Topological entropy for differentiable maps of intervals (Q5937402)

The results of the paper generalize the known relation between the topological entropy \(h(f)\) and the growth rate of the number of periodic points: \(h(f)\leq \lim\sup_{n \to\infty} {1\over n}\log\# \text{Per}(f,n)\), where \(\text{Per} (f,n)\) is the set of fixed points of \(f^n\), \(n\geq 1\), and \(\#A\) is the number of elements of \(A\). Let \(f:I\to I\) be a \(C^{1+ \varepsilon}\) map of a compact interval \(I\). A periodic point \(p\) of \(f\) with period \(n\) is a source if \(\nu(p)= |(f^n)'(p) |^{1/p}>1\).NEWLINENEWLINENEWLINEMain notations:NEWLINENEWLINENEWLINEa) \(\text{Per} (f,n,\nu,\delta)= \{p\in\text{Per}(f,n): \nu(p)\geq \nu\), \(|f'(f^i(p))|\geq\delta\), \(0\leq i\leq n-1\}\),NEWLINENEWLINENEWLINEb) \(\overline {TH(p)}\) is the transversal homoclinic closure of \(p\),NEWLINENEWLINENEWLINEc) \(H(p,m,\delta) =\{q\in W^u_{\text{loc}}(p): f^{(m)}(q)=p\), \(|f'(f^i(q)) |\geq \delta,0\leq i\leq m-1\}\).NEWLINENEWLINENEWLINEMain results of the paper:NEWLINENEWLINENEWLINE1) \(h(f)=\max \{0,\lim_{\nu\to 1} \lim_{\delta\to 0} \limsup_{n\to\infty} {1\over n} \log\text{Per}(f,n,\nu,\delta)\).NEWLINENEWLINENEWLINE2) If \(h(f)>0\) then \(h(f)= \sup\{h(f |_{\overline{TH(p)}}\})=\max\{0,\lim_{\delta\to 0}\limsup_{m\to\infty}{1\over m}\log H(p,m, \delta)\}\).NEWLINENEWLINENEWLINE3) The following statements are equivalent: (i) \(h(f)>0\), (ii) \(f\) has a transversal homoclinic point of source, (iii) \(f\) has a homoclinic point of a periodic point.