Orbit and Strong Orbit Realization for Minimal Homeomorphisms: Ormes’ Results

2018 ◽  
pp. 479-483
Author(s):  
Thierry Giordano ◽  
David Kerr ◽  
N. Christopher Phillips ◽  
Andrew Toms
Keyword(s):  
Minimal Homeomorphisms

Minimal homeomorphisms on low-dimensional tori

2009 ◽  
Vol 29 (5) ◽  
pp. 1515-1528
Author(s):  
N. M. DOS SANTOS ◽  
R. URZÚA-LUZ
Keyword(s):  
Affine Transformation ◽  
Homology Group ◽  
Arbitrary Dimension ◽  
Linear Part ◽  
Linear Mapping ◽  
Sufficient Condition ◽  
Skew Product ◽  
Lefschetz Fixed Point Theorem ◽  
Low Dimensional ◽  
Minimal Homeomorphisms

AbstractWe study minimal homeomorphisms (all orbits are dense) of the tori Tn, n≤4. The linear part of a homeomorphism φ of Tn is the linear mapping L induced by φ on the first homology group of Tn. It follows from the Lefschetz fixed point theorem that 1 is an eigenvalue of L if φ minimal. We show that if φ is minimal and n≤4, then L is quasi-unipontent, that is, all of the eigenvalues of L are roots of unity and conversely if L∈GL(n,ℤ) is quasi-unipotent and 1 is an eigenvalue of L, then there exists a C∞ minimal skew-product diffeomorphism φ of Tn whose linear part is precisely L. We do not know whether these results are true for n≥5. We give a sufficient condition for a smooth skew-product diffeomorphism of a torus of arbitrary dimension to be smoothly conjugate to an affine transformation.

scholarly journals Full groups of minimal homeomorphisms and Baire category methods

2014 ◽  
Vol 36 (2) ◽  
pp. 550-573
Author(s):  
TOMÁS IBARLUCÍA ◽  
JULIEN MELLERAY
Keyword(s):  
Baire Category ◽  
Full Group ◽  
Group Topology ◽  
Cantor Space ◽  
Polish Group ◽  
Minimal Homeomorphisms

We study full groups of minimal actions of countable groups by homeomorphisms on a Cantor space$X$, showing that these groups do not admit a compatible Polish group topology and, in the case of$\mathbb{Z}$-actions, are coanalytic non-Borel inside$\text{Homeo}(X)$. We point out that the full group of a minimal homeomorphism is topologically simple. We also study some properties of the closure of the full group of a minimal homeomorphism inside$\text{Homeo}(X)$.

scholarly journals Dynamical simplices and Fraïssé theory

2018 ◽  
Vol 39 (11) ◽  
pp. 3111-3126 ◽  
Author(s):  
JULIEN MELLERAY
Keyword(s):  
Invariant Measures ◽  
Probability Measures ◽  
Choquet Simplex ◽  
Cantor Space ◽  
Orbit Equivalent ◽  
Minimal Homeomorphisms

We simplify a criterion (due to Ibarlucía and the author) which characterizes dynamical simplices, that is, sets $K$ of probability measures on a Cantor space $X$ for which there exists a minimal homeomorphism of $X$ whose set of invariant measures coincides with $K$ . We then point out that this criterion is related to Fraïssé theory, and use that connection to provide a new proof of Downarowicz’ theorem stating that any non-empty metrizable Choquet simplex is affinely homeomorphic to a dynamical simplex. The construction enables us to prove that there exist minimal homeomorphisms of a Cantor space which are speedup equivalent but not orbit equivalent, answering a question of Ash.

scholarly journals Dynamical simplices and minimal homeomorphisms

2017 ◽  
Vol 145 (11) ◽  
pp. 4981-4994 ◽  
Author(s):  
Tomás Ibarlucía ◽  
Julien Melleray
Keyword(s):  
Minimal Homeomorphisms

There are no minimal homeomorphisms of the multipunctured plane

1992 ◽  
Vol 12 (1) ◽  
pp. 75-83 ◽  
Author(s):  
Michael Handel
Keyword(s):  
Periodic Points ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Dense Orbit ◽  
Finite Set ◽  
Minimal Homeomorphisms

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.

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