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In my talk I will present a multitude of new constructions of
diffeomorphisms with various specific ergodic, topological and
spectral properties. These are based on the ``Conjugation by
approximation''-method developed by D.V. Anosov and A. Katok in
[1]. In fact on every smooth compact connected manifold of
dimension \(m\geq 2\) admitting a non-trivial circle action
\(\mathcal{S} = \left\{S_t\right\}_{t \in \mathbb{S}^1}\) preserving
a smooth volume \(\nu\) this method enables the construction of
smooth diffeomorphisms with particular ergodic properties or
non-standard smooth realizations of measure preserving
systems. Moreover, one can deduce results on the genericity of
designed properties.
One main topic is the construction of weak mixing diffeomorphisms
preserving a measurable Riemannian metric in the restricted space
\(\mathcal{A}_{\alpha}\left(M\right) = \overline{\left\{h \circ
S_{\alpha} \circ h^{-1} \ : h \in \text{Diff}^{\infty}\left(M,
\nu \right)\right\}}^{C^{\infty}}\) for a given Liouvillean
number \(\alpha \in \mathbb{S}^1\). So we design maps with
predetermined rotation number. In addition, we examine the existence
of such diffeomorphisms on the \(m\)-dimensional torus
\(\mathbb{T}^m\), \(m\geq 2\), with prescribed number of ergodic
invariant measures, in particular uniquely ergodic ones.
Furthermore, we start to examine the problem of uniformly rigid and
simultaneously weak mixing maps, which is an up-to-date research
topic in measurable as well as topological dynamics, in the smooth
and beyond that even in the real-analytic category: Under sufficient
conditions on the growth rate of the rigidity sequence we are able
to construct uniformly rigid and weak mixing real-analytic as well
as \(C^{\infty}\)-diffeomorphisms on \(\mathbb{T}^2\).
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