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# 8 Fractional Time Evolution

[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]

###### Theorem 8.1.

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:

1. [214.1.7] Either or there exists a number such that

 (20)
2. [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.