Lipschitz Homeomorphism Research Articles

On the structure of the group of Lipschitz homeomorphisms and its subgroups

We consider the group of Lipschitz homeomorphisms of a Lipschitz manifold and its subgroups. First we study properties of Lipschitz homeomorphisms and show the local contractibility and the perfectness of the group of Lipschitz homeomorphisms. Next using this result we can prove that the identity component of the group of equivariant Lipschitz homeomorphisms of a principal G-bundle over a closed Lipschitz manifold is perfect when G is a compact Lie group.

On the structure of the group of Lipschitz homeomorphisms and its subgroups

We consider the group of Lipschitz homeomorphisms of a Lipschitz manifold and its subgroups. First we study properties of Lipschitz homeomorphisms and show the local contractibility and the perfectness of the group of Lipschitz homeomorphisms. Next using this result we can prove that the identity component of the group of equivariant Lipschitz homeomorphisms of a principal G-bundle over a closed Lipschitz manifold is perfect when G is a compact Lie group.

On commutators of foliation preserving Lipschitz homeomorphisms

We consider the group of foliation preserving Lipschitz homeomorphisms of a Lipschitz foliated manifold. First we show that the identity component of the group of leaf preserving Lipschitz homeomorphisms of a Lipschitz foliated manifold is perfect. Next using this result we compute the first homology of the group of foliation preserving Lipschitz homeomorphisms of a codimension one $C^{2}$-foliated manifold. Then we have results which are different from those of topological and differentiable cases.

Lipschitz image of a measure-null set can have a null complement

We give two examples which show that in infinite dimensional Banach spaces the measure-null sets are not preserved by Lipschitz homeomorphisms. There exists a closed setD ⊂ l2 which contains a translate of any compact set in the unit ball of l2 and a Lipschitz isomorphismF of l2 onto l2 so thatF(D) is contained in a hyperplane. LetX be a Banach space with an unconditional basis. There exists a Borel setA⊂X and a Lipschitz isomorphismF ofX onto itself so that the setsX/A andF(A) are both Haar null.

Slow entropy type invariants and smooth realization of commuting measure-preserving transformations

We define invariants for measure-preserving actions of discrete amenable groups which characterize various subexponential rates of growth for the number of “essential” orbits similarly to the way entropy of the action characterizes the exponential growth rate. We obtain above estimates for these invariants for actions by diffeomorphisms of a compact manifold (with a Borel invariant measure) and, more generally, by Lipschitz homeomorphisms of a compact metric space of finite box dimension. We show that natural cutting and stacking constructions alternating independent and periodic concatenation of names produce ℤ 2 actions with zero one-dimensional entropies in all (including irrational) directions which do not allow either of the above realizations.

Losing Hausdorff dimension while generating pseudogroups

Considering different finite sets of maps generating a pseudogroup G of locally Lipschitz homeomorphisms between open subsets of a compact metric space X we arrive at a notion of a Hausdorff dimension $dim_H G$ of G. Since $dim_H G ≤ dim_H X$, the dimensi

Small commutators of piecewise linear homeomorphisms of the real line

IN THIS PAPER we study the group P&(R) of piecewise linear (PL) homeomorphisms of the real line R with compact support. This group has been studied by many people. As for the homology of this group, Epstein showed that the group is a perfect group and hence it is a simple group [2]. The lower-dimensional homology of this group was determined by Greenberg [S]. In particular, his result says that the 2-dimensional homology is R @Jz R and it is easy to see that the canonical bilinear map R Oz R + R is nothing but the discrete Godbillon-Vey class described in [6] and [4]. In [22], we determined all the homology group using Greenberg’s description [8] of the classifying space for transversely PL foliations. Our study on the group PL,(R) was motivated by an intention to use these results on the homology of PL,(R) to understand the homology of groups of Lipschitz homeomorphisms of R. In particular, since the (discrete) Godbillon-Vey 2-cocycle for PL,(R) was completely understood, we tried to understand the Godbillon-Vey 2-cocycle for groups of Lipschitz homeomorphisms of R using approximations by elements of the group PL,(R). The Godbillon-Vey invariant was first defined for codimension-1 C2 foliations of closed oriented 3-manifolds [7]. This invariant is the only known non-trivial invariant for C2 foliated cobordism and it varies continuously under the deformation of the foliation [ 141. For transversely oriented codimension-1 C’ foliations, the classifying space for them is contractible [ 1 S] and such foliations of closed oriented 3-manifolds are all cobordant. This fact has already been shown for transversely oriented Lipschitz foliations [l 11. Hence the Godbillon-Vey invariant cannot be defined for codimension-1 C’ or Lipschitz foliations. There are, however, a lot of classes of foliations between C’ or Lipschitz and C2. In fact, the Godbillon-Vey invariant was extended to the foliations of class C’ +a (CI > f) by Hurder and Katok [9], and to the transversely piecewise linear (or piecewise C*) by Ghys and Sergiescu [6,4]. We defined the Godbillon-Vey invariant for the foliations of class CLvyn with (1 I fl l/2) and the transversely PL foliations. Since each cobordism class of the foliations of class CL’% (1 5 j? < 2) of closed oriented 3-manifolds contains a representative which is a 2-cycle for the group GfvYn(R) of diffeomorphisms of class C ‘,% of the real line R with compact support [12], the key problem is to understand the Godbillon-Vey invariant on 2-cycles of the group G:*“(R). For a real number j3 (p 2 l), the group G, L*vn(R) of CL’Yn diffeomorphisms of R with compact support is defined as follows. An element fof G, L*Y#(R) is a Lipschitz homeomorphism with compact support such that logf’(x 0) is with bounded p-variation (see

Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle

Homological and dynamical study on certain groups of Lipschitz homeomorphisms of the circle

Structural Analysis of Nonsmooth Mappings, Inverse Functions, and Metric Projections

It is known that a locally Lipschitz continuous function ƒ: R n → R n which is strongly B-differentiable at a point x 0 ∈ R n is a Lipschitz homeomorphism in a neighborhood of x 0 if and only if its B-derivative at x 0 is a Lipschitz homeomorphism. We show that strongly B-differentiability is a rather restrictive requirement for piecewise differentiable functions. However, it turns out that generically a piecewise differentiable function can be locally transformed into a strongly B-differentiable function by means of a piecewise differentiable homeomorphism. The corresponding criteria yield structural inverse function theorems for piecewise differentiable functions. The results are applied to the metric projection onto a convex set.

Stability of Lyapunov exponents

AbstractWe consider small random perturbations of matrix cocycles over Lipschitz homeomorphisms of compact metric spaces. Lyapunov exponents are shown to be stable provided that our perturbations satisfy certain regularity conditions. These results are applicable to dynamical systems, particularly to volume-preserving diffeomorphisms.

An embedding space triple of the unit interval into a graph and its bundle structure

Let l 2 denote a Hilbert space, and let l 2 Q ={(x i )∈l 2 |sup|i.x i |<∞} and l 2 f ={(x i )∈l 2 |x i =0 except for finitely many i}. We show that the triple (H(X), H LIP (X), H PL (X)) of spaces of homeomorphisms, of Lipschitz homeomorphisms, and of PL homeomorphisms of a finite graph X onto itself is an (l 2 ,L 2 Q , l 2 f )-manifold triple, and that the triple (E(I,X), E LIP (I,X), E PL 5I,X)) of spaces of embeddings, of Lipschitz embeddings, and of PL embeddings of I=[0,1] into a graph X is an (l 2 ,l 2 Q ,l 2 4 )-manifold triple

Universal scaling in circle maps

We discuss scaling, renormalization, and universality for critical and subcritical circle maps. Using the structure of the rationals provided by the Farey tree, we construct a pair of renormalization group operators for circle maps with all values of the winding number. In this general formulation, the renormalization equations define a universal invariant set which determines all universal quantities for all cubic maps with all irrational winding numbers. As expected, the manifold of pure rotations defines the subcritical attracting set. We show that the invariant set governing the critical circle maps cannot be embedded in less than 3 dimensions. However, we are able to obtain a differentiable parametrization of the unstable manifold of the critical invariant set, which we believe is universal up to Lipschitz homeomorphisms; the derivative of this function describes the scaling structure of all small gaps in the devil's staircase for all critical maps. Our work can be viewed as providing numerical evidence for Lanford's conjectures on the strange set underlying renormalizations of critical circle maps.

On the space of Lipschitz homeomorphisms of a compact polyhedron

Let X be a positive dimensional compact Euclidean polyhedron. Let H(X), HUP{X) and HPL(X) be respectively the space of homeomorphisms, the space of Lipschitz homeomorphisms and the space of piecewise-lin ear homeomorphisms of X onto itself. In this paper, we establish a homeomorphism taking the triple (H(X), HUp(X), H?L(X)) onto the triple (H(X) x s, HUp(X) x Σ, HFL(X) X σ), where 5 = (-1, l)ω, Σ = {(*,) e s\sup\Xi < 1} and σ = {(*,) e s\x t = 0 except for finitely many /}. As a consequence we prove that when X is a PL manifold with dim c Φ 4 and dX = 0, in case dimZ = 5, (H(X),HUP(X)) is an (s, Σ)-manifold pair if H(X) is an s-manifold. We also prove that if dim* = 1 or 2, then (H(X),HPL(X)) is an (5, σ)-manifold pair and (H(X),HUp(X)) is an (s, Σ)-manifold.

A Metric Form of Microtransitivity

We prove that every homogeneous compact metrizable space $X$ has a compatible metric for which $X$ is Lipschitz homogeneous and for which the group $L(X)$ of Lipschitz homeomorphisms of $X$ acts Lipschitz microtransitively on $X$.

A metric form of microtransitivity

We prove that every homogeneous compact metrizable space X X has a compatible metric for which X X is Lipschitz homogeneous and for which the group L ( X ) L(X) of Lipschitz homeomorphisms of X X acts Lipschitz microtransitively on X X .

On Lipschitz Homogeneity of the Hilbert Cube

The main contribution of this paper is to prove the conjecture of [Vä] that the Hilbert cube $Q$ is Lipschitz homogeneous for any metric ${d_s}$, where $s$ is a decreasing sequence of positive real numbers ${s_k}$ converging to zero, ${d_s}(x,y) = \sup \{ {s_k}|{x_k} - {y_k}|:k \in N\}$, and $R(s) = \sup \{ {s_k}/{s_{k + 1}}:k \in N\} < \infty$. In addition to other results, we shall show that for every Lipschitz homogeneous compact metric space $X$ there is a constant $\lambda < \infty$ such that $X$ is homogeneous with respect to Lipschitz homeomorphisms whose Lipschitz constants do not exceed $\lambda$. Finally, we prove that the hyperspace ${2^I}$ of all nonempty closed subsets of the unit interval is not Lipschitz homogeneous with respect to the Hausdorff metric.

On Lipschitz homogeneity of the Hilbert cube

The main contribution of this paper is to prove the conjecture of [Vä] that the Hilbert cube Q Q is Lipschitz homogeneous for any metric d s {d_s} , where s s is a decreasing sequence of positive real numbers s k {s_k} converging to zero, d s ( x , y ) = sup { s k | x k − y k | : k ∈ N } {d_s}(x,y) = \sup \{ {s_k}|{x_k} - {y_k}|:k \in N\} , and R ( s ) = sup { s k / s k + 1 : k ∈ N } &gt; ∞ R(s) = \sup \{ {s_k}/{s_{k + 1}}:k \in N\} &gt; \infty . In addition to other results, we shall show that for every Lipschitz homogeneous compact metric space X X there is a constant λ &gt; ∞ \lambda &gt; \infty such that X X is homogeneous with respect to Lipschitz homeomorphisms whose Lipschitz constants do not exceed λ \lambda . Finally, we prove that the hyperspace 2 I {2^I} of all nonempty closed subsets of the unit interval is not Lipschitz homogeneous with respect to the Hausdorff metric.

Lipschitz homeomorphisms of the Hilbert cube

We study the question whether the Hilbert cube Q is Lipschitz homogeneous. The answer depends on the metric of Q. For example, setting d( x, y)= sup j | x j - y j |/ j we obtain a Lipschitz homogeneous metric, but if the last j is replaced by j!, the answer is negative.