Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
    
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. –  Vaughn Climenhaga Jun 16 '11 at 11:36
    
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. –  Vaughn Climenhaga Jun 16 '11 at 11:45
    
Hi, dear Climenhaga, thank you for your answers. –  ljjpfx Jun 17 '11 at 3:08
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.