Jumps of entropy in one dimension.

*(English)*Zbl 0694.54019The author investigates the continuity properties of topological entropy as a function on continuous maps of the interval. The author has previously proved that in this case topological entropy is lower semi- continuous. In order to make the problem meaningful we must restrict ourselves to the piecewise monotone maps and make arbitrary small \(C^ 0\) perturbations which do not enlarge the number of turning points.

The following notation is needed: A finite cover of the interval is called f-mono if its members are intervals (possibly points) and f is monotone on each element of the cover. Then define \(c_ n(f)=\min \{Card {\mathcal A}\); \({\mathcal A}\) is an \(f^ n\)-mono cover\(\}\). Now let U(f) be the space of all continuous piecewise monotone maps g: \(I\to I\) for which \(c_ 1(g)\leq c_ 1(f)\). Define \(\alpha (f)=\limsup_{g\to f}h(g)\), \(g\in U(f)\), and \(\beta (f)=\max \{\frac{p}{q}\log 2\); there exists a periodic orbit of f of period q with p turning points\(\}\). The author’s main theorem states that \(\alpha (f)=\max (h(f),\beta (f))\). Thus if h(f)\(\geq \beta (f)\) then the topological entropy is continuous at f. He also investigates the function \(\beta\) and shows that it is upper semi- continuous on U(f).

If f has only one turning point, then it is called weakly unimodal. For such maps, his second theorem proves that the topological entropy is continuous at all maps where it is positive. Finally he deduces some results for kneading invariants.

The following notation is needed: A finite cover of the interval is called f-mono if its members are intervals (possibly points) and f is monotone on each element of the cover. Then define \(c_ n(f)=\min \{Card {\mathcal A}\); \({\mathcal A}\) is an \(f^ n\)-mono cover\(\}\). Now let U(f) be the space of all continuous piecewise monotone maps g: \(I\to I\) for which \(c_ 1(g)\leq c_ 1(f)\). Define \(\alpha (f)=\limsup_{g\to f}h(g)\), \(g\in U(f)\), and \(\beta (f)=\max \{\frac{p}{q}\log 2\); there exists a periodic orbit of f of period q with p turning points\(\}\). The author’s main theorem states that \(\alpha (f)=\max (h(f),\beta (f))\). Thus if h(f)\(\geq \beta (f)\) then the topological entropy is continuous at f. He also investigates the function \(\beta\) and shows that it is upper semi- continuous on U(f).

If f has only one turning point, then it is called weakly unimodal. For such maps, his second theorem proves that the topological entropy is continuous at all maps where it is positive. Finally he deduces some results for kneading invariants.

Reviewer: M.Sears

##### MSC:

54C70 | Entropy in general topology |