Minimal Sets on Graphs and Dendrites

In this paper, we describe in detail the global and cocycle attractors related to nonautonomous scalar differential equations with diffusion. In particular, we investigate reaction–diffusion equations with almost-periodic coefficients. The associated semiflows are strongly monotone which allow us to give a full characterization of the cocycle attractor. We prove that, when the upper Lyapunov exponent associated to the linear part of the equations is positive, the flow is persistent in the positive cone, and we study the stability and the set of continuity points of the section of each minimal set in the global attractor for the skew product semiflow. We illustrate our result with some nontrivial examples showing the richness of the dynamics on this attractor, which in some situations shows internal chaotic dynamics in the Li–Yorke sense. We also include the sublinear and concave cases in order to go further in the characterization of the attractors, including, for instance, a nonautonomous version of the Chafee–Infante equation. In this last case we can show exponentially forward attraction to the cocycle (pullback) attractors in the positive cone of solutions.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127495001113 ◽

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.

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Dynamics of homeomorphisms on minimal sets generated by triangular mappings

Bulletin of the Australian Mathematical Society ◽

10.1017/s000497270003255x ◽

1999 ◽

Vol 59 (1) ◽

pp. 1-20 ◽

Cited By ~ 34

Author(s):

Gian Luigi Forti ◽

Luigi Paganoni ◽

Jaroslav Smítal

Keyword(s):

Topological Entropy ◽

Minimal Set ◽

Minimal Sets ◽

Continuous Maps ◽

Triangular Maps ◽

Parametric Class

The main goal of the paper is the construction of a triangular mapping F of the square with zero topological entropy, possessing a minimal set M such that F|M is a strongly chaotic homeomorphism, as well as other properties that are impossible for continuous maps on an interval.To do this we define a parametric class of triangular maps on Q × I, where Q is an infinite minimal set on the interval, which are extendable to continuous triangular maps F: I2 → I2. This class can be used to create other examples.

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Planar Lattice Subsets with Minimal Vertex Boundary

The Electronic Journal of Combinatorics ◽

10.37236/10055 ◽

2021 ◽

Vol 28 (3) ◽

Author(s):

Radhika Gupta ◽

Ivan Levcovitz ◽

Alexander Margolis ◽

Emily Stark

Keyword(s):

Integer Lattice ◽

Geometric Characterization ◽

Planar Lattice ◽

Minimal Set ◽

Efficient Sets ◽

Minimal Sets ◽

Large Size ◽

Maximal Size ◽

Infinite Component

A subset of vertices of a graph is minimal if, within all subsets of the same size, its vertex boundary is minimal. We give a complete, geometric characterization of minimal sets for the planar integer lattice $X$. Our characterization elucidates the structure of all minimal sets, and we are able to use it to obtain several applications. We show that the neighborhood of a minimal set is minimal. We characterize uniquely minimal sets of $X$: those which are congruent to any other minimal set of the same size. We also classify all efficient sets of $X$: those that have maximal size amongst all such sets with a fixed vertex boundary. We define and investigate the graph $G$ of minimal sets whose vertices are congruence classes of minimal sets of $X$ and whose edges connect vertices which can be represented by minimal sets that differ by exactly one vertex. We prove that G has exactly one infinite component, has infinitely many isolated vertices and has bounded components of arbitrarily large size. Finally, we show that all minimal sets, except one, are connected.

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Sums of standard uniform random variables

Journal of Applied Probability ◽

10.1017/jpr.2019.52 ◽

2019 ◽

Vol 56 (3) ◽

pp. 918-936 ◽

Cited By ~ 1

Author(s):

Tiantian Mao ◽

Bin Wang ◽

Ruodu Wang

Keyword(s):

Random Variables ◽

Complete Characterization ◽

Partial Characterization ◽

Challenging Problem ◽

Marginal Distributions ◽

Characterization Result ◽

Full Characterization

AbstractIn this paper, we analyse the set of all possible aggregate distributions of the sum of standard uniform random variables, a simply stated yet challenging problem in the literature of distributions with given margins. Our main results are obtained for two distinct cases. In the case of dimension two, we obtain four partial characterization results. For dimension greater than or equal to three, we obtain a full characterization of the set of aggregate distributions, which is the first complete characterization result of this type in the literature for any choice of continuous marginal distributions.

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Minimality, transitivity, mixing and topological entropy on spaces with a free interval

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385712000442 ◽

2012 ◽

Vol 33 (6) ◽

pp. 1786-1812 ◽

Cited By ~ 11

Author(s):

MATÚŠ DIRBÁK ◽

ĽUBOMÍR SNOHA ◽

VLADIMÍR ŠPITALSKÝ

Keyword(s):

Topological Entropy ◽

Open Subset ◽

Topological Structure ◽

Open Interval ◽

Topological Transitivity ◽

Topological Mixing ◽

Minimal Sets ◽

Free Interval ◽

Continuous Maps ◽

Metrizable Spaces

AbstractWe study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e. an open subset homeomorphic to an open interval). Special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.

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A Thorough Theoretical Exploration of Intriguing Characteristics of Cyclo[18]carbon: Geometry, Bonding Nature, Aromaticity, Weak Interaction, Reactivity, Excited States, Vibrations, Molecular Dynamics and Various Molecular Properties

10.26434/chemrxiv.11320130.v1 ◽

2019 ◽

Cited By ~ 2

Author(s):

Tian Lu ◽

Qinxue Chen ◽

Zeyu Liu

Keyword(s):

Molecular Dynamics ◽

Excited States ◽

Electronic Excitation ◽

Ring Structure ◽

Molecular Properties ◽

Molecular Vibration ◽

Full Characterization ◽

Long Time ◽

Almost All

Although cyclo[18]carbon has been theoretically and experimentally investigated since long time ago, only very recently it was prepared and directly observed by means of STM/AFM in condensed phase (Kaiser et al., Science, 365, 1299 (2019)). The unique ring structure and dual 18-center π delocalization feature bring a variety of unusual characteristics and properties to the cyclo[18]carbon, which are quite worth to be explored. In this work, we present an extremely comprehensive and detailed investigation on almost all aspects of the cyclo[18]carbon, including (1) Geometric characteristics (2) Bonding nature (3) Electron delocalization and aromaticity (4) Intermolecular interaction (5) Reactivity (6) Electronic excitation and UV/Vis spectrum (7) Molecular vibration and IR/Raman spectrum (8) Molecular dynamics (9) Response to external field (10) Electron ionization, affinity and accompanied process (11) Various molecular properties. We believe that our full characterization of the cyclo[18]carbon will greatly deepen researchers' understanding of this system, and thereby help them to utilize it in practice and design its various valuable derivatives.

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A Thorough Theoretical Exploration of Intriguing Characteristics of Cyclo[18]carbon: Geometry, Bonding Nature, Aromaticity, Weak Interaction, Reactivity, Excited States, Vibrations, Molecular Dynamics and Various Molecular Properties

10.26434/chemrxiv.11320130 ◽

2019 ◽

Cited By ~ 2

Author(s):

Tian Lu ◽

Qinxue Chen ◽

Zeyu Liu

Keyword(s):

Molecular Dynamics ◽

Excited States ◽

Electronic Excitation ◽

Ring Structure ◽

Molecular Properties ◽

Molecular Vibration ◽

Full Characterization ◽

Long Time ◽

Almost All

Although cyclo[18]carbon has been theoretically and experimentally investigated since long time ago, only very recently it was prepared and directly observed by means of STM/AFM in condensed phase (Kaiser et al., Science, 365, 1299 (2019)). The unique ring structure and dual 18-center π delocalization feature bring a variety of unusual characteristics and properties to the cyclo[18]carbon, which are quite worth to be explored. In this work, we present an extremely comprehensive and detailed investigation on almost all aspects of the cyclo[18]carbon, including (1) Geometric characteristics (2) Bonding nature (3) Electron delocalization and aromaticity (4) Intermolecular interaction (5) Reactivity (6) Electronic excitation and UV/Vis spectrum (7) Molecular vibration and IR/Raman spectrum (8) Molecular dynamics (9) Response to external field (10) Electron ionization, affinity and accompanied process (11) Various molecular properties. We believe that our full characterization of the cyclo[18]carbon will greatly deepen researchers' understanding of this system, and thereby help them to utilize it in practice and design its various valuable derivatives.

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