On the ω-limit sets of tent maps

Andrew D. Barwell; Gareth Davies; Chris Good

Displaying similar documents to “On the ω-limit sets of tent maps”

Shadowing and internal chain transitivity

Jonathan Meddaugh, Brian E. Raines (2013)

Fundamenta Mathematicae

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The main result of this paper is that a map f: X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval...

Direct summands of systems of continuous linear transformations

Uri Fixman, Frank A. Zorzitto

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CONTENTSIntroduction............................................................................................................ 51. The category of $C^{N}$ -systems........................................................................... 82. The problem of split monomorphisms................................................................ 103. Internal hom and tensor product........................................................................... 134. Characterizations of split monomorphisms..........................................................

A classification of inverse limit spaces of tent maps with periodic critical points

Lois Kailhofer (2003)

Fundamenta Mathematicae

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We work within the one-parameter family of symmetric tent maps, where the slope is the parameter. Given two such tent maps $f_{a}$ , $f_{b}$ with periodic critical points, we show that the inverse limit spaces $(_{a}, f_{a})$ and $(_{b}, g_{b})$ are not homeomorphic when a ≠ b. To obtain our result, we define topological substructures of a composant, called “wrapping points” and “gaps”, and identify properties of these substructures preserved under a homeomorphism.

On cusps and flat tops

Neil Dobbs (2014)

Annales de l’institut Fourier

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Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to $C^{1 + ϵ}$ . The critical points are not required to verify a non-flatness condition, so the results are applicable to $C^{1 + ϵ}$ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.

Acknowledgment to the paper: “Another proof that a chain of non-empty $H$ -closed subspaces has a non-empty intersection’

J. Mioduszewski (1971)

Colloquium Mathematicae

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More classes of non-orbit-transitive operators

Carl Pearcy, Lidia Smith (2010)

Studia Mathematica

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In [JKP] and its sequel [FPS] the authors initiated a program whose (announced) goal is to eventually show that no operator in ℒ(ℋ) is orbit-transitive. In [JKP] it is shown, for example, that if T ∈ ℒ(ℋ) and the essential (Calkin) norm of T is equal to its essential spectral radius, then no compact perturbation of T is orbit-transitive, and in [FPS] this result is extended to say that no element of this same class of operators is weakly orbit-transitive. In the present note we show...

Submultiplicative functions and operator inequalities

Hermann König, Vitali Milman (2014)

Studia Mathematica

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Let T: C¹(ℝ) → C(ℝ) be an operator satisfying the “chain rule inequality” T(f∘g) ≤ (Tf)∘g⋅Tg, f,g ∈ C¹(ℝ). Imposing a weak continuity and a non-degeneracy condition on T, we determine the form of all maps T satisfying this inequality together with T(-Id)(0) < 0. They have the form Tf = ⎧ $(H \circ f / H) f^{' p}$ , f’ ≥ 0, ⎨ ⎩ $- A (H \circ f / H) | f^{'} |^{p}$ , f’ < 0, with p > 0, H ∈ C(ℝ), A ≥ 1. For A = 1, these are just the solutions of the chain rule operator equation. To prove this, we characterize the submultiplicative, measurable...

Diagonal points having dense orbit

T. K. Subrahmonian Moothathu (2010)

Colloquium Mathematicae

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Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, $f \times f ² \times \dots \times f^{m} : X^{m} \to X^{m}$ is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in $X^{m}$ under the action of $f \times f ² \times \dots \times f^{m}$ . We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying...

On Surjective Bing Maps

Hisao Kato, Eiichi Matsuhashi (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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In [7], M. Levin proved that the set of all Bing maps of a compact metric space to the unit interval is a dense $G_{δ}$ -subset of the space of all maps. In [6], J. Krasinkiewicz independently proved that the set of all Bing maps of a compact metric space to an n-dimensional manifold (n ≥ 1) is a dense $G_{δ}$ -subset of the space of maps. In [9], J. Song and E. D. Tymchatyn, solving some problems of J. Krasinkiewicz ([6]), proved that the set of all Bing maps of a compact metric space to a nondegenerate...

Kneading sequences for double standard maps

Michael Benedicks, Ana Rodrigues (2009)

Fundamenta Mathematicae

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We investigate the symbolic dynamics for the double standard maps of the circle onto itself, given by $f_{a, b} (x) = 2 x + a + (b / π) s i n (2 π x) (m o d 1)$ , where b = 1 and a is a real parameter, 0 ≤ a < 1.

Real $C^{k}$ Koebe principle

Weixiao Shen, Michael Todd (2005)

Fundamenta Mathematicae

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We prove a $C^{k}$ version of the real Koebe principle for interval (or circle) maps with non-flat critical points.

Parapuzzle of the multibrot set and typical dynamics of unimodal maps

Artur Avila, Mikhail Lyubich, Weixiao Shen (2011)

Journal of the European Mathematical Society

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We study the parameter space of unicritical polynomials $f_{c} : z \mapsto z^{d} + c$ . For complex parameters, we prove that for Lebesgue almost every $c$ , the map $f_{c}$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$ , the map $f_{c}$ is either hyperbolic, or Collet–Eckmann, or infinitely renormalizable. These results are based on controlling the spacing between consecutive elements in the “principal nest” of parapuzzle pieces.

On the central limit theorem for some birth and death processes

Tymoteusz Chojecki (2011)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Suppose that ${X n : n \geq 0}$ is a stationary Markov chain and $V$ is a certain function on a phase space of the chain, called an observable. We say that the observable satisfies the central limit theorem (CLT) if $Y_{n} : = N^{- 1 / 2} \sum_{n = 0}^{N} V (X_{n})$ converge in law to a normal random variable, as $N \to + \infty$ . For a stationary Markov chain with the $L^{2}$ spectral gap the theorem holds for all $V$ such that $V (X_{0})$ is centered and square integrable, see Gordin [7]. The purpose of this article is to characterize a family of observables $V$ for which the CLT holds...

Evaluating default priors with a generalization of Eaton’s Markov chain

Brian P. Shea, Galin L. Jones (2014)

Annales de l'I.H.P. Probabilités et statistiques

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We consider evaluating improper priors in a formal Bayes setting according to the consequences of their use. Let $𝛷$ be a class of functions on the parameter space and consider estimating elements of $𝛷$ under quadratic loss. If the formal Bayes estimator of every function in $𝛷$ is admissible, then the prior is strongly admissible with respect to $𝛷$ . Eaton’s method for establishing strong admissibility is based on studying the stability properties of a particular Markov chain associated with...

Linking and the Morse complex

Michael Usher (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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For a Morse function $f$ on a compact oriented manifold $M$ , we show that $f$ has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in $M$ whose components have nontrivial linking number, such that the minimal value of $f$ on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of $f$ in terms of the Betti numbers of $M$ and the behavior of $f$ with respect...

Longer chains of idempotents in βG

Neil Hindman, Dona Strauss, Yevhen Zelenyuk (2013)

Fundamenta Mathematicae

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Given idempotents e and f in a semigroup, e ≤ f if and only if e = fe = ef. We show that if G is a countable discrete group, p is a right cancelable element of G* = βG∖G, and λ is a countable ordinal, then there is a strictly decreasing chain $⟨ q_{σ} ⟩_{σ < λ}$ of idempotents in $C_{p}$ , the smallest compact subsemigroup of G* with p as a member. We also show that if S is any infinite subsemigroup of a countable group, then any nonminimal idempotent in S* is the largest element of such a strictly decreasing...

Distortion bounds for $C^{2 + η}$ unimodal maps

Mike Todd (2007)

Fundamenta Mathematicae

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We obtain estimates for derivative and cross-ratio distortion for $C^{2 + η}$ (any η > 0) unimodal maps with non-flat critical points. We do not require any “Schwarzian-like” condition. For two intervals J ⊂ T, the cross-ratio is defined as the value B(T,J): = (|T| |J|)/(|L| |R|) where L,R are the left and right connected components of T∖J respectively. For an interval map g such that $g_{T} : T \to ℝ$ is a diffeomorphism, we consider the cross-ratio distortion to be B(g,T,J): = B(g(T),g(J))/B(T,J). We prove...

C¹-Stably Positively Expansive Maps

Kazuhiro Sakai (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

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The notion of C¹-stably positively expansive differentiable maps on closed $C^{\infty}$ manifolds is introduced, and it is proved that a differentiable map f is C¹-stably positively expansive if and only if f is expanding. Furthermore, for such maps, the ε-time dependent stability is shown. As a result, every expanding map is ε-time dependent stable.

Insensitivity analysis of Markov chains

Kocurek, Martin

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Sensitivity analysis of irreducible Markov chains considers an original Markov chain with transition probability matix $P$ and modified Markov chain with transition probability matrix $P$ . For their respective stationary probability vectors $π, \tilde{π}$ , some of the following charactristics are usually studied: $∥ π - \tilde{π} ∥_{p}$ for asymptotical stability [3], $| π_{i} - {\tilde{π}}_{i} |, \frac{| π_{i} - {\tilde{π}}_{i} |}{π_{i}}$ for componentwise stability or sensitivity [1]. For functional transition probabilities, $P = P (t)$ and stationary probability vector $π (t)$ , derivatives are also used...

Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets

Dae San Kim, Hyun Kwang Kim

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Let P be a poset on the set [m]×[n], which is given as the disjoint sum of posets on ’columns’ of [m]×[n], and let P̌ be the dual poset of P. Then P is called a generalized Niederreiter-Rosenbloom-Tsfasman poset (gNRTp) if all further posets on columns are weak order posets of the ’same type’. Let G (resp. Ǧ) be the group of all linear automorphisms of the space $_{q}^{m \times n}$ preserving the P-weight (resp. P̌-weight). We define two partitions of $_{q}^{m \times n}$ , one consisting of ’P-orbits’ and the other of...

More on tie-points and homeomorphism in ℕ*

Alan Dow, Saharon Shelah (2009)

Fundamenta Mathematicae

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A point x is a (bow) tie-point of a space X if X∖x can be partitioned into (relatively) clopen sets each with x in its closure. We denote this as $X = A ⋈_{x} B$ where A, B are the closed sets which have a unique common accumulation point x. Tie-points have appeared in the construction of non-trivial autohomeomorphisms of βℕ = ℕ* (by Veličković and Shelah Steprans) and in the recent study (by Levy and Dow Techanie) of precisely 2-to-1 maps on ℕ*. In these cases the tie-points have been the unique...

Ordinal remainders of classical ψ-spaces

Alan Dow, Jerry E. Vaughan (2012)

Fundamenta Mathematicae

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Let ω denote the set of natural numbers. We prove: for every mod-finite ascending chain $T_{α} : α < λ$ of infinite subsets of ω, there exists $ℳ \subset {[ω]}^{ω}$ , an infinite maximal almost disjoint family (MADF) of infinite subsets of the natural numbers, such that the Stone-Čech remainder βψ∖ψ of the associated ψ-space, ψ = ψ(ω,ℳ ), is homeomorphic to λ + 1 with the order topology. We also prove that for every λ < ⁺, where is the tower number, there exists a mod-finite ascending chain $T_{α} : α < λ$ , hence a ψ-space with...

Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Lan Zeng, Chun Lei Tang (2016)

Annales Polonici Mathematici

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We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $- [a + b (\int_{Ω} {| \nabla u | ² d x)}^{m} {] Δ u = f (x, u) + | u |}^{2 * - 2} u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.