scholarly journals Chain recurrence and structure of \begin{document}$ \omega $\end{document}-limit sets of multivalued semiflows

2020 ◽  
Vol 19 (4) ◽  
pp. 2197-2217
Author(s):  
Olexiy V. Kapustyan ◽  
◽  
Pavlo O. Kasyanov ◽  
José Valero ◽  
◽  
...  
Keyword(s):  
Chain Recurrence ◽  
Limit Sets

scholarly journals Chain recurrence and discretisation

1997 ◽  
Vol 55 (1) ◽  
pp. 63-71 ◽  
Author(s):  
Barnabas M. Garay ◽  
Josef Hofbauer
Keyword(s):  
Lower Semicontinuity ◽  
Variable Stepsize ◽  
Chain Recurrence ◽  
Limit Sets ◽  
Chain Recurrent Set ◽  
Numerical Dynamics ◽  
Upper And Lower Semicontinuity ◽  
Recurrent Set

Upper and lower semicontinuity results for the chain recurrent set are shown to remain valid in numerical dynamics with constant stepsizes. It is also pointed out that the chain recurrent set contains numerical ω–limit sets for discretisations with a variable stepsize sequence approaching zero.

Chain recurrence in flows on the Klein bottle

1999 ◽  
Vol 127 (1) ◽  
pp. 83-91 ◽  
Author(s):  
ATHANASSIOS ALEXELLIS ◽  
KONSTANTIN ATHANASSOPOULOS
Keyword(s):  
Klein Bottle ◽  
Finite Graph ◽  
Invariant Chain ◽  
Dimensional Chain ◽  
Chain Recurrence ◽  
Limit Sets ◽  
Regular Covering ◽  
Carry Over ◽  
Object Of Study ◽  
Negative Time

In the classical Poincaré–Bendixson theory the object of study are the limit sets of a continuous flow on the 2-sphere S2 and the behaviour of the orbits near them (see [7, 9]). In [2] the second author proved that an assertion similar to the Poincaré–Bendixson theorem is true in the wider class of the 1-dimensional invariant (internally) chain recurrent continua of flows on S2. On the other hand, it is known that among the closed 2-manifolds, the 2-sphere S2, the projective plane RP2 and the Klein bottle K2 are the only ones for which the Poincaré–Bendixson theorem is true (see [1, 8, 11]).The motivation of the present paper was to examine to what extent the main results of [2] carry over to flows on RP2 and K2. A first attempt to study chain recurrent sets of flows on closed 2-manifolds other than the 2-sphere was [3]. As one expects, the results of [2] carry over easily to RP2, since chain recurrence behaves well with respect to regular covering maps of compact manifolds, as we show in Section 3. The situation with K2 is quite different, since it is doubly covered by the 2-torus T2, where we have no Poincaré–Bendixson theorem. Actually, the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. For example, identifying suitably the boundary periodic orbits of a 2-dimensional Reeb flow on a closed annulus (see [7, chapter III, 2·6]) we get a flow on K2 with a 1-dimensional invariant chain recurrent continuum consisting of the unique periodic orbit and another orbit, which spirals against it in positive and negative time. As we prove in Theorem 4·4, this situation, or concatenations of it, is the only one where the Poincaré–Bendixson theorem for 1-dimensional invariant chain recurrent continua of flows on K2 is not true. Then, we are concerned with the topological structure of the 1-dimensional chain components of a flow on K2 with finitely many singularities. In Proposition 4·6 we find when such a set consists of finitely many orbits and is homeomorphic to a finite graph. An example shows that the hypothesis of Proposition 4·6 is essential. Finally, in Theorem 4·9 we give a description of the structure of the 1-dimensional chain components of a flow on K2 with finitely many singular points.

Length distortion and the Hausdorff dimension of limit sets

2000 ◽  
Vol 122 (3) ◽  
pp. 465-482 ◽  
Author(s):  
Martin Bridgeman ◽  
Edward C. Taylor
Keyword(s):  
Hausdorff Dimension ◽  
Limit Sets ◽  
Length Distortion

scholarly journals A non-Borel special alpha-limit set in the square

2021 ◽  
pp. 1-11
Author(s):  
STEPHEN JACKSON ◽  
BILL MANCE ◽  
SAMUEL ROTH
Keyword(s):  
Dynamical Systems ◽  
Limit Set ◽  
Limit Sets ◽  
Unit Square

Abstract We consider the complexity of special $\alpha $ -limit sets, a kind of backward limit set for non-invertible dynamical systems. We show that these sets are always analytic, but not necessarily Borel, even in the case of a surjective map on the unit square. This answers a question posed by Kolyada, Misiurewicz, and Snoha.

scholarly journals Limit Behavior of Solutions of Ordinary Linear Differential Equations

1994 ◽  
Vol 1 (3) ◽  
pp. 315-323
Author(s):  
František Neuman
Keyword(s):  
Differential Equations ◽  
Oscillatory Behavior ◽  
Linear Differential Equations ◽  
Limit Behavior ◽  
Limit Sets ◽  
Equivalent Linear ◽  
Linear Differential ◽  
Ordinary Linear Differential Equations

Abstract A classification of classes of equivalent linear differential equations with respect to ω-limit sets of their canonical representatives is introduced. Some consequences of this classification to the oscillatory behavior of solution spaces are presented.

scholarly journals A Full Description of ω-Limit Sets of Cournot Maps Having Non-Empty Interior and Some Economic Applications

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 452
Author(s):  
Antonio Linero-Bas ◽  
María Muñoz-Guillermo
Keyword(s):  
Economic Models ◽  
Full Description ◽  
Cournot Model ◽  
Empty Interior ◽  
Limit Sets ◽  
Economic Applications

Given a continuous Cournot map F(x,y)=(f2(y),f1(x)) defined from I2=[0,1]×[0,1] into itself, we give a full description of its ω-limit sets with non-empty interior. Additionally, we present some partial results for the empty interior case. The distribution of the ω-limits with non-empty interior gives information about the dynamics and the possible outputs of each firm in a Cournot model. We present some economic models to illustrate, with examples, the type of ω-limits that appear.

scholarly journals Omega-limit sets and robust stability for switched systems with distinct equilibria

2020 ◽  
Vol 53 (2) ◽  
pp. 2039-2044
Author(s):  
Matina Baradaran ◽  
Andrew R. Teel
Keyword(s):  
Robust Stability ◽  
Switched Systems ◽  
Limit Sets

scholarly journals Graph directed Markov systems on Hilbert spaces

2009 ◽  
Vol 147 (2) ◽  
pp. 455-488 ◽  
Author(s):  
R. D. MAULDIN ◽  
T. SZAREK ◽  
M. URBAŃSKI
Keyword(s):  
Hausdorff Measure ◽  
Iterated Function System ◽  
Sufficient Conditions ◽  
Topological Pressure ◽  
Limit Set ◽  
Covering Theorem ◽  
Limit Sets ◽  
Markov Systems ◽  
Polish Spaces ◽  
Iterated Function

AbstractWe deal with contracting finite and countably infinite iterated function systems acting on Polish spaces, and we introduce conformal Graph Directed Markov Systems on Polish spaces. Sufficient conditions are provided for the closure of limit sets to be compact, connected, or locally connected. Conformal measures, topological pressure, and Bowen's formula (determining the Hausdorff dimension of limit sets in dynamical terms) are introduced and established. We show that, unlike the Euclidean case, the Hausdorff measure of the limit set of a finite iterated function system may vanish. Investigating this issue in greater detail, we introduce the concept of geometrically perfect measures and provide sufficient conditions for geometric perfectness. Geometrical perfectness guarantees the Hausdorff measure of the limit set to be positive. As a by–product of the mainstream of our investigations we prove a 4r–covering theorem for all metric spaces. It enables us to establish appropriate co–Frostman type theorems.

scholarly journals A variational approach to the alternating projections method

2021 ◽  
Author(s):  
Carlo Alberto De Bernardi ◽  
Enrico Miglierina
Keyword(s):  
Convex Sets ◽  
Nonempty Intersection ◽  
Feasibility Problem ◽  
Alternating Projections ◽  
Limit Sets ◽  
Von Neumann ◽  
Convex Feasibility ◽  
Starting Point ◽  
Method Of Alternating Projections ◽  
Alternating Projections Method

AbstractThe 2-sets convex feasibility problem aims at finding a point in the nonempty intersection of two closed convex sets A and B in a Hilbert space H. The method of alternating projections is the simplest iterative procedure for finding a solution and it goes back to von Neumann. In the present paper, we study some stability properties for this method in the following sense: we consider two sequences of closed convex sets $$\{A_n\}$$ { A n } and $$\{B_n\}$$ { B n } , each of them converging, with respect to the Attouch-Wets variational convergence, respectively, to A and B. Given a starting point $$a_0$$ a 0 , we consider the sequences of points obtained by projecting on the “perturbed” sets, i.e., the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } given by $$b_n=P_{B_n}(a_{n-1})$$ b n = P B n ( a n - 1 ) and $$a_n=P_{A_n}(b_n)$$ a n = P A n ( b n ) . Under appropriate geometrical and topological assumptions on the intersection of the limit sets, we ensure that the sequences $$\{a_n\}$$ { a n } and $$\{b_n\}$$ { b n } converge in norm to a point in the intersection of A and B. In particular, we consider both when the intersection $$A\cap B$$ A ∩ B reduces to a singleton and when the interior of $$A \cap B$$ A ∩ B is nonempty. Finally we consider the case in which the limit sets A and B are subspaces.

scholarly journals Free vs. locally free Kleinian groups

2019 ◽  
Vol 2019 (746) ◽  
pp. 149-170
Author(s):  
Pekka Pankka ◽  
Juan Souto
Keyword(s):  
Hausdorff Dimension ◽  
Kleinian Group ◽  
Cantor Set ◽  
Cantor Sets ◽  
Kleinian Groups ◽  
The Other ◽  
Limit Set ◽  
Limit Sets ◽  
Other Hand

Abstract We prove that Kleinian groups whose limit sets are Cantor sets of Hausdorff dimension < 1 are free. On the other hand we construct for any ε > 0 an example of a non-free purely hyperbolic Kleinian group whose limit set is a Cantor set of Hausdorff dimension < 1 + ε.

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