Minimal number of periodic points for smooth self-maps of S³

Grzegorz Graff; Jerzy Jezierski

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Displaying similar documents to “Minimal number of periodic points for smooth self-maps of S³”

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 the Extension of Certain Maps with Values in Spheres

Carlos Biasi, Alice K. M. Libardi, Pedro L. Q. Pergher, Stanisław Spież (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let E be an oriented, smooth and closed m-dimensional manifold with m ≥ 2 and V ⊂ E an oriented, connected, smooth and closed (m-2)-dimensional submanifold which is homologous to zero in E. Let $S^{n - 2} \subset S ⁿ$ be the standard inclusion, where Sⁿ is the n-sphere and n ≥ 3. We prove the following extension result: if $h : V \to S^{n - 2}$ is a smooth map, then h extends to a smooth map g: E → Sⁿ transverse to $S^{n - 2}$ and with $g^{- 1} (S^{n - 2}) = V$ . Using this result, we give a new and simpler proof of a theorem of Carlos Biasi related to the ambiental...

Existence and uniqueness of periodic solutions for odd-order ordinary differential equations

Yongxiang Li, He Yang (2011)

Annales Polonici Mathematici

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The paper deals with the existence and uniqueness of 2π-periodic solutions for the odd-order ordinary differential equation $u^{(2 n + 1)} = f (t, u, u^{'}, . . ., u^{(2 n)})$ , where $f : ℝ \times ℝ^{2 n + 1} \to ℝ$ is continuous and 2π-periodic with respect to t. Some new conditions on the nonlinearity $f (t, x ₀, x ₁, . . ., x_{2 n})$ to guarantee the existence and uniqueness are presented. These conditions extend and improve the ones presented by Cong [Appl. Math. Lett. 17 (2004), 727-732].

Three periodic solutions for a class of higher-dimensional functional differential equations with impulses

Yongkun Li, Changzhao Li, Juan Zhang (2010)

Annales Polonici Mathematici

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By using the well-known Leggett–Williams multiple fixed point theorem for cones, some new criteria are established for the existence of three positive periodic solutions for a class of n-dimensional functional differential equations with impulses of the form ⎧y’(t) = A(t)y(t) + g(t,yt), $t \neq t_{j}$ , j ∈ ℤ, ⎨ ⎩ $y (t ⁺_{j}) = y (t ¯_{j}) + I_{j} (y (t_{j}))$ , where $A (t) = {(a_{i j} (t))}_{n \times n}$ is a nonsingular matrix with continuous real-valued entries.

A class of $I_{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

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Periodic solutions of $x " + f (x) x^{' m} + g (x) = 0$

W. R. Utz (1971)

Annales Polonici Mathematici

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Existence of nonnegative periodic solutions in neutral integro-differential equations with functional delay

Imene Soulahia, Abdelouaheb Ardjouni, Ahcene Djoudi (2015)

Commentationes Mathematicae Universitatis Carolinae

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The fixed point theorem of Krasnoselskii and the concept of large contractions are employed to show the existence of a periodic solution of a nonlinear integro-differential equation with variable delay $x^{'} (t) = - \int_{t - τ (t)}^{t} a (t, s) g (x (s)) d s + \frac{d}{d t} Q (t, x (t - τ (t))) + G (t, x (t), x (t - τ (t))) .$ We transform this equation and then invert it to obtain a sum of two mappings one of which is completely continuous and the other is a large contraction. We choose suitable conditions for $τ$ , $g$ , $a$ , $Q$ and $G$ to show that this sum of mappings fits into the framework of a modification of...

On the uniqueness of periodic decomposition

Viktor Harangi (2011)

Fundamenta Mathematicae

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Let $a ₁, . . ., a_{k}$ be arbitrary nonzero real numbers. An $(a ₁, . . ., a_{k})$ -decomposition of a function f:ℝ → ℝ is a sum $f ₁ + \dots + f_{k} = f$ where $f_{i} : ℝ \to ℝ$ is an $a_{i}$ -periodic function. Such a decomposition is not unique because there are several solutions of the equation $h ₁ + \dots + h_{k} = 0$ with $h_{i} : ℝ \to ℝ a_{i}$ -periodic. We will give solutions of this equation with a certain simple structure (trivial solutions) and study whether there exist other solutions or not. If not, we say that the $(a ₁, . . ., a_{k})$ -decomposition is essentially unique. We characterize those periods for which essential...

Addenda to “Periodic solutions of $x " + f (x) x^{' 2 n} + g (x) = 0$ with arbitrarily large period”

J. W. Heidel (1973)

Annales Polonici Mathematici

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Periodic solutions of $x " + f (x) x^{' 2 n} + g (x) = 0$ with arbitrarily large period

J. W. Heidel (1971)

Annales Polonici Mathematici

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The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian

Jean Mawhin (2006)

Journal of the European Mathematical Society

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We prove an Ambrosetti–Prodi type result for the periodic solutions of the equation $(| u^{'} |^{p - 2} u^{'} {))}^{'} + f (u) u^{'} + g (x, u) = t$ , when $f$ is arbitrary and $g (x, u) \to + \infty$ or $g (x, u) \to - \infty$ when $| u | \to \infty$ . The proof uses upper and lower solutions and the Leray–Schauder degree.

Some remarks providing discontinuous maps on some $C_{p} (X)$ spaces

S. Moll (2008)

Banach Center Publications

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Let X be a completely regular Hausdorff topological space and $C_{p} (X)$ the space of continuous real-valued maps on X endowed with the pointwise topology. A simple and natural argument is presented to show how to construct on the space $C_{p} (X)$ , if X contains a homeomorphic copy of the closed interval [0,1], real-valued maps which are everywhere discontinuous but continuous on all compact subsets of $C_{p} (X)$ .

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...

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.

Stability of periodic stationary solutions of scalar conservation laws with space-periodic flux

Anne-Laure Dalibard (2011)

Journal of the European Mathematical Society

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This article investigates the long-time behaviour of parabolic scalar conservation laws of the type $\partial_{t} u + {div}_{y} A (y, u) - Δ_{y} u = 0$ , where $y \in ℝ^{N}$ and the flux $A$ is periodic in $y$ . More specifically, we consider the case when the initial data is an $L^{1}$ disturbance of a stationary periodic solution. We show, under polynomial growth assumptions on the flux, that the difference between u and the stationary solution behaves in $L^{1}$ norm like a self-similar profile for large times. The proof uses a time and space change of variables...

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.

Global Attractor for the Convective Cahn-Hilliard Equation in $H^{k}$

Xiaopeng Zhao, Ning Duan (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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We consider the convective Cahn-Hilliard equation with periodic boundary conditions. Based on the iteration technique for regularity estimates and the classical theorem on existence of a global attractor, we prove that the convective Cahn-Hilliard equation has a global attractor in $H^{k}$ .

Are All Smooth Solutions of Euclidean $S U (2)$ -Yang-Mills Equations Selfdual or Antiselfdual?

R. Flume (1979)

Recherche Coopérative sur Programme n°25

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On the classification of inverse limits of tent maps

Louis Block, Slagjana Jakimovik, Lois Kailhofer, James Keesling (2005)

Fundamenta Mathematicae

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Let $f_{s}$ and $f_{t}$ be tent maps on the unit interval. In this paper we give a new proof of the fact that if the critical points of $f_{s}$ and $f_{t}$ are periodic and the inverse limit spaces $(I, f_{s})$ and $(I, f_{t})$ are homeomorphic, then s = t. This theorem was first proved by Kailhofer. The new proof in this paper simplifies the proof of Kailhofer. Using the techniques of the paper we are also able to identify certain isotopies between homeomorphisms on the inverse limit space.

Minimal models for $ℤ^{d}$ -actions

Bartosz Frej, Agata Kwaśnicka (2008)

Colloquium Mathematicae

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We prove that on a metrizable, compact, zero-dimensional space every $ℤ^{d}$ -action with no periodic points is measurably isomorphic to a minimal $ℤ^{d}$ -action with the same, i.e. affinely homeomorphic, simplex of measures.

On dimensionally restricted maps

H. Murat Tuncali, Vesko Valov (2002)

Fundamenta Mathematicae

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Let f: X → Y be a closed n-dimensional surjective map of metrizable spaces. It is shown that if Y is a C-space, then: (1) the set of all maps g: X → ⁿ with dim(f △ g) = 0 is uniformly dense in C(X,ⁿ); (2) for every 0 ≤ k ≤ n-1 there exists an $F_{σ}$ -subset $A_{k}$ of X such that $d i m A_{k} \leq k$ and the restriction $f | (X ∖ A_{k})$ is (n-k-1)-dimensional. These are extensions of theorems by Pasynkov and Toruńczyk, respectively, obtained for finite-dimensional spaces. A generalization of a result due to Dranishnikov and Uspenskij...

Multiple positive solutions of a nonlinear fourth order periodic boundary value problem

Lingbin Kong, Daqing Jiang (1998)

Annales Polonici Mathematici

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The fourth order periodic boundary value problem $u^{(4)} - m ⁴ u + F (t, u) = 0$ , 0 < t < 2π, with $u^{(i)} (0) = u^{(i)} (2 π)$ , i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of $\pm 10^{- 7}$ .

$C^{1}$ self-maps on closed manifolds with finitely many periodic points all of them hyperbolic

Jaume Llibre, Víctor F. Sirvent (2016)

Mathematica Bohemica

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Let $X$ be a connected closed manifold and $f$ a self-map on $X$ . We say that $f$ is almost quasi-unipotent if every eigenvalue $λ$ of the map $f_{* k}$ (the induced map on the $k$ -th homology group of $X$ ) which is neither a root of unity, nor a zero, satisfies that the sum of the multiplicities of $λ$ as eigenvalue of all the maps $f_{* k}$ with $k$ odd is equal to the sum of the multiplicities of $λ$ as eigenvalue of all the maps $f_{* k}$ with $k$ even. We prove that if $f$ is $C^{1}$ having finitely many periodic points all of them...

One-dimensional symmetry of periodic minimizers for a mean field equation

Chang-Shou Lin, Marcello Lucia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider on a two-dimensional flat torus $T$ defined by a rectangular periodic cell the following equation $Δ u + ρ (\frac{e^{u}}{\int_{T} e^{u}} - \frac{1}{| T |}) = 0, \int_{T} u = 0 .$ It is well-known that the associated energy functional admits a minimizer for each $ρ \leq 8 π$ . The present paper shows that these minimizers depend actually only on one variable. As a consequence, setting $λ_{1} (T)$ to be the first eigenvalue of the Laplacian on the torus, the minimizers are identically zero whenever $ρ \leq min {8 π, λ_{1} (T) | T |}$ . Our results hold more generally for solutions that...

$Z ₂^{k}$ -actions with a special fixed point set

Pedro L. Q. Pergher, Rogério de Oliveira (2005)

Fundamenta Mathematicae

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Let Fⁿ be a connected, smooth and closed n-dimensional manifold satisfying the following property: if $N^{m}$ is any smooth and closed m-dimensional manifold with m > n and $T : N^{m} \to N^{m}$ is a smooth involution whose fixed point set is Fⁿ, then m = 2n. We describe the equivariant cobordism classification of smooth actions $(M^{m}; Φ)$ of the group $G = Z ₂^{k}$ on closed smooth m-dimensional manifolds $M^{m}$ for which the fixed point set of the action is a submanifold Fⁿ with the above property. This generalizes a result of F....