Related papers: Global Saddles for Planar Maps
The Nielsen-Thurston theory of surface diffeomorphisms shows that useful dynamical information can be obtained about a surface diffeomorphism from a finite collection of periodic orbits.In this paper, we extend these results to homoclinic…
In this paper, we investigate the existence and uniqueness of fixed points for self-mappings defined on bipolar metric spaces using a new class of contractive conditions, namely polynomial-type contractions. Our main results establish…
There are many tools for studying local dynamics. An important problem is how this information can be used to obtain global information. We present examples for which local stability does not carry on globally. To this purpose we construct,…
This paper contains theory on two related topics relevant to manifolds of normally hyperbolic singularities. First, theorems on the formal and $ C^k $ normal forms for these objects are proved. Then, the theorems are applied to give…
This paper presents a generalization to image matching of the Hamiltonian approach for planar curve matching developed in the context of group of diffeomorphisms. We propose an efficient framework to deal with discontinuous images in any…
A fixed point theorem is proved for inverse transducers, leading to an automata-theoretic proof of the fixed point subgroup of an endomorphism of a finitely generated virtually free group being finitely generated. If the endomorphism is…
The main objective of this paper is to show that balls under invariant metrics on hyperbolic planar domains are finitely-connected. As applications, we give new and transparent proofs of classical results on conformal mappings of planar…
Inertial manifold theory, saddle point property and exponential dichotomy have been treated as different topics in the literature with different proofs. As a common feature, they all have the purpose of `splitting' the space to understand…
A point set $S \subseteq \mathbb{R}^2$ is universal for a class $\cal G$ if every graph of ${\cal G}$ has a planar straight-line embedding on $S$. It is well-known that the integer grid is a quadratic-size universal point set for planar…
We prove a generalization of the fixed point theorem of Cartwright and Littlewood. Namely, suppose $h : \mathbb{R}^2 \to\mathbb{R}^2$ is an orientation preserving planar homeomorphism, and let $C$ be a continuum such that $h^{-1}(C)\cup C$…
We prove that if $G\subset\text{Diff}^{1}(\mathbb{R}^2)$ is an Abelian subgroup generated by a family of commuting diffeomorphisms of the plane, all of which are $C^{1}$-close to the identity in the strong $C^{1}$-topology, and if there…
In this paper we consider maps on the plane which are similar to quadratic maps in that they are degree 2 branched covers of the plane. In fact, consider for $\alpha$ fixed, maps $f_c$ which have the following form (in polar coordinates):…
We consider semidifferentiable (possibly nonsmooth) maps, acting on a subset of a Banach space, that are nonexpansive either in the norm of the space or in the Hilbert's or Thompson's metric inherited from a convex cone. We show that the…
Given a saddle fixed point of a surface diffeomorphism, its stable and unstable curves $W^S$ and $W^U$ often form a homoclinic tangle. Given such a tangle, we use topological methods to find periodic points of the diffeomorphism, using only…
Let $(f_\lambda)_{\lambda\in \Lambda}$ be a holomorphic family of polynomial automorphisms of $\mathbb{C}^2$. Following previous work of Dujardin and Lyubich, we say that such a family is weakly stable if saddle periodic orbits do not…
A folklore result due to M.W. Hirsch states that most competitive maps admit a carrying simplex, i.e., an invariant hypersurface which attracts all nontrivial orbits. The common approach in the study of these maps is to focus on the…
We construct two planar homeomorphisms $f$ and $g$ for which the origin is a globally asymptotically stable fixed point whereas for $f \circ g$ and $g \circ f$ the origin is a global repeller. Furthermore, the origin remains a global…
A well-known result from Brouwer states that any orientation preserving homeomorphism of the plane with no fixed points has an empty non-wandering set. In particular, an invariant compact set implies the existence of a fixed point. In this…
In this paper, we study the existence of fixed points for mappings defined on complete metric space (X, d) satisfying a general contractive inequality of integral type depended on another function. This conditions is analogous of Banach…
Let $G$ be the group of orientation-preserving isometries of a rank-one symmetric space $X$ of non-compact type. We study local rigidity of certain actions of a solvable subgroup $\Gamma \subset G$ on the boundary of $X$, which is…