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Abstract

Let $f\colon\, [a,b]\to [a,b]$ be a continuous function of the compact real interval such that (i) $\mathop{\rm card} f^{-1}(y)\ge 2$ for every $y\in [a,b]$; (ii) for some $m\in\{\infty,2,3,\dots\}$ there is a countable set $L\subset [a,b]$ such that $\mathop{\rm card} f^{-1}(y)\ge m$ for every $y\in [a,b]\setminus L$. We show that the topological entropy of $f$ is greater than or equal to $\log m$. This generalizes our previous result for $m=2$.