[page 214, §1]
[214.1.1] The induced time evolution is obtained from
by iteration.
[214.1.2] According to its definition in eq. (16)
the induced measure preserving transformation
acts as a convolution in time,
![]() |
(17) |
where is a mixed state on
.
[214.1.3] Iterating
times gives
![]() |
(18) |
where
is the probability density of the sum
![]() |
(19) |
of independent
and identically with
distributed random
recurrence times
.
[214.1.4] Then the long time limit
for induced
measure preserving transformations on subsets of
small measure is generally governed by
well known local limit theorems for convolutions
[42, 43, 44, 45].
[214.1.5] Application to the case at hand yields
the following fundamental theorem of fractional
dynamics [1, 21]
Assume that is maximal in the sense that there is no larger
for which all recurrence
times lie in
.
[214.1.6] Then the following conditions are equivalent:
[214.1.7] Either
or there exists a number
such that
![]() |
(20) |
[214.1.8] There exist constants and
such that
![]() |
(21) |
where , if
,
and
otherwise.
[214.1.9] The function
vanishes for
, and is
![]() |
(22) |
for .
[page 215, §0]
[215.0.1] If the limit exists, and is nondegenerate, i.e. , then
the rescaling constants
have the form
![]() |
(23) |
where is a slowly varying function [46], i.e.
![]() |
(24) |
for all .
[215.1.1] The theorem shows that
![]() |
(25) |
holds for sufficiently large .
[215.1.2] The asymptotic behaviour of the iterated
induced measure preserving transformation
for
allows to remove the discretization,
and to find the induced continuous time evolution on
subsets
.
[215.1.3] First, the definition eq. (15)
is extended from the arithmetic progression
to
by linear interpolation.
[215.1.4] Let
denote the extended measure
defined for
.
[215.1.5] Using eq. (11) and setting
![]() |
(26) |
the summation in eq. (18) can
be approximated for sufficently large
by an integral.
[215.1.6] Then
,
where
![]() |
(27) |
is the induced continuous time evolution.[215.1.7]
is also called fractional time evolution.
[215.1.8] Laplace tranformation shows that
fulfills eq. (5).
[215.1.9] It is an example of subordination of semigroups
[47, 33, 7, 48].
[215.1.10] Indeed
![]() |
(28) |
where denotes right translations
on the interpolated measure.
[215.1.11] Because
and
,
eq. (26) implies
.
[215.1.12] As remarked in the introduction,
the induced time evolution
is in general not a translation (group or semigroup), but
a convolution semigroup.
[215.1.13] The fundamental classification parameter
![]() |
(29) |
[page 216, §0]
depends not only on the dynamical rule
and the subset
, but also on the discretization time
step
, i.e. on the time scale of interest.