scholarly journals Almost minimal systems and periodicity in hyperspaces

2017 ◽  
Vol 38 (6) ◽  
pp. 2158-2179 ◽  
Author(s):  
LEOBARDO FERNÁNDEZ ◽  
CHRIS GOOD ◽  
MATE PULJIZ
Keyword(s):  
Metric Space ◽  
Necessary Conditions ◽  
Periodic Points ◽  
Cantor Set ◽  
Full Description ◽  
Compact Metric Space ◽  
Symmetric Products ◽  
Interval Maps ◽  
Admissible Sets ◽  
Induced Maps

Given a self-map of a compact metric space $X$, we study periodic points of the map induced on the hyperspace of closed non-empty subsets of $X$. We give some necessary conditions on admissible sets of periods for these maps. Seemingly unrelated to this, we construct an almost totally minimal homeomorphism of the Cantor set. We also apply our theory to give a full description of admissible period sets for induced maps of the interval maps. The description of admissible periods is also given for maps induced on symmetric products.

scholarly journals Bowen’s equation in the non-uniform setting

2010 ◽  
Vol 31 (4) ◽  
pp. 1163-1182 ◽  
Author(s):  
VAUGHN CLIMENHAGA
Keyword(s):  
Hausdorff Dimension ◽  
Lyapunov Exponents ◽  
Metric Space ◽  
Periodic Points ◽  
Topological Pressure ◽  
Conformal Map ◽  
Arbitrary Subset ◽  
Compact Metric Space ◽  
Symbolic Space ◽  
Contraction Condition

AbstractWe show that Bowen’s equation, which characterizes the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents for maps with parabolic periodic points, and to relate the Hausdorff dimension to the topological entropy for arbitrary subsets of symbolic space with the appropriate metric.

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.

Perfect Images of Zero-Dimensional Separable Metric Spaces

1982 ◽  
Vol 25 (1) ◽  
pp. 41-47 ◽  
Author(s):  
Jan Van Mill ◽  
R. Grant Woods
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Dense Subset ◽  
Cantor Set ◽  
Locally Compact ◽  
Compact Metric Space ◽  
Continuous Surjection

AbstractLet Q denote the rationals, P the irrationals, C the Cantor set and L the space C − {p} (where p ∈ C). Let f : X → Y be a perfect continuous surjection. We show: (1) If X ∈ {Q, P, Q × P}, or if f is irreducible and X ∈ {C, L}, then Y is homeomorphic to X if Y is zero-dimensional. (2) If X ∈ {P, C, L} and f is irreducible, then there is a dense subset S of Y such that f|f ← [S] is a homeomorphism onto S. However, if Z is any σ-compact nowhere locally compact metric space then there is a perfect irreducible continuous surjection from Q × C onto Z such that each fibre of the map is homeomorphic to C.

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].

scholarly journals Consequences of the Pseudo Orbits Tracing Property and Expansiveness

1987 ◽  
Vol 43 (3) ◽  
pp. 301-313 ◽  
Author(s):  
Jerzy Ombach
Keyword(s):  
Metric Space ◽  
Periodic Points ◽  
Compact Metric Space ◽  
Stable And Unstable Manifolds ◽  
Hyperbolic Case ◽  
Expansive Homeomorphism ◽  
Unstable Manifolds

Let f be an expansive homeomorphism with the pseudo orbits tracing property on a compact metric space. There are stable and unstable “manifolds” with similar properties as in the hyperbolic case, and similar behavior near periodic points is observed. Per (f) = Ω(f) = CR(f). Mappings Ω and CR are continuous at f.

scholarly journals A counter example on common periodic points of functions

1998 ◽  
Vol 21 (4) ◽  
pp. 823-827 ◽  
Author(s):  
Aliasghar Alikhani-Koopaei
Keyword(s):  
Periodic Point ◽  
Metric Space ◽  
Periodic Points ◽  
Continuous Functions ◽  
Compact Metric Space ◽  
Counter Example

By a counter example we show that two continuous functions defined on a compact metric space satisfying a certain semi metric need not have a common periodic point.

scholarly journals A theorem on homeomorphism groups and products of spaces

1969 ◽  
Vol 1 (1) ◽  
pp. 137-141 ◽  
Author(s):  
A. R. Vobach
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Structure Theorem ◽  
Cantor Set ◽  
Compact Metric Space ◽  
Compact Metric Spaces ◽  
Group Of Homeomorphisms ◽  
Homeomorphism Groups

Let H(C) be the group of homeomorphisms of the Cantor set, C, onto itself. Let p: C → M be a map of C onto a compact metric space M, and let G(p, M) be is a group.The map p: C → M is standard, if for each (x, y) ∈ C × C such that p(x) = p(y), there is a sequence and a sequence such that xn → x and hn (xn) → y Standard maps and their associated groups characterize compact metric spaces in the sense that: Two such spaces, M and N, are homeomorphic if and only if, given p standard from C onto M, there is a standard q from C onto N for which G(p, M) = h−1G(q, N)h, for some h ∈ H(C) The present paper exhibits a structure theorem connecting these characterizing subgroups of H(C) and products of spaces: Let M1 and M2 be compact metric spaces. Then there are standard maps p: C → M1 × M2 and pi: C → Mi, i = 1, 2, such that G(p, M1 × M2) = G(p1, M1) ∩ G(p2, M2).

Weakly Confluent Mappings and Atriodic Suslinian Curves

1978 ◽  
Vol 30 (01) ◽  
pp. 32-44 ◽  
Author(s):  
H. Cook ◽  
A. Lelek
Keyword(s):  
Continuous Function ◽  
Branch Point ◽  
Metric Space ◽  
Related Result ◽  
Cantor Set ◽  
Topological Spaces ◽  
Compact Metric Space ◽  
Covering Theorem ◽  
The Given

There are theorems in which some classes of topological spaces are characterized by means of properties of mappings of these spaces into a single space. For example, it is well known that a compactum X is at most n-dimensional if and only if no mapping of X irto an (n + l)-cube has a stable value [5, Theorems VI. 1-2, pp. 75-77]. Also, a curve X is tree-like if and only if no mapping of X into a figure eight is homotopically essential [1, Theorem 1, pp. 74-75; 8, p. 91]. By a curve we mean any at most 1-dimensional continuum; a continuum is a connected compactum; a compactum is a compact metric space, and a mapping is a continuous function. The aim of the present paper is to prove another theorem of this type. We distinguish a class of curves and show that it is characterized by imposing the condition that no weakly confluent mapping [13] can transform the given curve onto a simple triod (see 2.4). A related result is applied to a generalized branch-point covering theorem (see 3.2). In addition, two results are obtained in which we establish some characterizations of weakly confluent images and preimages of the product of the Cantor set and an arc (see 1.1 and 2.2). Continua that are such images turn out to be identical with regular curves (see 1.3).

Sign in / Sign up

Export Citation Format

Share Document