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==Examples of minimal homeomorphisms==

''Example'' 1. Consider a homeomorphism of the <math>2</math>-torus, <math>S: \Bbb T \to \Bbb T\ ,</math>
of the form <math>S(x,y)=(x+\alpha, y+\beta)\ ,</math> where <math>1,\alpha, \beta \in
\Bbb R</math> are rationally independent and  <math>+ : \Bbb R / \Bbb Z \times
\Bbb R \to \Bbb R / \Bbb Z</math> is defined in the obvious way. Then <math>S</math> is
minimal (and ergodic with respect to Lebesgue measure). M. Rees [R1]
found a minimal homeomorphism <math>S_1</math> which is an extension of <math>S</math> (i.e.,
<math>\varphi \circ S_1 = S\circ \varphi</math>  for some continuous surjection
<math>\varphi</math>  of <math>\Bbb T</math>) such that <math>S_1</math> has positive
topological entropy. In fact every <math>n</math>-manifold, <math>n\geq 2\ ,</math> which carries a minimal homeomorphism also carries a minimal homeomorphism with positive topological entropy [BCLR].

''Example'' 2. Let <math>K=(k_n)_{n>0}</math> be a sequence of integers <math>k_n\geq 2\ .</math> Let <math>\Sigma_{K}</math> be the set of all one-sided infinite sequences <math>(i_n)_{n\geq 1}</math> for which <math>0\leq i_n \leq k_n-1\ .</math> Think of these sequences as 'integers' in multibase notation, the base of the <math>n^{th}</math> digit <math>i_n</math> being <math>k_n\ .</math> With the natural (product) topology, <math>\Sigma_{K}</math> is homeomorphic to the Cantor set. Define a map <math> \alpha_{K}: \Sigma_{K} \to \Sigma_{K}</math> which informally may be described as 'add 1 and carry' where the addition is performed at the leftmost term <math>i_1</math> and the carry proceeds to the right in multibase notation. Then <math>\alpha_{K}</math> is a minimal homeomorphism and is called a ''generalized adding machine'' or an ''odometer'' (as a general reference see e.g. [Dow]).

In general it is difficult to construct a minimal homeomorphism, see for instance the examples of minimal homeomorphisms on the Klein bottle in [Ell] and
[Par]. For some methods of constructions of minimal homeomorphisms see [AnK], [Ell], [GW] and [FK].