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12 Jun 2000
Hi All
Orbital measures and connected exercises , references, commentaries died at
12 PM
11 of June.
Regards,
Gabor
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Dettmann's response:
Dear Gabor,
I am disappointed. I think orbital measures are an important contribution
to the subject, in that they
(a) Provide an alternative way of looking at periodic orbit formulas
(b) Are simpler to understand, and so benefit the reader
(c) Are simpler to calculate, for example applications in statistical
mechanics
(d) Appear in quite a lot of literature, which would otherwise be quite
confusing to the readership of the book.
I am aware that their application has so been almost all restricted to
trace formulas, which are very much less convergent than zeta functions or
spectral determinants in the best case (uniform hyperbolicity, finite
topology), and so treatments of periodic orbit theory using only trace
formula based orbital measures are very incomplete, however
(i) There is ample discussion of these other methods in the book, so the
reader will never get the impression that orbital measures are all there
is.
(ii) For weaker chaotic properties (the extreme example would be the
randomly oriented Ehrenfest model, which is an infinite polygonal billiard
with diffusive properties intimately related to the periodic orbits,
nlin.CD/0001062), it is not clear how to extract the most information from
the periodic orbits, and trace fromulas may do as well as other methods.
(iii) Zeta function and spectral determinant expressions for averages are
linear combinations of the observable computed on the periodic orbits, so
they also correspond to orbital measures, albeit more complicated than
those from trace formulas; thus the concept of "orbital measure" is not
restricted to trace formulas and their slow convergence.
In terms of other "commentary" related to orbital measures, the part about
periodic orbits being dense is surely essential.
I hope orbital measures will live again in the virtual pages of the
webbook, for they are very much alive elsewhere.
Regards,
Carl
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16 Jun 2000
Hi!
In DasBook OMs are not used very much. All the OM texts were
just repetitions of stuff already derived. So, I admit that
they are useful research tools etc. but probably it is yet another
book :-). I kept the dicussion on why POs are dense.
Regards,
Gabor
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21 Jun 2000
Dear Predrag,
Vattay correspondence... I disagree, but leave it to you
people to decide what you want in DasBook, after all, I don't have time
at the moment to contribute to this book, let alone "yet another book".
Regards,
Carl
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