On criteria for the positiveness of the topological entropy of continuous mappings on dendrites

By continuum we mean a compact connected metric space. A dendrite is a locally connected continuum that does not contain subsets homeomorphic to a circle.
Let $X$ be a dendrite, point $p\in X$. We will say that
$\bullet$ $p$ – branch point of $X$ dendrite if the number of connected sets in $X\setminus\{p\}$ is greater than two;
$\bullet$ $p$ is the endpoint of the dendrite $X$ if $X\setminus\{p\}$ is connected.
A dendrite with a finite number of endpoints is called a finite tree.
Note that any two points $x,\,y$ in the dendrite $X$ can be connected by a single arc; the set of branch points in $X$ is at most countable and the number of connected components of the set $X\setminus\{p\}$ is at most countable for any point $p\in X$.
For a continuous mapping $f:X\to X$, where $X$ is a segment or a finite tree, the following statements are equivalent:
(1) the topological entropy $f$ is positive;
(2) for some natural number $n\ge1$ $f^n$ has a horseshoe;
(3) $f$ has homoclinic points;
(4) $f$ has a strongly sensitive point in $\overline{Per(f)}\setminus Per(f)$.
The paper studies the connections between statements (1) – (4) for continuous mappings on dendrites that are not finite trees. It has been established that the implications between (1) – (4) depend on the structure of the dendrite.