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Jan 13, 2013 at 14:53 comment added R W I see what you mean. It's not a problem, because the entropy of $h_1$-ergodic components is constant along $H$-orbits.
Jan 13, 2013 at 13:42 comment added Roger Weilik The point in the question is that the measures (the ergodic components of $\mu$) may be different with respect to which one calculated entropy in 1) and 2). An $H$-ergodic measure does not need to be $h_1$-ergodic.
Jan 13, 2013 at 13:36 history answered R W CC BY-SA 3.0