It is well known that any continuous piecewise monotone interval map f with positive topological entropy $h_{top}\left(f\right)$ is semiconjugate to some piecewise affine map with constant slope $e^{h_{top}\left(f\right)}$. We prove this result for a class of Markov countably piecewise monotone continuous interval maps.

In this paper we explore topological factors in between the Kronecker factor and the maximal equicontinuous factor of a system. For this purpose we introduce the concept of sequence entropy $n$-tuple for a measure and we show that the set of sequence entropy tuples for a measure is contained in the set of topological sequence entropy tuples [H- Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we introduce a weak notion and a strong notion of complexity pair for a...

We prove that ${𝒞}^r$ maps with $r\>1$ on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].