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
1  
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

1 Answer 1

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.

share|improve this answer

Your Answer

 
discard

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

Not the answer you're looking for? Browse other questions tagged or ask your own question.