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Let G be a group and let V be an algebraic variety over an algebraically closed field K. Let A denote the set of K -points of V. We introduce algebraic sofic subshifts ${\Sigma \subset A^G}$ and study endomorphisms $\tau \colon \Sigma \to \Sigma $. We generalize several results for dynamical invariant sets and nilpotency of $\tau $ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $\tau $ is nilpotent if and only if its limit set, that is, the intersection of the images of its iterates, is a singleton. If moreover G is infinite, finitely generated and $\Sigma $ is topologically mixing, we show that $\tau $ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values. [ABSTRACT FROM AUTHOR]

We consider the 3-sphere S³ seen as the boundary at infinity of the complex hyperbolic plane H²C. It comes equipped with a contact structure and two classes of special curves. First, R-circles are boundaries at infinity of totally real totally geodesic subspaces and are tangent to the contact distribution. Second, C-circles are boundaries of complex totally geodesic subspaces and are transverse to the contact distribution. We define a quantitative notion, called slimness, that measures to what extent a continuous path in the sphere S³ is near to being an R-circle. We analyse the classical foliation of the complement of an R-circle by arcs of C-circles. Next, we consider deformations of this situation where the R-circle becomes a slim curve. We apply these concepts to the particular case where the slim curve is the limit set of a quasi-Fuchsian subgroup of PU.2; 1/. As an application, we describe a class of spherical CR uniformisations of certain cusped 3-manifolds. [ABSTRACT FROM AUTHOR]

This paper tightens the classical Poincaré–Bendixson theory for a positively invariant, simply-connected compact set $\mathcal M$ in a continuously differentiable planar vector field by further characterizing for any point $p\in \mathcal M$, the composition of the limit sets $\omega (p)$ and $\alpha(p)$ after counting separately the fixed points on $\mathcal M$'s boundary and interior. In particular, when $\mathcal M$ contains finitely many boundary but no interior fixed points, $\omega (p)$ contains only a single fixed point, and when $\mathcal M$ may have infinitely many boundary but no interior fixed points, $\omega (p)$ can, in addition, be a continuum of fixed points. When $\mathcal M$ contains only one interior and finitely many boundary fixed points, $\omega (p)$ or $\alpha (p)$ contains exclusively a fixed point, a closed orbit or the union of the interior fixed point and homoclinic orbits joining it to itself. When $\mathcal M$ contains in general a finite number of fixed points and neither $\omega (p)$ nor $\alpha (p)$ is a closed orbit or contains just a fixed point, at least one of $\omega (p)$ and $\alpha (p)$ excludes all boundary fixed points and consists only of a number of the interior fixed points and orbits connecting them.

A class consisting of subharmonic functions in the unit disk such that their superpositions with some families of linear fractional automorphisms of the disk form normal families is considered. A theorem stating that for any function of class the set of points of the unit circle can be represented as a union of a set of Fatou points, a set of generalized Plesner points, and a set of measure zero is proved. [ABSTRACT FROM AUTHOR]

We study the isolated singularities of functions satisfying (E) (− Δ) s v ± | v | p − 1 v = 0 in Ω ∖ { 0 } , v = 0 in R N ∖ Ω , where 0 < s < 1 , p > 1 and Ω is a bounded domain containing the origin. We use the Caffarelli-Silvestre extension to R + × R N. We emphasize the obtention of a priori estimates and analyse the set of self-similar solutions via energy methods to characterize the singularities. [ABSTRACT FROM AUTHOR]

Given a non-empty bounded subset of hyperbolic space and a Kleinian group acting on that space, the orbital set is the orbit of the given set under the action of the group. We may view orbital sets as bounded (often fractal) subsets of Euclidean space. We prove that the upper box dimension of an orbital set is given by the maximum of three quantities: the upper box dimension of the given set, the Poincaré exponent of the Kleinian group, and the upper box dimension of the limit set of the Kleinian group. Since we do not make any assumptions about the Kleinian group, none of the terms in the maximum can be removed in general. We show by constructing an explicit example that our assumption that the given set is bounded (in the hyperbolic metric) cannot be removed in general. [ABSTRACT FROM AUTHOR]

The Assouad dimension of the limit set of a geometrically finite Kleinian group with parabolics may exceed the Hausdorff and box dimensions. The Assouad spectrum is a continuously parametrised family of dimensions which 'interpolates' between the box and Assouad dimensions of a fractal set. It is designed to reveal more subtle geometric information than the box and Assouad dimensions considered in isolation. We conduct a detailed analysis of the Assouad spectrum of limit sets of geometrically finite Kleinian groups and the associated Patterson–Sullivan measure. Our analysis reveals several novel features, such as interplay between horoballs of different rank not seen by the box or Assouad dimensions. [ABSTRACT FROM AUTHOR]

In the present work, we study the Hausdorff dimension of the limit set of Schottky groups on the boundary of the complex hyperbolic group via the Eigenvalue algorithm as in [17]. The visual geometry on H C 2 ¯ allows a direct application of the Eigenvalue algorithm, but at the same time, it hardens the computations. Using the Heisenberg structure on ∂ H C 2 \ { ∞ } and the Cygan metric, we propose a Markov partition associated with the group that simplifies the application of the Eigenvalue algorithm. [ABSTRACT FROM AUTHOR]

The study of multivariate extremes is dominated by multivariate regular variation, although it is well known that this approach does not provide adequate distinction between random vectors whose components are not always simultaneously large. Various alternative dependence measures and representations have been proposed, with the most well-known being hidden regular variation and the conditional extreme value model. These varying depictions of extremal dependence arise through consideration of different parts of the multivariate domain, and particularly through exploring what happens when extremes of one variable may grow at different rates from other variables. Thus far, these alternative representations have come from distinct sources, and links between them are limited. In this work we elucidate many of the relevant connections through a geometrical approach. In particular, the shape of the limit set of scaled sample clouds in light-tailed margins is shown to provide a description of several different extremal dependence representations. [ABSTRACT FROM AUTHOR]

In the paper we study what sets can be obtained as α α -limit sets of backward trajectories in graph maps. We show that in the case of mixing maps, all those α α -limit sets are ω ω -limit sets and for all but finitely many points x x , we can obtain every ω ω -limits set as the α α -limit set of a backward trajectory starting in x x. For zero entropy maps, every α α -limit set of a backward trajectory is a minimal set. In the case of maps with positive entropy, we obtain a partial characterization which is very close to complete picture of the possible situations. [ABSTRACT FROM AUTHOR]

Let be a flow of homeomorphisms of a connected -manifold and let be its limit set. The flow is said to be strongly reversed by a reflection if for all . In this paper, we study the dynamics of positively equicontinuous strongly reversible flows. If is nonempty, we discuss the existence of symmetric periodic orbits, and for we prove that such flows must be periodic. If is empty, we show that positively equicontinuous implies strongly reversible and strongly reversible implies parallelizable with global section the fixed point set . [ABSTRACT FROM AUTHOR]

In this paper, we show that any convex affine domain with a nonempty limit set on the boundary under the action of the identity component of the automorphism group cannot cover a compact affine manifold with a parallel volume, which is a positive answer to the Markus conjecture for convex case. Consequently, we show that the Markus conjecture is true for convex affine manifolds of dimension ≤ 5. [ABSTRACT FROM AUTHOR]

Let Γ be a discrete subgroup of PU(1, n) . In this work we look at the induced action of Γ on the Grassmannian Gr k (CP n) of all k-planes in CP n for each 0 ≤ k < n . We prove various analytic, geometric and dynamical properties of this action. In particular, we show that the equicontinuity region Eq k , n (Γ) is the complement of the union of all k-planes that either pass through a point p in the Chen–Greenberg limit set L (Γ) or are contained in the orthogonal hyperplane p ⊥ , generalizing a known theorem for the case k = 0 . We also prove that the action of Γ on Eq k , n (Γ) is properly discontinuous. [ABSTRACT FROM AUTHOR]

In this paper we classify the phase portrait and the limit sets of a special class of piecewise smooth vector fields in the plane with different configurations of straight lines as switching manifolds. We study the behavior at infinity through a Poincaré compactification as well as the relations between canonical regions and vector fields which are defined over the switching regions. Results addressing the global behavior of trajectories, tangency points, vector fields over the switching manifolds, and equilibrium and pseudo-equilibrium points at the finite and the infinite part are stated. In particular, we prove the existence of 123 distinct phase portraits for the class of piecewise smooth vector fields that we consider. [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamical system of a discontinuous Volterra operator mapping S² to itself. We find all fixed points of the operator. Besides we show that the limit sets of trajectories lie on the boundary of the simplex for some values of parameters and this set is infinite if the initial point is an inner point of the simplex. For some values of parameters we found 3-periodic points. [ABSTRACT FROM AUTHOR]

This paper deals with optimality conditions and duality theory for vector optimization involving non-convex set-valued maps. Firstly, under the assumption of nearly cone-subconvexlike property for set-valued maps, the necessary and sufficient optimality conditions in terms of limit sets are derived for local weak minimizers of a set-valued constraint optimization problem. Then, applications to Mond-Weir type and Wolfe type dual problems are presented.

This paper is devoted to the study of efficient elements for set-valued maps. We propose two new notions of relative weak ϵ -efficient element and strict relative weak ϵ -efficient element of set-valued maps and provide new necessary optimality conditions for the proposed concepts. We provide existence results for efficient elements. The critical ingredients for the existence results for efficient elements are the well-known separation arguments and Fan's lemma. As an application of the existence results, we derive relationships between the efficiency concepts and the local optimizers of certain optimization problems. [ABSTRACT FROM AUTHOR]

We study the kernel function of the operator u ↦ L μ u = − Δ u + μ | x | 2 u in a bounded smooth domain Ω ⊂ R + N such that 0 ∈ ∂ Ω , where μ ≥ − N 2 4 is a constant. We show the existence of a Poisson kernel vanishing at 0 and a singular kernel with a singularity at 0. We prove the existence and uniqueness of weak solutions of L μ u = 0 in Ω with boundary data ν + k δ 0 , where ν is a Radon measure on ∂ Ω ∖ { 0 } , k ∈ R and show that this boundary data corresponds in a unique way to the boundary trace of positive solution of L μ u = 0 in Ω. [ABSTRACT FROM AUTHOR]

It is constructed the polynomial system in R3 with the w-limit set, consisting of the union of 2m parallel straight lines. [ABSTRACT FROM AUTHOR]

Large samples from a light-tailed distribution often have a well-defined shape. This paper examines the implications of the assumption that there is a limit shape. We show that the limit shape determines the upper quantiles for a large class of random variables. These variables may be described loosely as continuous homogeneous functionals of the underlying random vector. They play an important role in evaluating risk in a multivariate setting. The paper also looks at various coefficients of tail dependence and at the distribution of the scaled sample points for large samples. The paper assumes convergence in probability rather than almost sure convergence. This results in an elegant theory. In particular, there is a simple characterization of domains of attraction. [ABSTRACT FROM AUTHOR]

The farthest point map sends a point in a compact metric space to the set of points farthest from it. We focus on the case when this metric space is a convex centrally symmetric polyhedral surface, so that we can compose the farthest point map with the antipodal map. The purpose of this work is to study the properties of their composition. We show that: 1. the map has no generalized periodic points; 2. its limit set coincides with its generalized fixed point set; 3. each of its orbit converges; 4. its limit set is contained in a finite union of hyperbolas. We will define some of these terminologies later. [ABSTRACT FROM AUTHOR]

We study the dynamics of piecewise conformal maps in the Riemann sphere. The normality and chaotic regions are defined and we state several results and properties of these sets. We show that the stability of these piecewise maps is related to the Kleinian group generated by their transformations under certain hypotheses. The general motivation of the article is to compare the dynamics of piecewise conformal maps and those of the Kleinian groups and iterations of rational maps. [ABSTRACT FROM AUTHOR]

The aim of this paper is to show that α-limit sets in Lorenz maps do not have to be completely invariant. This highlights unexpected dynamical behavior in these maps, showing gaps existing in the literature. Similar result is obtained for unimodal maps on [0,1]. On the basis of provided examples, we also present how the performed study on the structure of α-limit sets is closely connected with the calculation of the topological entropy.

In topological dynamics, the generic limit set is the smallest closed subset which has a comeager realm of attraction. We study some of its topological properties, and the links with equicontinuity and sensitivity. We emphasize the case of cellular automata, for which the generic limit set is included in all subshift attractors, and discuss directional dynamics, as well as the link with measure-theoretical similar notions. [ABSTRACT FROM AUTHOR]

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Abstract We consider the energy landscape of a dissipative Klein–Gordon lattice with a ϕ 4 on-site potential. Our analysis is based on suitable energy arguments, combined with a discrete version of the Łojasiewicz inequality, in order to justify the convergence to a single, nontrivial equilibrium for all initial configurations of the lattice. Then, global bifurcation theory is explored, to illustrate that in the discrete regime all linear states lead to nonlinear generalizations of equilibrium states. Direct numerical simulations reveal the rich structure of the equilibrium set, consisting of non-trivial topological (kink-shaped) interpolations between the adjacent minima of the on-site potential, and the wealth of dynamical convergence possibilities. These dynamical evolution results also provide insight on the potential stability of the equilibrium branches, and glimpses of the emerging global bifurcation structure, elucidating the role of the interplay between discreteness, nonlinearity and dissipation. Highlights • Analysis of the dynamical transitions between equilibria in a dissipative Klein–Gordon lattice. • Application of a discrete variant of Łojasiewicz inequality. • Applications of the global bifurcation theorem in discrete set-ups. • Numerical simulations reveal the rich structure of the equilibrium set and relevant bifurcations. [ABSTRACT FROM AUTHOR]

In this paper we give a generalization of the Conze-Guivarc’h limit set. With this definition the limit set has very similar properties to those of the limit set in hyperbolic spaces. Moreover, we prove a relation between this new limit set and the Kulkarni limit set. Additionally we show that some closed subsets can be approximated by the Conze-Guivarc’h limit set. This is a result in the theory of classic Kleinian groups. [ABSTRACT FROM AUTHOR]

We present properties of sets of invariant lines for Brouwer homeomorphisms which are not necessarily embeddable in a flow. Using such lines we describe the structure of equivalence classes of the codivergency relation. We also obtain a result concerning the set of regular points. [ABSTRACT FROM AUTHOR]

In this article we provide a classification of the projective transformations in P S L ( n + 1 , C ) considered as automorphisms of the complex projective space P C n . Our classification is an interplay between algebra and dynamics. Just as in the case of isometries of C A T ( 0 ) -spaces, this is given by means of three types of transformations, namely: elliptic, parabolic and loxodromic. We describe the dynamics in each case, more precisely we determine the corresponding Kulkarni's limit set, the equicontinuity region, minimal sets, the discontinuity region and maximal regions where the corresponding cyclic group acts properly discontinuously. We also provide, in each case, some equivalent ways to classify the projective transformations. [ABSTRACT FROM AUTHOR]

Considering the fact that the production and provision of some vaccines are ordered and governed by the government according to the history data of disease, a kind of SIR model with constant vaccination rate and impulsive state feedback control is presented. The dynamical properties of semi-continuous three-dimensional SIR system can be obtained by discussing the properties of the corresponding two-dimensional system in the limit set. The existence and uniqueness of order-1 periodic solution are discussed by using the successive function and the compression mapping theorem. A new theorem for the orbital stability of order-1 periodic solution is proved by geometric method. Finally, numerical simulations are given to verify the mathematical results and some conclusions are given. The results show that the disease can be controlled to a lower level by means of impulsive state feedback control strategy, but cannot be eradicated. [ABSTRACT FROM AUTHOR]

This paper deals with the Zhukovskij quasi-stability of an orbit in a planar impulsive dynamical system. We prove that if the positive limit set of an orbit is an asymptotically stable limit cycle (an isolated periodic orbit), then the orbit is uniformly asymptotically Zhukovskij quasi-stable. Also, we prove that if an orbit is not eventually periodic and its positive limit set is a periodic orbit, then it is asymptotically Zhukovskij quasi-stable. [ABSTRACT FROM AUTHOR]

In this study, we consider the dynamics of a fuzzy system extended by a crisp compact system, which has a mixing regular periodic decomposition. First, we describe the complete dynamics of Zadeh's extension of a crisp compact system, which has a mixing regular periodic decomposition, by splitting the space of fuzzy numbers on the compact space. Next, using the results obtained, we prove that the topological entropies of the crisp system and its Zadeh's extension to the space of fuzzy numbers are the same. [ABSTRACT FROM AUTHOR]

This paper presents the mathematical conditions and the associated design methodology of an active fault diagnosis technique for continuous-time linear systems. Given a set of faults known a priori, the system is modeled by a finite family of linear time-invariant systems, accounting for one healthy and several faulty configurations. By assuming bounded disturbances and using a residual generator, an invariant set and its projection in the residual space (i.e., its limit set) are computed for each system configuration. Each limit set, related to a single system configuration, is parameterized with respect to the system input. Thanks to this design, active fault isolation can be guaranteed by the computation of a test input, either constant or periodic, such that the limit sets associated with different system configurations are separated, and the residual converges toward one limit set only. In order to alleviate the complexity of the explicit computation of the limit set, an implicit dual representation is adopted, leading to efficient procedures, based on quadratic programming, for computing the test input. The developed methodology offers a competent continuous-time solution to the optimization-based computation of the test input via Hahn-Banach duality. Simulation examples illustrate the application of the proposed active fault diagnosis methods and its efficiency in providing a solution, even in relatively large state-dimensional problems. [ABSTRACT FROM AUTHOR]

This paper is to introduce some analytical tools to characterize the properties of fractal basin boundaries for planar switched systems (with time-dependent switching). The characterizing methods are based on the view point of limit sets and prime ends. By constructing the auxiliary dynamical system, the fractal basin boundaries of planar switched systems can be proved if every diverging path in the basin of associated auxiliary system has the entire basin boundary as its limit set. Fractal property is also verified if every prime end that is defined in the basin of associated auxiliary system is a prime end of type 3 and all other prime ends are of type 1. Bifurcations of fractal basin boundary are investigated by analyzing what types of prime ends in the basin are involved. The fractal basin boundary of switched system is also described by the indecomposable continuum. [ABSTRACT FROM AUTHOR]

For continuously differentiable vector fields, we characterize the ω limit set of a trajectory converging to a compact curve Γ ⊂ R n . In particular, the limit set is either a fixed point or a continuum of fixed points if Γ is a simple open curve; otherwise, the limit set can in addition be either a closed orbit or a number of fixed points with compatibly oriented orbits connecting them. An implication of the result is a tightened-up version of the Poincaré–Bendixson theorem. [ABSTRACT FROM AUTHOR]

In this note we prove the following law of the iterated logarithm for the Grenander estimator of a monotone decreasing density: If f ( t 0 ) > 0 , f ′ ( t 0 ) < 0 , and f ′ is continuous in a neighborhood of t 0 , then b l a lim sup n → ∞ ( n 2 log log n ) 1 / 3 ( f ̂ n ( t 0 ) − f ( t 0 ) ) = | f ( t 0 ) f ′ ( t 0 ) / 2 | 1 / 3 2 M almost surely where M ≡ sup g ∈ G T g = ( 3 / 4 ) 1 / 3 and T g ≡ argmax u { g ( u ) − u 2 } ; here G is the two-sided Strassen limit set on R . The proof relies on laws of the iterated logarithm for local empirical processes, Groeneboom’s switching relation, and properties of Strassen’s limit set analogous to distributional properties of Brownian motion; see Strassen [26] . [ABSTRACT FROM AUTHOR]