The relation between a flow and its discretization (original) (raw)
On iteration groups of singularity-free homeomorphisms of the plane
1994
Let D be a simply connected region on the plane. We prove that a continuous iteration group of homeomorphisms {/':<€ %• } defined on D is of the form f'(x) = v > 1 + tej) for z € D, t e 1, where e\ =(1,0) and <p is a homeomorphism mapping D onto 1, if and only if / is a singularity-free homeomorphism, i.e. /> =: / has the property that for every Jordan domain B C D there exists an integer n 0 such that Bn/"[S] = 0for |n| >n 0 , n € Z.
Stable and unstable manifolds of the H�non mapping
Journal of Statistical Physics, 1981
By using a parametric representation of the stable and unstable manifolds, we prove that for some given values of the parameter (in particular in the case first investigated by H~non) the H~non mapping has a transversal homoclinic orbit.
has proved a theorem about local homeomorphism of mappings with bounded distortion with the coefficient of distortion close to identity. Independently, this result was established by O. Martio, C. Rickman and J. Väisälä . In the present paper, we use ideas of V. M. Goldshtein, Yu. G. Reshetnyak, O. Martio, C. Rickman, J. Väisälä and prove a theorem about local homeomorphism provided that a nonconstant mapping with bounded distortion and the coefficient of distortion 1 on the Carnot group is a homeomorphism. The problem is that the structure of 1-quasiregular mappings is unknown on an arbitrary Carnot group. It is known ] that 1-quasiregular mappings on the one-point compactification of the Heisenberg group H n are generated by left translations, dilations, actions of a group element in SU (1, n), and the inversion The problem is still open for an arbitrary Carnot group. 1991 Mathematics Subject Classification. 22E30, 30C65, 43A80.
A structure theorem for semi-parabolic H 'enon maps
2014
Consider the parameter space mathcalPlambdasubsetmathbbC2\mathcal{P}_{\lambda}\subset \mathbb{C}^{2}mathcalPlambdasubsetmathbbC2 of complex H\'enon maps Hc,a(x,y)=(x2+c+ay,ax),aneq0H_{c,a}(x,y)=(x^{2}+c+ay,ax),\ \ a\neq 0Hc,a(x,y)=(x2+c+ay,ax),aneq0 which have a semi-parabolic fixed point with one eigenvalue lambda=e2piip/q\lambda=e^{2\pi i p/q}lambda=e2piip/q. We give a characterization of those H\'enon maps from the curve mathcalPlambda\mathcal{P}_{\lambda}mathcalPlambda that are small perturbations of a quadratic polynomial ppp with a parabolic fixed point of multiplier lambda\lambdalambda. We prove that there is an open disk of parameters in mathcalPlambda\mathcal{P}_{\lambda}mathcalPlambda for which the semi-parabolic H\'enon map has connected Julia set JJJ and is structurally stable on JJJ and J+J^{+}J+. The Julia set J+J^{+}J+ has a nice local description: inside a bidisk mathbbDrtimesmathbbDr\mathbb{D}_{r}\times \mathbb{D}_{r}mathbbDrtimesmathbbDr it is a trivial fiber bundle over JpJ_{p}Jp, the Julia set of the polynomial ppp, with fibers biholomorphic to mathbbDr\mathbb{D}_{r}mathbbDr. The Julia set JJJ is homeomorphic to a quotiented solenoid.
Connections between the stability of a Poincare map and boundedness of certain associate sequences
2011
Let m ≥ 1 and N ≥ 2 be two natural numbers and let U = {U (p, q)} p≥q≥0 be the N-periodic discrete evolution family of m × m matrices, having complex scalars as entries, generated by L(C m)-valued, N-periodic sequence of m × m matrices (A n). We prove that the solution of the following discrete problem y n+1 = A n y n + e iµn b, n ∈ Z + , y 0 = 0 is bounded for each µ ∈ R and each m-vector b if the Poincare map U (N, 0) is stable. The converse statement is also true if we add a new assumption to the boundedness condition. This new assumption refers to the invertibility for each µ ∈ R of the matrix V µ := N ν=1 U (N, ν)e iµν. By an example it is shown that the assumption on invertibility cannot be removed. Finally, a strong variant of Barbashin's type theorem is proved.
On the group of homeomorphisms on R: A revisit
Applied General Topology
In this article, we prove that the group of all increasing homeomorphisms on R has exactly five normal subgroups, and the group of all homeomorphisms on R has exactly four normal subgroups. There are several results known about the group of homeomorphisms on R and about the group of increasing homeomorphisms on R ([2], [6], [7] and [8]), but beyond this there is virtually nothing in the literature concerning the topological structure in the aspects of topological dynamics. In this paper, we analyze this structure in some detail.
End Behaviour and Ergodicity for Homeomorphisms of Manifolds with Finitely Many Ends
Canadian Journal of Mathematics, 1987
The recent paper of Berlanga and Epstein [5] demonstrated the significant role played by the "ends" of a noncompact manifold M in answering questions relating homeomorphisms of M to measures on M. In this paper we show that an analysis of the end behaviour of measure preserving homeomorphisms of a manifold also leads to an understanding of some of their ergodic properties, and allows results previously obtained for compact manifolds to be extended (with qualifications) to the noncompact case. We will show that ergodicity is typical (dense G s) with respect to various compact-open topology closed subsets of the space Jrif = J(?(M, fx) consisting of all homeomorphisms of a manifold M which preserve a measure /A. It may be interesting for topologists to note that we prove when M is a a-compact connected «-manifold, n = 2, then M is the countable union of an increasing family of compact connected manifolds. If M is a PL or smooth manifold, this is well known and easy. If M is just, however, a topological «-manifold then we apply the recent results [9] and [12] to prove the result. The Borel measure ju, is taken to be nonatomic, locally finite, positive on open sets, and zero for the manifold boundary of M. The study of the space Jf(M, fi) for compact connected manifolds was initiated by Oxtoby and Ulam [10] who showed that ergodicity is typical. Later Katok and Stepin [8] and Alpern [1, 2] extended the ergodic theoretic generality of the Oxtoby-Ulam Theorem by showing that ergodicity could be replaced by weak mixing or indeed by any property which is typical in the purely measure theoretical context (i.e., for automorphisms of a Lebesgue space, with the coarse topology). This paper also seeks to generalise the Oxtoby-Ulam Theorem, but in a different direction; namely by removing the assumption that the underlying manifold is compact. A complete generalisation in this direction was shown to be impossible by Alpern's example [4] of the unit shift h(0, r) = (0, r + 1) along the two dimensional cylinder M = S x X R. Any homeomorphism h' which is sufficiently close to h in the compact-open topology will map
Fixed Point Theorems for Plane Continua with Applications
Memoirs of the American Mathematical Society, 2012
Part 1. Basic Theory Chapter 2. Preliminaries and outline of Part 1 2.1. Index 2.2. Variation 2.3. Classes of maps 2.4. Partitioning domains Chapter 3. Tools 3.1. Stability of Index 3.2. Index and variation for finite partitions 3.3. Locating arcs of negative variation 3.4. Crosscuts and bumping arcs 3.5. Index and Variation for Carathéodory Loops 3.6. Prime Ends 3.7. Oriented maps 3.8. Induced maps of prime ends Chapter 4. Partitions of domains in the sphere 4.1. Kulkarni-Pinkall Partitions 4.2. Hyperbolic foliation of simply connected domains 4.3. Schoenflies Theorem 4.4. Prime ends Part 2. Applications of basic theory Chapter 5. Description of main results of Part 2 5.1. Outchannels 5.2. Fixed points in invariant continua 5.3. Fixed points in non-invariant continua-the case of dendrites 5.4. Fixed points in non-invariant continua-the planar case 5.5. The polynomial case Chapter 6. Outchannels and their properties 6.1. Outchannels v vi CONTENTS 6.2. Uniqueness of the Outchannel Chapter 7. Fixed points 7.1. Fixed points in invariant continua 7.2. Dendrites 7.3. Non-invariant continua and positively oriented maps of the plane 7.4. Maps with isolated fixed points 7.