Let $(X,d)$ be a compact metric space and f a continuous transformation on X. f has the specification if one can always find a single orbit to interpolate between different pieces of orbits, up to a pre-assigned error. We call the map $\mu \to h_{\mu}(f)$ (Kolmogorov-Sinai entropy) the entropy map . Does specification implies that entropy map is upper semicontinuous?
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2$\begingroup$ Your definition of specification sounds more like a definition of topological transitivity -- the key function of specification is to ensure that the time you spend going from one orbit segment to the next when you approximate is uniformly bounded. $\endgroup$– Vaughn ClimenhagaCommented Jun 16, 2011 at 11:36
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$\begingroup$ This sounds a little bit like a homework exercise (my apologies if it's not). A natural thing to do would be to write down a couple examples of systems whose entropy map is not upper semi-continuous, and see if you can find something from that list with specification. $\endgroup$– Vaughn ClimenhagaCommented Jun 16, 2011 at 11:45
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$\begingroup$ Hi, dear Climenhaga, thank you for your answers. $\endgroup$– ljjpfxCommented Jun 17, 2011 at 3:08
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Expansivity implies that the entropy map is upper semicontinuous. See P. walters book Thm 8.2. I think that if the shadowing orbit in your definition of specification is unique then the map f is expansive.
