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### Open Access Journals

PROC

Given a cellular automaton $F:A^{\ZZ} \to A^{\ZZ}$, we define its
small quasi-attractor $\Qq_F$ as the nonempty intersection
of all shift-invariant attractors of all $F^q\sigma^p$, where $q>0$ and
$p\in\ZZ$. The measure attractor $\Mm_F$ is the closure of the supports
of the members of the unique
attractor of $F:\MMM_{\sigma}(A^{\ZZ}) \to \MMM_{\sigma}(A^{\ZZ})$ in the
space of shift-invariant Borel probability measures.

PROC

We study the parameter space of an iterative system consisting
of two hyperbolic disc Möbius transformations. We identify several classes of
parameters which yield discrete groups whose fundamental polygons have sides
at the Euclidean boundary. It follows that these system are not minimal.

DCDS

We investigate iterative systems consisting of Möbius
transformations on the extended real line. We characterize
systems with unique attractor and give some sufficient
conditions for minimality.

DCDS

We modify the idea of a previous article [8] and introduce polynomial
and exponential

*dynamically defined recurrence dimensions*, topological invariants which express how the Poincaré recurrence time of a set grows when the diameter of the set shrinks. We introduce also the concept of polynomial entropy which applies in the case that topological entropy is zero and complexity function is polynomial. We compare recurrence dimensions with topological and polynomial entropies, evaluate recurrence dimensions of Sturmian subshifts and show some examples with Toeplitz subshifts.## Year of publication

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