scholarly journals Minimal set of periods for continuous self-maps of the eight space

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jaume Llibre ◽  
Ana Sá
Keyword(s):  
Quotient Space ◽  
Single Point ◽  
Minimal Set ◽  
Circle Maps ◽  
Continuous Map ◽  
Branching Point

AbstractLet $G_{k}$ G k be a bouquet of circles, i.e., the quotient space of the interval $[0,k]$ [ 0 , k ] obtained by identifying all points of integer coordinates to a single point, called the branching point of $G_{k}$ G k . Thus, $G_{1}$ G 1 is the circle, $G_{2}$ G 2 is the eight space, and $G_{3}$ G 3 is the trefoil. Let $f: G_{k} \to G_{k}$ f : G k → G k be a continuous map such that, for $k>1$ k > 1 , the branching point is fixed.If $\operatorname{Per}(f)$ Per ( f ) denotes the set of periods of f, the minimal set of periods of f, denoted by $\operatorname{MPer}(f)$ MPer ( f ) , is defined as $\bigcap_{g\simeq f} \operatorname{Per}(g)$ ⋂ g ≃ f Per ( g ) where $g:G_{k}\to G_{k}$ g : G k → G k is homological to f.The sets $\operatorname{MPer}(f)$ MPer ( f ) are well known for circle maps. Here, we classify all the sets $\operatorname{MPer}(f)$ MPer ( f ) for self-maps of the eight space.

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 A CHARACTERIZATION OF THE SET Ω (f)\ ω (f) FOR CONTINUOUS MAPS OF THE INTERVAL WITH ZERO TOPOLOGICAL ENTROPY

1995 ◽  
Vol 05 (05) ◽  
pp. 1433-1435
Author(s):  
F. BALIBREA ◽  
J. SMÍTAL
Keyword(s):  
Topological Entropy ◽  
Minimal Set ◽  
Continuous Map ◽  
Continuous Maps ◽  
Limit Points ◽  
Infinite Set ◽  
Nonwandering Points

We give a characterization of the set of nonwandering points of a continuous map f of the interval with zero topological entropy, attracted to a single (infinite) minimal set Q. We show that such a map f can have a unique infinite minimal set Q and an infinite set B ⊂ Ω (f)\ ω (f) (of nonwandering points that are not ω-limit points) attracted to Q and such that B has infinite intersections with infinitely many disjoint orbits of f.

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 preimage entropy, folding entropy and stable entropy

2020 ◽  
pp. 1-33 ◽  
Author(s):  
WEISHENG WU ◽  
YUJUN ZHU
Keyword(s):  
Dynamical Systems ◽  
Variational Principle ◽  
Invariant Measures ◽  
Single Point ◽  
Nonequilibrium Statistical Mechanics ◽  
Stat Phys ◽  
Metric Entropy ◽  
Compact Metric Space ◽  
Continuous Map ◽  
Partially Hyperbolic

For non-invertible dynamical systems, we investigate how ‘non-invertible’ a system is and how the ‘non-invertibility’ contributes to the entropy from different viewpoints. For a continuous map on a compact metric space, we propose a notion of pointwise metric preimage entropy for invariant measures. For systems with uniform separation of preimages, we establish a variational principle between this version of pointwise metric preimage entropy and pointwise topological entropies introduced by Hurley [On topological entropy of maps. Ergod. Th. & Dynam. Sys.15 (1995), 557–568], which answers a question considered by Cheng and Newhouse [Pre-image entropy. Ergod. Th. & Dynam. Sys.25 (2005), 1091–1113]. Under the same condition, the notion coincides with folding entropy introduced by Ruelle [Positivity of entropy production in nonequilibrium statistical mechanics. J. Stat. Phys.85(1–2) (1996), 1–23]. For a $C^{1}$ -partially hyperbolic (non-invertible and non-degenerate) endomorphism on a closed manifold, we introduce notions of stable topological and metric entropies, and establish a variational principle relating them. For $C^{2}$ systems, the stable metric entropy is expressed in terms of folding entropy (namely, pointwise metric preimage entropy) and negative Lyapunov exponents. Preimage entropy could be regarded as a special type of stable entropy when each stable manifold consists of a single point. Moreover, we also consider the upper semi-continuity for both of pointwise metric preimage entropy and stable entropy and give a version of the Shannon–McMillan–Breiman theorem for them.

scholarly journals Equicontinuity of iterates of circle maps

1998 ◽  
Vol 21 (3) ◽  
pp. 453-458 ◽  
Author(s):  
Antonios Valaristos
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Circle Maps ◽  
Continuous Map ◽  
The Family ◽  
Necessary And Sufficient

Letfbe a continuous map of the circle to itself. Necessary and sufficient conditions are given for the family ofiterates{fn}n=1∞to be equicontinuous.

Proper minimal sets on compact connected 2-manifolds are nowhere dense

2008 ◽  
Vol 28 (3) ◽  
pp. 863-876 ◽  
Author(s):  
SERGII˘ KOLYADA ◽  
L’UBOMÍR SNOHA ◽  
SERGEI˘ TROFIMCHUK
Keyword(s):  
Dynamical System ◽  
Recent Result ◽  
Dense Subset ◽  
Klein Bottle ◽  
Dimensional Manifold ◽  
Two Dimensional ◽  
Minimal Set ◽  
Minimal Sets ◽  
Continuous Map

AbstractLet $\mathcal {M}^2$ be a compact connected two-dimensional manifold, with or without boundary, and let $f:{\mathcal {M}}^2\to \mathcal {M}^2$ be a continuous map. We prove that if $M \subseteq \mathcal {M}^2$ is a minimal set of the dynamical system $(\mathcal {M}^2,f)$ then either $M = \mathcal {M}^2$ or M is a nowhere dense subset of $\mathcal {M}^2$. Moreover, we add a shorter proof of the recent result of Blokh, Oversteegen and Tymchatyn, that in the former case $\mathcal {M}^2$ is a torus or a Klein bottle.

Fine-scale taxonomic and temporal variability in the energy density of invertebrate prey of juvenile Chinook salmon Oncorhynchus tshawytscha

2020 ◽  
Vol 655 ◽  
pp. 185-198
Author(s):  
J Weil ◽  
WDP Duguid ◽  
F Juanes
Keyword(s):  
Energy Density ◽  
Chinook Salmon ◽  
Temporal Variability ◽  
Energy Content ◽  
Single Point ◽  
Single Species ◽  
Fine Scale ◽  
Temporal Differences ◽  
Juvenile Chinook Salmon ◽  
Juvenile Chinook

Variation in the energy content of prey can drive the diet choice, growth and ultimate survival of consumers. In Pacific salmon species, obtaining sufficient energy for rapid growth during early marine residence is hypothesized to reduce the risk of size-selective mortality. In order to determine the energetic benefit of feeding choices for individuals, accurate estimates of energy density (ED) across prey groups are required. Frequently, a single species is assumed to be representative of a larger taxonomic group or related species. Further, single-point estimates are often assumed to be representative of a group across seasons, despite temporal variability. To test the validity of these practices, we sampled zooplankton prey of juvenile Chinook salmon to investigate fine-scale taxonomic and temporal differences in ED. Using a recently developed model to estimate the ED of organisms using percent ash-free dry weight, we compared energy content of several groups that are typically grouped together in growth studies. Decapod megalopae were more energy rich than zoeae and showed family-level variability in ED. Amphipods showed significant species-level variability in ED. Temporal differences were observed, but patterns were not consistent among groups. Bioenergetic model simulations showed that growth rate of juvenile Chinook salmon was almost identical when prey ED values were calculated on a fine scale or on a taxon-averaged coarse scale. However, single-species representative calculations of prey ED yielded highly variable output in growth depending on the representative species used. These results suggest that the latter approach may yield significantly biased results.

Modeling and design of role engineering in development of access control for dynamic information systems

2013 ◽  
Vol 61 (3) ◽  
pp. 569-579 ◽  
Author(s):  
A. Poniszewska-Marańda
Keyword(s):  
Information Systems ◽  
Dynamic Information ◽  
Minimal Set ◽  
Distributed Information ◽  
Heterogeneous Information ◽  
System Protection ◽  
Level Model ◽  
Model Independent ◽  
Role Concept ◽  
High Level

Abstract Nowadays, the growth and complexity of functionalities of current information systems, especially dynamic, distributed and heterogeneous information systems, makes the design and creation of such systems a difficult task and at the same time, strategic for businesses. A very important stage of data protection in an information system is the creation of a high level model, independent of the software, satisfying the needs of system protection and security. The process of role engineering, i.e. the identification of roles and setting up in an organization is a complex task. The paper presents the modeling and design stages in the process of role engineering in the aspect of security schema development for information systems, in particular for dynamic, distributed information systems, based on the role concept and the usage concept. Such a schema is created first of all during the design phase of a system. Two actors should cooperate with each other in this creation process, the application developer and the security administrator, to determine the minimal set of user’s roles in agreement with the security constraints that guarantee the global security coherence of the system.

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