1. CMB 2017 (vol 61 pp. 812)
 Medini, Andrea; van Mill, Jan; Zdomskyy, Lyubomyr S.

Infinite Powers and Cohen Reals
We give a consistent example of a zerodimensional separable
metrizable space $Z$ such that every homeomorphism of $Z^\omega$
acts like a permutation of the coordinates almost everywhere.
Furthermore, this permutation varies continuously. This shows
that a result of Dow and Pearl is sharp, and gives some insight
into an open problem of Terada. Our example $Z$ is simply the
set of $\omega_1$ Cohen reals, viewed as a subspace of $2^\omega$.
Keywords:infinite power, zerodimensional, firstcountable, homogeneous, Cohen real, hhomogeneous, rigid Categories:03E35, 54B10, 54G20 

2. CMB 2015 (vol 58 pp. 334)
 Medini, Andrea

Countable Dense Homogeneity in Powers of Zerodimensional Definable Spaces
We show that, for a coanalytic subspace $X$ of $2^\omega$, the
countable dense homogeneity of $X^\omega$ is equivalent to $X$
being Polish. This strengthens a result of HruÅ¡Ã¡k and Zamora
AvilÃ©s. Then, inspired by results of HernÃ¡ndezGutiÃ©rrez,
HruÅ¡Ã¡k and van Mill, using a technique of Medvedev, we
construct a nonPolish subspace $X$ of $2^\omega$ such that $X^\omega$
is countable dense homogeneous. This gives the first $\mathsf{ZFC}$ answer
to a question of HruÅ¡Ã¡k and Zamora AvilÃ©s. Furthermore,
since our example is consistently analytic, the equivalence result
mentioned above is sharp. Our results also answer a question
of Medini and Milovich. Finally, we show that if every countable
subset of a zerodimensional separable metrizable space $X$ is
included in a Polish subspace of $X$ then $X^\omega$ is countable
dense homogeneous.
Keywords:countable dense homogeneous, infinite power, coanalytic, Polish, $\lambda'$set Categories:54H05, 54G20, 54E52 
