## Some unsolved problems on the prevalence of ergodicity, instability, and algebraic independence.(English)Zbl 0846.28006

Let $$G$$ be a complete metric group. A set $$X\subset G$$ is said to be shy if there exists a probability measure $$m$$ on $$G$$ such that $$m(gXh)= 0$$ for all $$g,h\in G$$. It is called prevalent if the set $$G_s X$$ is shy. Let $$H$$ denote the group of all measure-preserving autohomeomorphisms of the $$n$$-dimensional cube $$I^n$$ with the distance $$d$$ defined as follows: $d(f, g)= \max\Biggl( \max_{x\in I^n}|f(x)- g(x)|,\;\max_{x\in I^n} |f^{- 1}(x)- g^{- 1}(x)|\Biggr).$ Let $$G$$ be the additive group of real continuous functions, defined on the Cantor set, equipped with the Chebyshev metric.
The paper contains a series of open problems connected with the prevalence of (a) the subset of $$H$$ consisting of ergodic mappings and the subset consisting of unstable mappings, and (b) the subset of $$G$$ consisting of functions which avoid null sets and the subset consisting of so-called clever functions.

### MSC:

 28D05 Measure-preserving transformations