Horseshoe effect and topological entropy of one-dimensional maps

In this paper, for any continuous functionf : [0, 1] → [0, 1], some entropiesh _s (f) andh _c (f) are defined. When the topological entropyh(f) is positive, horseshoe effect is observed, and the following formula is proved: $$h(f) = mathop {sup}limits_n h_c (f^n )/n = mathop {inf}limits_n h_s (f^n )/n = h_s (f^tau )/tau $$ whereτ is any integer satisfyingr > log 3/h(f).