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1. CMB Online first
Countable dense homogeneity in powers of zero-dimensional 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¡ndez-GutiÃ©rrez,
HruÅ¡Ã¡k and van Mill, using a technique of Medvedev, we
construct a non-Polish 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 zero-dimensional 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 |
2. CMB 2012 (vol 56 pp. 477)
Hypercyclic Abelian Groups of Affine Maps on $\mathbb{C}^{n}$ We give a characterization of hypercyclic abelian group
$\mathcal{G}$ of affine maps on $\mathbb{C}^{n}$. If $\mathcal{G}$
is finitely generated, this characterization is explicit. We prove
in particular
that no abelian group generated by $n$ affine maps on $\mathbb{C}^{n}$ has a dense orbit.
Keywords:affine, hypercyclic, dense, orbit, affine group, abelian Categories:37C85, 47A16 |
3. CMB 2012 (vol 56 pp. 860)
On Countable Dense and $n$-homogeneity We prove that a connected, countable dense homogeneous space is
$n$-homogeneous for every $n$, and strongly 2-homogeneous provided it
is locally connected. We also present an example of a connected and
countable dense homogeneous space which is not strongly
2-homogeneous. This answers Problem 136 of Watson in the Open Problems
in Topology Book in the negative.
Keywords:countable dense homogeneous, connected, $n$-homogeneous, strongly $n$-homogeneous, counterexample Categories:54H15, 54C10, 54F05 |
4. CMB 2012 (vol 56 pp. 241)
Versions of Schwarz's Lemma for Condenser Capacity and Inner Radius We prove variants of Schwarz's lemma involving monotonicity
properties of condenser capacity and inner radius. Also, we
examine when a similar monotonicity property holds for the
hyperbolic metric.
Keywords:condenser capacity, inner radius, hyperbolic metric, Schwarz's lemma Categories:30C80, 30F45, 31A15 |