Interval maps of given topological entropy and Sharkovskii's type
It is known that the topological entropy of a continuous interval map
f is positive if and only if the type of f for Sharkovskii's order
is 2d p for some odd integer p ≥ 3 and some d ≥ 0; and
in this case the topological entropy of f is greater than or equal to
log(λp) / 2d, where λp
is the unique positive root of Xp - 2Xp-2 - 1.
For every odd p ≥ 3, every d ≥ 0
and every λ ≥ λp, we build a piecewise monotone
continuous interval map that is of type 2d p for Sharkovskii's order
and whose topological entropy is log (λ) / 2d. This shows that,
for a given type, every possible finite entropy above the minimum
can be reached provided the type allows the map to have positive entropy.
Moreover, if d = 0 the map we build is topologically mixing.