Minimal Homeomorphisms Research Articles

The conley index and non-existence of minimal homeomorphisms

The conley index and non-existence of minimal homeomorphisms

The Relationship between Entropy and Strong Orbit Equivalence for the Minimal Homeomorphisms (II)

Every minimal homeomorphism of a Cantor set is strongly orbit equivalent to a homeomorphism of infinite entropy.

Strong orbit realization for minimal homeomorphisms

LetXbe a Cantor set,S a minimal self-homeomorphism ofX, and Μ anS-invariant Borel probability. LetT be an ergodic automorphism of a non-atomic Lebesgue probability space(Y,Ν). Then there is a minimal homeomorphismS′ with the same orbits asS such that (S′, Μ ) is measurably conjugate to (T, Ν). HereS′ can be chosen strongly orbit equivalent toS if and only if the periodic spectrum ofS is contained in the discrete spectrum ofT. Corollaries of these results generalize Dye’s Theorem and the Jewett-Krieger Theorem.

Orbit equivalence, flow equivalence and ordered cohomology

We study self-homeomorphisms of zero dimensional metrizable compact Hausdorff spaces by means of the ordered first cohomology group, particularly in the light of the recent work of Giordano Putnam, and Skau on minimal homeomorphisms. We show that flow equivalence of systems is analogous to Morita equivalence between algebras, and this is reflected in the ordered cohomology group. We show that the ordered cohomology group is a complete invariant for flow equivalence between irreducible shifts of finite type; it follows that orbit equivalence implies flow equivalence for this class of systems. The cohomology group is the (pre-ordered) Grothendieck group of the C*-algebra crossed product, and we can decide when the pre-ordering is an ordering, in terms of dynamical properties.

WHEN AN INFINITELY-RENORMALIZABLE ENDOMORPHISM OF THE INTERVAL CAN BE SMOOTHED

Let K be a closed subset of a smooth manifold M, and let f: K→K be a continuous self-map of K. We say that f is smoothable if it is conjugate to the restriction of a smooth map by a homeomorphism of the ambient space M. We give a necessary condition for the smoothability of the faithfully infinitely interval-renormalizable homeomorphisms of Cantor sets in the unit interval. This class contains, in particular, all minimal homeomorphisms of Cantor sets in the line which extend to continuous maps of an interval with zero topological entropy.

Entropy versus orbit equivalence for minimal homeomorphisms

1. Introduction and definitions. We construct, using the lexicographic or adic maps of Vershik (as modified by Herman, Putnam, and Skau [P], [HPS], [Sk]), examples of minimal homeomorphisms of all possible (topological) entropies (including infinite) which are orbit equivalent to the dyadic adding machine. In particular, this answers in a decisively negative way the question as to whether orbit equivalent homeomorphisms have the same entropy. We also show that any minimal homeomorphism of the Cantor set is strongly orbit equivalent to one of zero entropy. We recall some definitions. A source for these is the survey article of Skau [Sk]. A Bratteli diagram B is a directed graph whose vertex set decomposes into finite subsets, levels or rows, Bk (k = 0,1,...), together with edges from vertices in Bk to vertices in Bk+X additionally, every vertex of Bk is joined to a vertex of Bk+X. The Bratteli diagram is simple if for all k, there exists k1> k such that for every vertex v in Bk and every vertex v' in Bk>, there is a path from v Xo v'. The Bratteli diagram is pointed if |2f o| = 1, that is, there is a distinguished top vertex. The set of infinite paths, usually denoted X, is called the Bratteli compactum, and is a compact zero

There are no minimal homeomorphisms of the multipunctured plane

The main results of this paper are the following theorem and its corollary.Theorem 0.1. Suppose that f: S2 → S2 is an orientation-preserving homeomorphism of the two-dimensional sphere and that Fix (f) is a finite set containing at least three points. If f has a dense orbit then the number of periodic points of period n for some iterate of f grows exponentially in n.

TheC∗-algebras associated with minimal homeomorphisms of the Cantor set

The<i>C</i>^∗-algebras associated with minimal homeomorphisms of the Cantor set

On the construction of minimal skew products

It is shown that under fairly general conditions on a compact metric spaceY there are minimal homeomorphisms onZ×Y of the form(z,y)→(σz, hz(y)) where (Z, σ) is a arbitrary metric minimal flow andz→hz is a continuous map fromZ to the space of homeomorphisms ofY. Similar results are obtained for strict ergodicity, topolotical weak mixing and some relativized concepts.