PERIODS FOR MAPS OF THE FIGURE-EIGHT SPACE

1995 ◽  
Vol 05 (05) ◽  
pp. 1357-1368 ◽  
Author(s):  
CHRISTIAN GILLOT ◽  
JAUME LLIBRE
Keyword(s):  
Topological Space ◽  
Periodic Points ◽  
Continuous Map ◽  
Branching Point

Let Per (f) denote the set of periods of all periodic points of a map f from a topological space into itself. Let 8 be the figure-eight space. We extend to the 8 the following theorem from the circle due to Block [1981]. Let [Formula: see text] be the circle. For every map [Formula: see text] with Per (f) ∩ {1, 2, …, n} = {1, n} and n > 2 we have Per (f) = {1, n, n+1, n+2, …}. Conversely, for every n ∈ ℕ with n > 2 there exists a map [Formula: see text] such that Per (f) = {1, n, n+1, n+2, …}. For the space 8 we prove the following. Let f: 8 → 8 be a continuous map having the branching point fixed and such that Per (f) ∩ {1, 2, …, n} = {1, n} with n > 4. Then Per (f) is either {1, n, n+1, n+2, …}, or {1, n, n+2, n+4, …} with n even, or {1, n, n+2, n+4, …}∪ {2n+2, 2n+4, 2n+6, …} with n odd. Conversely, for every n ∈ ℕ with n > 4, if A (n) is one of the above three subsets of ℕ, then there is a continuous map f: 8 → 8 having the branching point fixed and such that Per (f) = A (n).

On the full periodicity kernel for one-dimensional maps

1999 ◽  
Vol 19 (1) ◽  
pp. 101-126
Author(s):  
M. CARME LESEDUARTE ◽  
JAUME LLIBRE
Keyword(s):  
Topological Space ◽  
Periodic Points ◽  
One Dimensional ◽  
Fixed Set ◽  
Branching Point ◽  
One Dimensional Maps

Let $\bpropto$ be the topological space obtained by identifying the points 1 and 2 of the segment $[0,3]$ to a point. Let $\binfty$ be the topological space obtained by identifying the points 0, 1 and 2 of the segment $[0,2]$ to a point. An $\bpropto$ (respectively $\binfty$) map is a continuous self-map of $\bpropto$ (respectively $\binfty$) having the branching point fixed. Set $E\in\{\bpropto,\binfty\}$. Let $f$ be an $E$ map. We denote by $\Per(f)$ the set of periods of all periodic points of $f$. The set $K \subset{\mathbb N}$ is the full periodicity kernel of $E$ if it satisfies the following two conditions: (1) if $f$ is an $E$ map and $K\subset \Per(f)$, then $\Per(f)={\mathbb N}$; (2) for each $k\in K$ there exists an $E$ map $f$ such that $\Per(f)={\mathbb N}\setminus\{ k\}$. In this paper we compute the full periodicity kernel of $\bpropto$ and $\binfty$.

On Certain Onto Maps

1962 ◽  
Vol 14 ◽  
pp. 461-466 ◽  
Author(s):  
Isaac Namioka
Keyword(s):  
Topological Space ◽  
Closed Subspace ◽  
Dimensional Space ◽  
Extension Theorem ◽  
Normal Space ◽  
Continuous Map ◽  
Image Position

Let Δn (n > 0) denote the subset of the Euclidean (n + 1)-dimensional space defined byA subset σ of Δn is called a face if there exists a sequence 0 ≤ i1 ≤ i2 ≤ … < im ≤ n such thatand the dimension of σ is defined to be (n — m). Let denote the union of all faces of Δn of dimensions less than n. A topological space Y is called solid if any continuous map on a closed subspace A of a normal space X into Y can be extended to a map on X into Y. By Tietz's extension theorem, each face of Δn is solid. The present paper is concerned with a generalization of the following theorem which seems well known.

Zero-Dimensional Compactifications of Locally Compact Spaces

1974 ◽  
Vol 26 (4) ◽  
pp. 920-930 ◽  
Author(s):  
R. Grant Woods
Keyword(s):  
Topological Space ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Dense Subspace ◽  
Locally Compact ◽  
Partially Order ◽  
Compact Spaces ◽  
Hausdorff Topological Space ◽  
Continuous Map ◽  
Locally Compact Spaces

Let X be a locally compact Hausdorff topological space. A compactification of X is a compact Hausdorff space which contains X as a dense subspace. Two compactifications αX and γX of X are equivalent if there is a homeomorphism from αX onto γX that fixes X pointwise. We shall identify equivalent compactifications of a given space. If is a family of compactifications of X, we can partially order by saying that αX ≦ γX if there is a continuous map from γX onto αX that fixes X pointwise.

scholarly journals ω-limit sets for maps of the interval

1986 ◽  
Vol 6 (3) ◽  
pp. 335-344 ◽  
Author(s):  
Louis Block ◽  
Ethan M. Coven
Keyword(s):  
Periodic Points ◽  
Compact Interval ◽  
Minimal Set ◽  
Limit Sets ◽  
Piecewise Monotone ◽  
Continuous Map ◽  
Limit Points

AbstractLet 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)≠.

scholarly journals Chain recurrent points of a tree map

1999 ◽  
Vol 59 (2) ◽  
pp. 181-186 ◽  
Author(s):  
Tao Li ◽  
Xiangdong Ye
Keyword(s):  
Topological Entropy ◽  
Periodic Points ◽  
Limit Set ◽  
Recurrent Point ◽  
Continuous Map ◽  
Tree Map

We generalise a result of Hosaka and Kato by proving that if the set of periodic points of a continuous map of a tree is closed then each chain recurrent point is a periodic one. We also show that the topological entropy of a tree map is zero if and only if thew-limit set of each chain recurrent point (which is not periodic) contains no periodic points.

scholarly journals On Perturbations of Continuous Maps

2013 ◽  
Vol 56 (1) ◽  
pp. 92-101
Author(s):  
Benoît Jacob
Keyword(s):  
Perturbation Theory ◽  
Topological Space ◽  
Small Perturbation ◽  
Sufficient Conditions ◽  
Matrix Perturbation ◽  
Continuous Map ◽  
Matrix Perturbation Theory ◽  
Continuous Maps

AbstractWe give sufficient conditions for the following problem: given a topological space X, ametric space Y, a subspace Z of Y, and a continuous map f from X to Y, is it possible, by applying to f an arbitrarily small perturbation, to ensure that f(X) does not meet Z? We also give a relative variant: if f(X') does not meet Z for a certain subset X'⊂ X, then we may keep f unchanged on X'. We also develop a variant for continuous sections of fibrations and discuss some applications to matrix perturbation theory.

ON HAUSDORFF DIMENSION AND TOPOLOGICAL ENTROPY

Fractals ◽  
2010 ◽  
Vol 18 (03) ◽  
pp. 363-370 ◽  
Author(s):  
DONGKUI MA ◽  
MIN WU
Keyword(s):  
Variational Principle ◽  
Hausdorff Dimension ◽  
Topological Space ◽  
Topological Entropy ◽  
Continuous Map ◽  
Compact Topological Space ◽  
Metric Function

Let f: X → X be a continuous map of a compact topological space. If there exists a metric function on X and it satisfies some restricted conditions, we obtain some relationships between Hausdorff dimension and topological entropy for any Z ⊆ X. Using those results, we also obtain a variational principle of dimensions, generalize some known results and give some examples.

Open and Proper Maps Characterized by Continuous Setvalued Maps

1981 ◽  
Vol 33 (4) ◽  
pp. 929-936 ◽  
Author(s):  
Eva Lowen- Colebunders
Keyword(s):  
Topological Space ◽  
Hausdorff Topological Space ◽  
Closed Sets ◽  
Continuous Map ◽  
Topological Convergence ◽  
Proper Maps ◽  
Convergence Of Sequences ◽  
Convergence Of Sets

In the first part of the paper, given a continuous map f from a Hausdorff topological space X onto a Hausdorff topological space Y, we consider the reciprocal map f* from Y into the collection of closed subsets of X, which maps y ∈ Y to . is endowed with the pseudotopological structure of convergence of closed sets. We will use the filter description of this convergence, as defined by Choquet and Gähler [2], [5], which is equivalent to the “topological convergence” of sets as introduced by Frolík and Mrówka [4], [10]. These notions in fact generalize the convergence of sequences of sets defined by Hausdorff [6]. We show that the continuity of f* is equivalent to the openness of f. On f*(Y), the set of fibers of f, we consider the pseudotopological structure induced by the closed convergence on .

scholarly journals The chain recurrent set, attractors, and explosions

1985 ◽  
Vol 5 (3) ◽  
pp. 321-327 ◽  
Author(s):  
Louis Block ◽  
John E. Franke
Keyword(s):  
Compact Space ◽  
Metric Space ◽  
Periodic Points ◽  
Limit Set ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Special Case ◽  
Equivalent Conditions ◽  
Chain Recurrent Set ◽  
Recurrent Set

AbstractCharles Conley has shown that for a flow on a compact metric space, a point x is chain recurrent if and only if any attractor which contains the & ω-limit set of x also contains x. In this paper we show that the same statement holds for a continuous map of a compact metric space to itself, and additional equivalent conditions can be given. A stronger result is obtained if the space is locally connected.It follows, as a special case, that if a map of the circle to itself has no periodic points then every point is chain recurrent. Also, for any homeomorphism of the circle to itself, the chain recurrent set is either the set of periodic points or the entire circle. Finally, we use the equivalent conditions mentioned above to show that for any continuous map f of a compact space to itself, if the non-wandering set equals the chain recurrent set then f does not permit Ω-explosions. The converse holds on manifolds.

Weakly Almost Periodic Points and Some Chaotic Properties of Dynamical Systems

2015 ◽  
Vol 25 (09) ◽  
pp. 1550115 ◽  
Author(s):  
Jiandong Yin ◽  
Zuoling Zhou
Keyword(s):  
Dynamical Systems ◽  
Metric Space ◽  
Sufficient Conditions ◽  
Periodic Points ◽  
Full Measure ◽  
Almost Periodic ◽  
Countable Base ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Weakly Almost Periodic

Let X be a compact metric space and f : X → X be a continuous map. In this paper, ergodic chaos and strongly ergodic chaos are introduced, and it is proven that f is strongly ergodically chaotic if f is transitive but not minimal and has a full measure center. In addition, some sufficient conditions for f to be Ruelle–Takens chaotic are presented. For instance, we prove that f is Ruelle–Takens chaotic if f is transitive and there exists a countable base [Formula: see text] of X such that for each i > 0, the meeting time set N(Ui, Ui) for Ui with respect to itself has lower density larger than [Formula: see text].

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