while we are deciding what to do and whether to move on, I will tie up a loose end where there was some disagreement about terminology at Christine's blog. Yesterday I posted this where I used the word "ergodic"
marcus said:
I just watched Lecture 1 again and I think the only main homework or thing to check is the fact about a certain set of moves being ergodic (in this case meaning that you can get from any trivalent graph to any other by repeatedly applying just those moves)
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In this situation that is simply what ergodic means. that you can get from any graph to any other graph by doing enough of these. We just showed that in the homework. Because if you can reduce any graph to a particular one, say theta, then you can get back from theta TO that graph by reversing the moves.
So you can get from A to B by collapsing A down to theta and then expanding theta out to B. There will certainly be OTHER ways to do it but that shows there is at least one.
The mental image I have of ergodic transformations is shuffling a deck of cards---which you do by repeating elementary moves. If the elementary shuffle moves are really ERGODIC then that means that if you do elementary shuffle moves enough times you can get ANY ordering of the deck.
Sorry, have to go, be back to finish later
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my wife is reading a book about Medieval monks and the relics they had at their monasteries. and the book is about other things too, but Chapter 8 is about things like
FURTA SACRA which is adventure stories about how bold and crafty monks from monastery A were sent to steal the relics at monastery B. quite a lot of that happened. there were spies and moles, sometimes a heist operation took years to prepare and put into effect.
and she tells me that at one point they were confronted with the problem that there were two heads of John the Baptist, at two different competing monasteries, and they resolved the logical dilemma by declaring that both heads were authentic, simply that one was his head when he was a young man and the other was when he was older. Of course we all know the story of how Salome danced for Herod and got him to have John the Baptist's head chopped off and given her as a present---which would have been the origin of the second of the two relics. this book is "Off the Road" by an editor of Harper's magazine, named Jack Hitt.
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Well we still have to draw the connection between THIS kind of ergodic----moves which effectively mix things around by getting from any configuration to any other configuration---and the OTHER kind of ergodic that people are used to where there is a probability measure on a set of points and a transformation of the set.
http://www.cscs.umich.edu/~crshalizi/notebooks/ergodic-theory.html
According to that defintion the transformation is ergodic, for that probability measure, if any invariant sets have either probability 1 or zero. In effect that means that the only way a nontrivial set can be invariant is that (up to sets of measure zero) it is the WHOLE THING.
But if you think about it that just means the transformation thoroughly moves things around because you can start at any point and applying the transformation over and over will eventually get you anywhere else. If it didnt, you could have an invariant set which was only a part of the whole.
an ergodic transformation, if you keep repeating it, explores the whole set.
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those University of Michigan notes say that it was Ludwig Boltzmann who coined the term ergodic and what he originally meant by it was what SMOLIN means---our meaning here! Boltzmann thought ergodic meant that it will take you from any point to any other point if you do it enough times.
funny, going by the Greek root ergos, work energy, ergodic should mean energetic, vigorous, hard-working. I wonder if that was what the mighty Ludwig meant---I would call it THOROUGHLY MIXING.
