Equi-topological entropy curves for skew tent maps in the square

Year, pages: 2017, 1577 - 1594

We consider skew tent maps $T_{{α}, {β}}(x)$ such that $(\aaa,\bbb)\in[0,1]²$ is the turning point of $TT_{{α}, {β}}$, that is, $T_{{α}, {β}}=frac{{β}}{{α}}x$ for $0≤ x ≤ {α}$ and $T_{{α}, {β}}x)=\frac{{β}}{1- {α}}(1-x)$ for ${α}< x ≤ 1$. We denote by $\uM=K(\aaa,\bbb)$ the kneading sequence of $TT_{{α}, {β}}$ and by $h(\aaa,\bbb)$ its topological entropy. For a given kneading squence $\uM$ we consider equi-kneading, (or equi-topological entropy, or isentrope) curves $(\aaa,\fff_\uM(\aaa))$ such that $K(\aaa,\fff_\uM(\aaa))=\uM$. To study the behavior of these curves an auxiliary function $\TTT_\uM(\aaa,\bbb)$ is introduced. For this function $\TTT_\uM(\aaa,\fff_\uM(\aaa))=0$, but it may happen that for some kneading sequences $\TTT_\uM(\aaa,\bbb)=0$ for some $\bbb< \fff_\uM(\aaa)$ with $(\aaa,\bbb)$ still in the dynamically interesting quarter of the unit square. Using $\TTT_\uM$ we show that the curves $(\aaa,\fff_\uM(\aaa))$ hit the diagonal $\{(\bbb,\bbb): 0.5 <\bbb < 1\}$ almost perpendicularly if $(\bbb,\bbb)$ is close to $(1,1)$. Answering a question asked by M. Misiurewicz at a conference we show that these curves are not necessarily exactly orthogonal to the diagonal, for example for $\uM=RLLRC$ the curve $(\aaa,\fff_\uM(\aaa))$ is not orthogonal to the diagonal. On the other hand, for $\uM=RLC$ it is. With different parametrization properties of equi-kneading maps for skew tent maps were considered by J. C. Marcuard, M. Misiurewicz and E. Visinescu.

Buczolich, Z., Keszthelyi, G. 2017. Equi-topological entropy curves for skew tent maps in the square. In Mathematica Slovaca, vol. 67, no.6, pp. 1577-1594. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0072

Buczolich, Z., Keszthelyi, G. (2017). Equi-topological entropy curves for skew tent maps in the square. Mathematica Slovaca, 67(6), 1577-1594. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0072