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Abstract
The dynamics of the transformation $F: (x,y)\rightarrow (x(4-x-y),xy)$
defined on the plane triangle $\Delta$ of vertices $(0,0)$, $(0,4)$ and $(4,0)$ plays an important role in the behaviour of the Lotka--Volterra map.
In 1993, A. N. SharkovskiĬ (Proc. Oberwolfach 20/1993) stated some problems on it, in particular a question
about the trasitivity of $F$ was posed.
The main aim of this paper is to prove that for every non--empty open set $\mathcal{U} \subset \Delta$
there is an integer $n_{0}$ such that for each $n>n_{0}$ it is
$F^{n}(\mathcal{U}) \supseteq \Delta \setminus P_{\varepsilon}$,
where $P_{\varepsilon} = \{ (x,y) \in D : y<\beta, \mbox{ $where$ F(t,\varepsilon)=(\alpha,\beta) \mbox{ and } t \in[0,2] \}$
and $\varepsilon \rightarrow 0$ for $n \rightarrow \infty$.
Consequently, we show that the map $F$ is transitive, it is not topologically exact and it is almost topologically exact.
Additionally, we prove that the union of all preimages of the point $(1,2)$ is a dense subset of $\Delta$.
Mathematics Subject Classification: Primary: 37B99, 37B10.
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