic shifts T satisfying this theorem are characterized, as are those mixing sofic shifts for which Krieger's Embedding Theorem holds. These and other results rest on a general method for extending shift-commuting continuous maps into mixing subshifts of finite type.
115 citations
TL;DR: In this article, a Lie group acting in Hamiltonian fashion on a symplectic manifold M with moment map Φ:M → g* is shown to form a complete integrable system.
Abstract: Let G be a Lie group acting in Hamiltonian fashion on a symplectic manifold M with moment map Φ:M → g*. A function of the form ƒ∘Φ where ƒ is a function on g* is called ‘collective’. We obtain necessary conditions on the G action for there to exist enough Poisson commuting functions on g* so that the corresponding collective functions on M form a completely integrable system. For the case G = O(n) or U(n) these conditions are sufficient. This explains Thimm's proof [17] of the complete integrability of the geodesic flow on the real and complex grassmanians. We also discuss related questions in the geometry of the moment map.
86 citations
TL;DR: For a homeomorphism of a compact metrizable space X, this paper showed 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 AFalgebra.
Abstract: 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.
85 citations
TL;DR: In this paper, the authors give sufficient conditions for an expansive diffeomorphism of a compact manifold to be such that every neighbouring diffEomorphism shows, roughly, all the dynamical features of the manifold.
Abstract: 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.
66 citations
TL;DR: In this article, the authors introduce the notions of induced regular homomorphisms and backward regular homomorphic structures, which are associated with every homomorphism between strongly connected graphs whose global map is finite-to-one and onto.
Abstract: 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.
TL;DR: In this article, it was shown that if A and B are positive 2×2 integral matrices of non-negative determinant and are similar over the integers, then they are strongly shift equivalent.
Abstract: 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.
TL;DR: In this paper, the Julia set B for an N'th degree polynomial T and its equilibrium electrostatic measure μ are considered and the unique balanced measure on B is shown to be μ.
Abstract: The Julia set B for an N'th degree polynomial T and its equilibrium electrostatic measure μ are considered. The unique balanced measure on B is shown to be μ. Integral properties of μ and of the monic polynomials orthogonal with respect to μ, Pn, n = 0, 1, 2, …, are derived. Formulae relating orthogonal polynomials of the second kind of different degrees are displayed. The measure μ is recovered both in the limit from the zeros and from the poles of the [Nn − 1/Nn] Pade approximant to the moment generating function to μ. For infinitely many polynomials of each degree N the zeros and poles all lie on an increasing sequence of trees of analytic arcs contained in B. The properties of these Pade approximant sequences support conjectures of George Baker which have not previously been tested on measures supported on sets nearly as complicated as Julia sets spread out in the complex plane.
TL;DR: In this article, the authors consider a Riemannian manifold M with no focal points such that the universal cover contains a geodesic which does not bound a flat geodesically embedded half plane.
Abstract: We consider a Riemannian manifold M with no focal points such that the universal cover contains a geodesic which does not bound a flat totally geodesically embedded half plane. It is shown that
(i) If the non-wandering set of the geodesic flow is the whole of SM , then the closed orbits are dense in SM .
(ii) If M is compact then the geodesic flow is ergodic and Bernoulli with respect to the Liouville measure on SM.
TL;DR: In this paper, the maximal and minimal characteristic exponents for dynamical systems in metric spaces are introduced. But they do not cover invariant sets, and they are not invariant to the smooth case.
Abstract: We introduce for dynamical systems in metric spaces some numbers which in the case of smooth dynamical systems turn out to be the maximal and the minimal characteristic exponents. These numbers have some properties similar to the smooth case. Analogous quantities are defined also for invariant sets.
TL;DR: For continuous maps of the circle to itself, this paper showed that the set of nonwandering points coincides with that of n for every odd n, and that every non-wandering point is periodic.
Abstract: For continuous maps ƒ of the circle to itself, we show: (A) the set of nonwandering points of ƒ coincides with that of ƒn for every odd n; (B) ƒ has a horseshoe if and only if it has a non-wandering homoclinic point; (C) if the set of periodic points is closed and non-empty, then every non-wandering point is periodic.
TL;DR: In this article, it was shown that there is a residual subset of the set of C1 diffeomorphisms on any compact manifold at which the map is continuous, which allows one to conclude that if f is in this residual set and X is an isolated chain component for f, then a neighbourhood U of X which isolates it from the rest of the chain recurrent set of f, and all g sufficiently close to f have precisely one chain component in U, and these chain components approach X as g approaches f.
Abstract: We show that there is a residual subset of the set of C1 diffeomorphisms on any compact manifold at which the mapis 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.
TL;DR: In this article, the existence of a homoclinic periodic orbit with given period is shown to be a dynamical property of quasi-unimodal maps of a compact interval to itself.
Abstract: The space of ‘quadratic-like’ (unimodal) maps of a compact interval to itself is shown to decompose in a ‘nice’ way (stratify) according to a dynamical property of such maps (the existence of a homoclinic periodic orbit with given period). This decomposition is refined by that discovered by Sarkovskii. Orbit structure and bifurcation properties are also discussed.
TL;DR: In this article, it was shown that the Euclidean de Rham factor of a Riemannian manifold M is a constant multiple of the metric of the manifold M and that the number and dimensions of the local de rham factor are the same for M and M* with rank at least 2.
Abstract: Let M, M* denote compact, connected manifolds of non-positive sectional curvature whose fundamental groups are isomorphic and whose Euclidean de Rham factors are trivial. We prove that: if M is a compact irreducible quotient of a reducible symmetric space H, then M and M* are isometric up to a constant multiple of the metric; and that the number and dimensions of the local de Rham factors are the same for M and M*. Gromov has independently proved the first result in the more general case that M is locally symmetric and globally irreducible with rank at least two. 0. Introduction A basic problem in Riemannian geometry is to determine the extent to which the geometry of the Riemannian metric and the topology of the underlying manifold influence each other. Restrictions on the curvature and topology are usually necessary to obtain reasonable results, and we shall confine our attention to compact connected manifolds of nonpositive sectional curvature. We define a geometric property of such manifolds to be a rigid property if whenever it holds for a manifold M it also holds for any manifold M* that is homotopically equivalent to M. Our goal is to look for rigid properties. In a previous paper [7] we showed that certain geometric properties of a free homotopy class of closed curves are rigid properties. In this paper we have two main results, the first of which is half of an independent result of Gromov. Before stating them we define a Riemannian manifold X to be reducible if some finite Riemannian cover X splits as a nontrivial Riemannian product X\\ x X2. If X is simply connected and reducible, then X itself is a nontrivial Riemannian product THEOREM A. Let M, M* denote compact connected Riemannian manifolds of nonpositive sectional curvature whose fundamental groups are isomorphic and whose universal Riemannian covering manifolds H, H* possess no Euclidean de Rham factor. Suppose that H* is a reducible symmetric space of noncompact type and M* is an irreducible quotient of H*. Then M and M* are isometric, provided that one multiplies the metric of M or M* by a suitable positive constant.
TL;DR: In this paper, the authors 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.
Abstract: 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.
TL;DR: In this paper, the authors consider a C*-algebra and τ:G → a compact abelian action such that the fixed point algebra τ is simple and the matrix inequality holds for all finite sequences X1, Xn in F.
Abstract: Let be a C*-algebra and τ:G → Aut a compact abelian action such that the fixed point algebra τ is simple. Denote by F the *-subalgebra of G-finite elements. Let H: F → be a *-operator commuting with τ such that and the matrix inequalityholds for all finite sequences X1, …, Xn in F. Then H is closable, and the closure is the generator of a strongly continuous semigroup {exp (−t): t ≥ 0} of completely positive contractions. Furthermore, there exists a convolution semigroup {μt: t ≥ 0} of probability measures on G such that.This result has various extensions and refinements.
TL;DR: In this article, a topological analogue to the measure-theoretic notion of a transformation having minimal self-joinings is proposed, based on the notion of minimal sets.
Abstract: 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. 0. Introduction Rudolph [9] introduced the concept of minimal self-joinings (MSJ) to build examples of exotic ergodic behaviour. His example of MSJ was not in the context of a concrete homeomorphism of a familiar compact metric space. Since then del Junco, Rahe and Swanson [3] have shown that Chacon's example [1] has MSJ. This example can be constructed by 'doubling the ones' in a certain symbolic almost automorphic minimal set whose maximal equicontinuous factor is the 3-adic integers. This construction is analogous to that used by Furstenberg, Keynes and Shapiro [5] to construct a proximal orbit dense (POD) flow. In fact, del Junco [2] has pointed out that Chacon's example is a POD flow. (The author was also aware of the existence of POD flows similar to the Chacon example [8].) These developments raise two natural questions. What is the relationship between MSJ and POD? What is the topological analogue of MSJ? Del Junco [2] has partially answered the first question. This paper is devoted to the second question and proceeds from the premise that minimal sets should be analogous to ergodic measures. This produces an interesting theory, but the results thus far are not as good as one might desire. The main thrust of our work is to study the dynamics of the obvious maps which commute with a countable power of a system with topological minimal self-joinings. In other words, we investigate Rudolph's U{ir, I) maps from a topological rather than measure theoretic viewpoint. Moreover, our analysis focuses on the minimal subsets of these maps. 1. Preliminaries Let X be a compact metric space. Given a homeomorphsim T of X onto itself, the pair (X, T) will be called a flow. The flow (X, T) is (topologically) ergodic if every proper closed invariant set is nowhere dense, and (topologically) weak mixing if
TL;DR: In this paper, an elementary proof of Krengel's stochastic ergodic theorem in the setting of superadditive processes is given for the case of multiparameter superadditives.
Abstract: An elementary proof is given of Krengel's stochastic ergodic theorem in the setting of multiparameter superadditive processes.
TL;DR: In this article, the question of which periods can occur as periods of periodic points of zero entropy surface homeomorphisms in a given isotopy class was dealt with, and new examples of isotopy classes for which there are non-trivial restrictions were given.
Abstract: This paper deals with the question of which periods can occur as periods of periodic points of zero entropy surface homeomorphisms in a given isotopy class. We give new examples of isotopy classes for which there are non-trivial restrictions and describe how the possible periods can be computed. Certain phenomena occur only for surfaces of large genus. These results have applications to the periodic data question for Morse–Smale maps.
TL;DR: In the paper as mentioned in this paper, the authors make some minor but misleading mistakes, such as: inequality (11) states:It should read:The first inequality is obviously false in general and the second one is an immediate corollary of Oseledec's theorem.
Abstract: In the paper ‘A proof of Pesin's formula’ (R. Mane, 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.