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The conservative sequence of a set $A$ under a transformation $T$ is the set of all $n in mathbb{Z}$ such that $T^n A cap A ot = varnothing$. By studying these sequences, we prove that given any countable collection of nonsingular transformations with no finite invariant measure ${T_i}$, there exists a rank-one transformation $S$ such that $T_i times S$ is not ergodic for all $i$. Moreover, $S$ can be chosen to be rigid or have infinite ergodic index. We establish similar results for $mathbb{Z}^d$ actions and flows. Then, we find sufficient conditions on rank-one transformations $T$ that guarantee the existence of a rank-one transformation $S$ such that $T times S$ is ergodic, or, alternatively, conditions that guarantee that $T times S$ is conservative but not ergodic. In particular, the infinite Chacon transformation satisfies both conditions. Finally, for a given ergodic transformation $T$, we study the Baire categories of the sets $E(T)$, $bar{E}C(T)$ and $bar{C}(T)$ of transformations $S$ such that $T times S$ is ergodic, ergodic but not conservative, and conservative, respectively.
Exponential dichotomy of a strongly continuous cocycle $bFi$ is proved to be equivalent to existence of a Ma~{n}e sequence either for $bFi$ or for its adjoint. As a consequence we extend some of the classical results to general Banach bundles. The dy
Iterated Function Systems (IFSs) have been at the heart of fractal geometry almost from its origin, and several generalizations for the notion of IFS have been suggested. Subdivision schemes are widely used in computer graphics and attempts have been
For $n$ and $k$ integers we introduce the notion of some partition of $n$ being able to generate another partition of $n$. We solve the problem of finding the minimum size partition for which the set of partitions this partition can generate contains
Monomial mappings, $xmapsto x^n$, are topologically transitive and ergodic with respect to Haar measure on the unit circle in the complex plane. In this paper we obtain an anologous result for monomial dynamical systems over $p-$adic numbers. The pro
We study aperiodic balanced sequences over finite alphabets. A sequence v of this type is fully characterised by a Sturmian sequence u and two constant gap sequences y and y. We show that the language of v is eventually dendric and we focus on return