For fpmtflag = 0, events registered in D2 modules with failed PMTs are
ignored, and the corresponding modules are also excluded in the computation of
the geometry function (cf. Appendix F). Conversely, for fpmtflag = 1 the
failure of the PMTs is ignored, and events from D2 modules are treated as if
the failed PMTs were still operating. Finally, for fpmtflag = 2, circular
exclusion regions are applied around the zones of the failed PMTs for dates
after their failure, as defined in Table 9. Events localised within these
regions are ignored and the regions are removed in the computation of the
geometry function (cf. Appendix F). Using circular exclusion regions was the
default for most of the COMPTEL analysis published in the literature in the
past. The impact of the fpmtflag value on the analysis results is illustrated
for the case of the spectral energy distribution of LS 5039 in Sect. 4.3.

Table 9: D2 module exclusion circles applied following the failure of PMTs at
the specified dates.

TJD | VP | Date | Module | $x_{e}$ | $y_{e}$ | $r_{e}$
---|---|---|---|---|---|---
8718 | 24.0 | 6-4-1992 | 13 | 9.0 | 41.2 | 9.0
8737 | 26.0 | 25-4-1992 | 11 | -51.7 | 8.7 | 9.0
8756 | 28.0 | 14-5-1992 | 14 | -34.7 | 49.0 | 9.0
8981 | 204.0 | 25-12-1992 | 2 | entire module
151515Module numbers are counted from 1. The circle centre locations $x$ and
$y$ and the circle radii $r$ are given in units of cm.

##  Appendix F Computation of the geometry function

The geometry function DRG is computed in GammaLib by the method
GCOMDri::compute_drg using

${\tt
DRG}(\chi,\psi,\bar{\varphi})=\frac{1}{N}\sum_{i\in\\{S\\}}\tilde{G}_{i}(\chi,\psi,\bar{\varphi})$
(44)

with

$\tilde{G}_{i}(\chi,\psi,\bar{\varphi})=\left\\{\begin{array}[]{l
l}\displaystyle G_{i}(\chi,\psi),&\mbox{if ${\rm EHA}(\chi,\psi)\geq{\rm
EHA}_{\rm min}(\bar{\varphi})$}\\\ \displaystyle
0,&\mbox{otherwise}\end{array}\right.,$ (45)

where ${\rm EHA}(\chi,\psi)$ is the distance between scatter direction
$(\chi,\psi)$ and the Earth horizon, ${\rm EHA}_{\rm min}(\bar{\varphi})$ is
given by Eq. (2), and $N$ is the number of selected superpackets.

The $G_{i}(\chi,\psi)$ is the geometry factor for a given superpacket and
corresponds to the area of the shadow that is cast by all active D1 modules on
all active D2 modules for a given scatter angle $(\chi,\psi)$ divided by the
total area of all D1 modules. The computation of the geometry factor is
implemented in GCOMDri::compute_geometry and calculated using

$G_{i}(\chi,\psi)=\frac{1}{7}\sum_{k=0}^{6}\delta^{{\rm
D1}}_{k}\sum_{l=0}^{13}\delta^{{\rm D2}}_{l}\tilde{o}_{kl}(\theta,\phi),$ (46)

where

$\delta^{{\rm D1}}_{k}=\left\\{\begin{array}[]{l l}\displaystyle 1,&\mbox{D1
module $k$ active}\\\ \displaystyle 0,&\mbox{otherwise}\\\ \end{array}\right.$
(47)

and

$\delta^{{\rm D2}}_{l}=\left\\{\begin{array}[]{l l}\displaystyle 1,&\mbox{D2
module $l$ active}\\\ \displaystyle 0,&\mbox{otherwise}\\\ \end{array}\right.$
(48)

are determined using the GCOMStatus class. In case of fpmtflag = 0, failed
modules according to Table 9 are also considered as inactive for superpacket
dates after the dates of PMT failure. Furthermore,

$\tilde{o}_{kl}(\theta,\phi)=\left\\{\begin{array}[]{l l}\displaystyle
0,&\mbox{if $d_{kl}\geq r_{1}+r_{2}$}\\\ \displaystyle
1-f_{kle}(\theta,\phi),&\mbox{if $d_{kl}\leq r_{2}-r_{1}+0.1$}\\\
\displaystyle o_{kl}(\theta,\phi)-f_{kle}(\theta,\phi),&\mbox{otherwise}\\\
\end{array}\right.,$ (49)

with $r_{1}=13.8$ cm being the radius of a D1 module and $r_{2}=14.085$ cm
being the radius of a D2 module. We note the margin of $0.1$ cm in Eq. (49),
which assures numerical stability with respect to rounding errors. The
$d_{kl}$ is the projected distance between the centres of the D1 and D2
modules, given by

$d_{kl}=\sqrt{(x_{l}-x_{k}+h\tan\theta\cos\phi)^{2}+(y_{l}-y_{k}+h\tan\theta\sin\phi)^{2}},$
(50)

where $x_{k}$ and $y_{k}$ are the geometric positions of the D1 modules and
$x_{l}$ and $y_{l}$ the positions of the D2 modules with respect to the
optical axis in cm, $h=158$ cm being the vertical separation between D1 and D2
modules, $(\theta,\phi)$ being the zenith and azimuth angles of the Compton
scatter direction $(\chi,\psi)$ with respect to the COMPTEL pointing
direction, and

$o_{kl}(\theta,\phi)=\frac{r_{1}^{2}(\alpha_{kl}-\sin\alpha_{kl}\cos\alpha_{kl})+r_{2}^{2}(\beta_{kl}-\sin\beta_{kl}\cos\beta_{kl})}{\pi
r_{1}^{2}}$ (51)

being the projected overlap of a D1 module and a D2 module with

$\cos\alpha_{kl}=\frac{d_{kl}^{2}+(r_{1}^{2}-r_{2}^{2})}{2d_{kl}r_{1}}$ (52)

and

$\cos\beta_{kl}=\frac{d_{kl}^{2}-(r_{1}^{2}-r_{2}^{2})}{2d_{kl}r_{2}}.$ (53)

The term $f_{kle}(\theta,\phi)$ accounts for the exclusion of circular regions
around failed PMTs and differs from zero only for fpmtflag = 2. It quantifies
the fractional overlap between the projected D1 module $k$ and the part of the
exclusion region for failed PMTs that is contained within D2 module $l$. The
exclusion region is circular, and specified by a geometric centre position
$x_{e}$ and $y_{e}$ and a radius $r_{e}$ as given in Table 9. Specifically,
for fpmtflag = 2

$f_{kle}(\theta,\phi)=\left\\{\begin{array}[]{l l}\displaystyle 0,&\mbox{if
$d_{kl}>r_{1}+r_{2}$ or $d_{ke}<r_{1}+r_{e}$}\\\ \displaystyle f^{\rm
contained}_{le}(\theta,\phi),&\mbox{if $d_{ke}+r_{e}<r_{1}$}\\\ \displaystyle
f^{\rm partial}_{kle}(\theta,\phi),&\mbox{otherwise}\\\ \end{array}\right.,$
(54)

with

$d_{ke}=\sqrt{(x_{e}-x_{k}+h\tan\theta\cos\phi)^{2}+(y_{e}-y_{k}+h\tan\theta\sin\phi)^{2}}$
(55)

being the distance between the projected D1 module centres $x_{k}$ and $y_{k}$
and the centre $x_{e}$ and $y_{e}$ of the exclusion region.

The $f^{\rm contained}_{le}(\theta,\phi)$ is the overlap for the case that the
exclusion circle is fully contained in the projected D1 module circumference.
In this case, the relevant overlap to take into account is the overlap between
the exclusion circle and the D2 module, given by

$f^{\rm contained}_{le}(\theta,\phi)=\left\\{\begin{array}[]{l l}\displaystyle
r_{2}^{2}/r_{1}^{2},&\mbox{if $r_{1}>d_{le}+r_{2}$}\\\ \displaystyle
1,&\mbox{if $r_{2}>d_{le}+r_{1}$}\\\ \displaystyle\tilde{f}^{\rm
contained}_{le}(\theta,\phi),&\mbox{otherwise}\\\ \end{array}\right.,$ (56)

with

$d_{le}=\sqrt{(x_{e}-x_{l})^{2}+(y_{e}-y_{l})^{2}}$ (57)

being the distance between the centres of the exclusion circle and the D2
module, and

$\tilde{f}^{\rm
contained}_{le}(\theta,\phi)=\frac{r_{1}^{2}(\alpha_{le}-\sin\alpha_{le}\cos\alpha_{le})+r_{2}^{2}(\beta_{le}-\sin\beta_{le}\cos\beta_{le})}{\pi
r_{1}^{2}}$ (58)

being the overlap of the exclusion region and the D2 module with

$\cos\alpha_{le}=\frac{d_{le}^{2}+(r_{1}^{2}-r_{2}^{2})}{2d_{le}r_{1}}$ (59)

and

$\cos\beta_{le}=\frac{d_{le}^{2}-(r_{1}^{2}-r_{2}^{2})}{2d_{le}r_{2}}.$ (60)

The evaluation of Eq. (56) is implemented by the method
GCOMDri::compute_surface.

Finally, $f^{\rm partial}_{kle}(\theta,\phi)$ specifies the fractional overlap
between the projected D1 module, the D2 module and the exclusion circle. This
quantity is evaluated numerically by testing a grid of $25\times 25$ $x$ and
$y$ positions around the exclusion circle. The numerical evaluation is
implemented by the method GCOMDri::compute_overlap.

##  Appendix G Response computation

The following sections describe some details of the response computations
implemented in GammaLib.

###  G.1 Efficiency factors

The efficiency factor $P_{\rm eff}(\varphi^{\prime}_{\rm geo},E_{\gamma})$ in
Eq. (6) is factorised according to

$\begin{split}P_{\rm eff}(\varphi^{\prime}_{\rm geo},E_{\gamma})=&P_{\rm
A1}(E_{\gamma})\times P_{\rm V1}(E_{\gamma})\times P_{\rm
D1}(E_{\gamma})\times P_{\rm C}(E_{\gamma})\,\times\\\ &P_{\rm
MH}(E_{\gamma})\times P_{\rm SV}(E_{\gamma})\,\times\\\ &P_{\rm
PSD}(\hat{E_{1}})\times P_{\rm A2}(\hat{E_{2}})\times P_{\rm
V23}(\hat{E_{2}})\times P_{\rm D2}(\hat{E_{2}})\,\times\\\ &P_{\rm
MS}(\varphi^{\prime}_{\rm geo},E_{\gamma})\end{split},$ (61)

with

$\hat{E_{2}}=\frac{E_{\gamma}}{\left(1-\cos\varphi^{\prime}_{\rm
geo}\right)\frac{E_{\gamma}}{m_{e}c^{2}}+1}$ (62)

being the energy of the photon entering the D2 module for a true scatter angle
$\varphi^{\prime}_{\rm geo}$ and an incident photon energy of $E_{\gamma}$,
and $\hat{E_{1}}=E_{\gamma}-\hat{E_{2}}$ being the true energy deposit in D1
for a single Compton scattering by an angle of $\varphi^{\prime}_{\rm geo}$.
The GammaLib methods that implement the computation of the efficiency factors
are summarised in Table 10.

Table 10: GammaLib methods that implement the computation of the efficiency factors.  Factor | Method
---|---
$P_{\rm eff}(\varphi^{\prime}_{\rm geo},E_{\gamma})$ | GCOMIaq::weight_iaq
$P_{\rm A1}(E_{\gamma})$ | GCOMInstChars::trans_D1
$P_{\rm V1}(E_{\gamma})$ | GCOMInstChars::trans_V1
$P_{\rm D1}(E_{\gamma})$ | GCOMInstChars::prob_D1inter
$P_{\rm C}(E_{\gamma})$ | GCOMIaq::weight_iaq
$P_{\rm MH}(E_{\gamma})$ | GCOMInstChars::prob_no_multihit
$P_{\rm SV}(E_{\gamma})$ | GCOMInstChars::prob_no_selfveto
$P_{\rm PSD}(\hat{E_{1}})$ | GCOMInstChars::psd_correction
$P_{\rm A2}(\hat{E_{2}})$ | GCOMInstChars::trans_D2
$P_{\rm V23}(\hat{E_{2}})$ | GCOMInstChars::trans_V23
$P_{\rm D2}(\hat{E_{2}})$ | GCOMInstChars::prob_D2inter
$P_{\rm MS}(\varphi^{\prime}_{\rm geo},E_{\gamma})$ | GCOMInstChars::multi_scatter
Table 11: Efficiency parameters as a function of energy.

$E$ (MeV) | $\mu_{\rm Veto}(E)$ | $\mu_{\rm Al}(E)$ | $\mu_{\rm D1}(E)$ | $\mu_{\rm D2}(E)$ | $P_{\rm MS}(E)$ | $P_{\rm SV}(E)$
---|---|---|---|---|---|---
0.05 | 0.205944 |  | 0.178262 |  |  |
0.1 | 0.169298 | 0.432 | 0.146770 | 5.9821 |  |
0.15 | 0.150704 | 0.3618 | 0.130690 | 2.15796 |  |
0.2 | 0.137444 | 0.324 | 0.119207 | 1.14871 | 1.0 |
0.25 | 0.127823 |  | 0.110860 |  |  |
0.3 | 0.119531 | 0.2781 | 0.103667 | 0.57986 | 1.0 |
0.35 | 0.112835 |  | 0.097861 |  |  |
0.4 | 0.106932 | 0.24894 | 0.092743 | 0.41471 | 1.0 |
0.45 | 0.101949 |  | 0.088426 |  |  |
0.5 | 0.097504 | 0.22707 | 0.084574 | 0.336906 | 1.0 |
0.55 | 0.093717 |  | 0.081289 |  |  |
0.6 | 0.090226 | 0.21006 | 0.078260 | 0.293233 | 1.0 |
0.65 | 0.087195 |  | 0.075629 |  |  |
0.7 | 0.084350 |  | 0.073160 |  |  |
0.75 | 0.081692 |  | 0.070853 |  |  |
0.8 | 0.079220 | 0.18468 | 0.068708 | 0.242954 | 1.0 |
0.85 | 0.077065 |  | 0.066840 |  |  |
0.9 | 0.075009 |  | 0.065057 |  |  |
0.95 | 0.073052 |  | 0.063360 |  |  |
1.0 | 0.071194 | 0.16578 | 0.061749 | 0.212493 | 0.993 |
1.25 | 0.064013 |  | 0.055524 |  |  |
1.5 | 0.058035 |  | 0.050340 | 0.170655 | 0.991 |
1.75 | 0.053580 |  | 0.046471 |  |  |
2.0 | 0.049689 | 0.1166 | 0.043092 | 0.150837 | 0.985 |
2.5 | 0.044222 |  | 0.038345 |  |  |
3.0 | 0.039767 | 0.0953 | 0.034476 | 0.134322 | 0.976 |
3.5 | 0.036580 |  | 0.031708 |  |  |
4.0 | 0.033893 | 0.0837 | 0.029373 | 0.128817 | 0.975 |
4.4 |  |  |  |  |  | 0.940
4.5 | 0.031814 |  | 0.027565 |  |  |
5.0 | 0.030016 | 0.07641 | 0.026002 | 0.127349 | 0.973 |
5.5 | 0.028562 |  | 0.024737 |  |  |
6.0 | 0.027268 | 0.07155 | 0.023611 | 0.127716 | 0.971 |
6.1 |  |  |  |  |  | 0.912
6.5 | 0.026254 |  | 0.022727 |  |  |
7.0 | 0.025319 |  | 0.021912 |  |  |
7.5 | 0.024462 |  | 0.021166 |  |  |
8.0 | 0.023684 | 0.06534 | 0.020488 | 0.130285 | 0.950 |
8.5 | 0.023043 |  | 0.019929 |  |  |
9.0 | 0.022442 |  | 0.019405 |  |  |
9.5 | 0.021880 |  | 0.018915 |  |  |
10.0 | 0.021357 | 0.0621 | 0.018459 | 0.134689 | 0.929 |
12.0 |  |  |  |  |  | 0.850
12.5 | 0.019618 |  | 0.016940 |  |  |
15.0 | 0.018280 | 0.05859 | 0.015770 | 0.147534 | 0.912 |
17.5 | 0.017477 |  | 0.015066 |  |  |
20.0 | 0.016805 | 0.05805 | 0.014475 | 0.158544 | 0.913 |
20.5 |  |  |  |  |  | 0.718
25.0 | 0.015988 |  | 0.013756 |  | 0.915 |
30.0 | 0.015407 | 0.05859 | 0.013242 | 0.176894 | 0.917 |
35.0 | 0.015158 |  | 0.013019 |  |  |
40.0 | 0.014948 | 0.06048 | 0.012830 | 0.190840 |  |
45.0 | 0.014756 |  | 0.012659 |  |  |
50.0 | 0.014646 | 0.06210 | 0.012558 | 0.201116 |  |
161616Interaction coefficients, $\mu$, are given in units of cm-1.

The $P_{\rm A1}(E_{\gamma})$ is the transmission probability for photons for
the material above D1, which is composed essentially of aluminium, and is
computed using

$P_{\rm A1}(E_{\gamma})=e^{-\mu_{\rm Al}(E_{\gamma})\,l_{\rm above}},$ (63)

where $\mu_{\rm Al}(E_{\gamma})$ is the energy-dependent interaction
coefficient of aluminium in units of cm-1 that is interpolated using a log-log
interpolation of the values given in Table 11 and $l_{\rm above}=0.147$ cm is
the thickness of the material above D1.

The $P_{\rm V1}(E_{\gamma})$ is the transmission probability for photons for
the first Veto dome and is computed using

$P_{\rm V1}(E_{\gamma})=e^{-\mu_{\rm Veto}(E_{\gamma})\,l_{\rm V1}},$ (64)

where $\mu_{\rm Veto}(E_{\gamma})$ is the energy-dependent interaction
coefficient for the Veto dome in units of cm-1 that is interpolated using a
log-log interpolation of the values given in Table 11 and $l_{\rm V1}=1.721$
cm is the thickness of the first Veto dome.

$P_{\rm D1}(E_{\gamma})$ is the interaction probability in D1 and is computed
using

$P_{\rm D1}(E_{\gamma})=1-e^{-\mu_{\rm D1}(E_{\gamma})\,l_{\rm D1}},$ (65)

where $\mu_{\rm D1}(E_{\gamma})$ is the energy-dependent D1 attenuation
coefficient in units of cm-1 that is interpolated using a log-log
interpolation of the values given in Table 11 and $l_{\rm D1}=8.5$ cm is the
thickness of the D1 modules.

$P_{\rm C}(E_{\gamma})$ is the fraction of Compton scatter interactions among
all possible photon interactions within D1 and is given by

$P_{\rm Compton}(E_{\gamma})=\left\\{\begin{array}[]{l l}\displaystyle
1.067-0.0295E_{\gamma}+3.4\,10^{-4}E^{2}_{\gamma},&\mbox{if $E_{\gamma}\geq 2$
MeV}\\\ \displaystyle 1,&\mbox{otherwise}\\\ \end{array}\right.,$ (66)

with the additional constraint of $0\leq P_{\rm Compton}(E_{\gamma})\leq 1$.

The $P_{\rm MH}(E_{\gamma})$ is the probability that there is no multi-hit.
This probability is computed using a log-log interpolation of the values of
Table 11. $P_{\rm SV}(E_{\gamma})$ is the probability that the photon was not
self-vetoed. This probability is computed using a linear interpolation of the
values of Table 11.

The $P_{\rm PSD}(\hat{E_{1}})$ is the D1 energy-dependent PSD correction and
is given by

$P_{\rm PSD}(\hat{E_{1}})=1-\frac{1}{1727.9\times\hat{E_{1}}^{2.53}+1},$ (67)

where $\hat{E_{1}}$ is in units of MeV. This correction applies to a standard
PSD selection of 0–110.

$P_{\rm A2}(\hat{E_{2}})$ gives the transmission probability of the aluminium
below the D1 modules and is computed using

$P_{\rm A2}(\hat{E_{2}})=e^{-\mu_{\rm Al}(\hat{E_{2}})\,l_{\rm T23}},$ (68)

where $\mu_{\rm Al}(\hat{E_{2}})$ is the energy-dependent interaction
coefficient for aluminium in units of cm-1 that is interpolated using a log-
log interpolation of the values given in Table 11 and $l_{\rm between}=0.574$
cm is the thickness of the aluminium plate.

$P_{\rm V23}(\hat{E_{2}})$ gives the transmission probability of the second
and third veto domes that are situated between the D1 and D2 modules, and is
computed using

$P_{\rm V23}(\hat{E_{2}})=e^{-\mu_{\rm Veto}(\hat{E_{2}})\,l_{\rm V23}},$ (69)

where $\mu_{\rm Veto}(\hat{E_{2}})$ is the energy-dependent interaction
coefficient of the Veto domes in units of cm-1 that is interpolated using a
log-log interpolation of the values given in Table 11 and $l_{\rm V23}=3.442$
cm is the combined thickness of the second and third veto domes.

$P_{\rm D2}(E_{\gamma})$ gives the interaction probability in D2 and is
computed using

$P_{\rm D2}(E_{\gamma})=1-e^{-\mu_{\rm D2}(\hat{E_{2}})\,l_{\rm D2}},$ (70)

where $\mu_{\rm D2}(\hat{E_{2}})$ is the energy-dependent D2 attenuation
coefficient in units of cm-1 that is interpolated using a log-log
interpolation of the values given in Table 11 and $l_{\rm D2}=7.525$ cm is the
thickness of the D2 modules.

Figure 21:  Energy dependence of the efficiency factor $P_{\rm
eff}(\varphi^{\prime}_{\rm geo},E_{\gamma})$ for the three geometrical scatter
angles $\varphi^{\prime}_{\rm geo}=5^{\circ}$, $25^{\circ}$, and $50^{\circ}$
(left) and the energy dependence of its components for $\varphi^{\prime}_{\rm
geo}=25^{\circ}$ (right).

The $P_{\rm MS}(E_{\gamma},\hat{E_{2}})$ is the probability that a photon that
has interacted in D1 leaves the D1 module without any further interaction.
This probability is computed using

$P_{\rm MS}(\varphi^{\prime}_{\rm geo},E_{\gamma})=\frac{\int_{0}^{r_{\rm
D1}}\int_{0}^{l_{\rm D1}}\mu_{\rm D1}(E_{\gamma})e^{-\mu_{\rm
D1}(E_{\gamma})z}\int_{0}^{\pi}e^{-\mu_{\rm
D1}(\hat{E_{2}})\,l(r,z,\phi)}\,d\phi\,dz\,dr}{\pi r^{2}_{\rm
D1}\left(1-e^{-\mu_{\rm D1}(E_{\gamma})\,l_{\rm D1}}\right)},$ (71)

which integrates all possible second interaction locations within a given D1
module assuming vertically incident photons, where $r_{\rm D1}=13.8$ cm and
$l_{\rm D1}=8.5$ cm is the radius and thickness of a D1 module. $l(r,z,\phi)$
gives the remaining interaction length for a photon that has interacted at
radius $r$, vertical depth $z$ and azimuth angle $\phi$ of the module and is
given by

$l(r,z,\phi)=\left\\{\begin{array}[]{l l}\displaystyle\frac{l_{\rm
D1}-z}{\cos\varphi^{\prime}_{\rm geo}},&\mbox{if
$l(r,z,\phi)\sin\varphi^{\prime}_{\rm geo}\leq R(r,\phi)$}\\\ \\\
\displaystyle\frac{R(r,\phi)}{\sin\varphi^{\prime}_{\rm
geo}},&\mbox{otherwise}\\\ \end{array}\right.$ (72)

with

$R(r,\phi)=\sqrt{r^{2}_{\rm D1}-r^{2}\sin^{2}\phi}-r\cos\phi.$ (73)

Figure 21 illustrates the energy dependence of the efficiency factor $P_{\rm
eff}(\varphi^{\prime}_{\rm geo},E_{\gamma})$ and its components.

###  G.2 Klein-Nishina contribution

$P_{\rm KN}(\varphi^{\prime}_{\rm
geo},E_{\gamma})=\frac{\int_{\varphi^{\prime}_{\rm
geo,min}}^{\varphi^{\prime}_{\rm geo,max}}\sigma_{\rm
KN}(\varphi^{\prime\prime}_{\rm geo},E_{\gamma})\,d\varphi^{\prime\prime}_{\rm
geo}}{\int_{0}^{\pi}\sigma_{\rm KN}(\varphi^{\prime\prime}_{\rm
geo},E_{\gamma})\,d\varphi^{\prime\prime}_{\rm geo}}$ (74)

is the contribution of the Klein-Nishina cross-section $\sigma_{\rm
KN}(\varphi^{\prime\prime}_{\rm geo},E_{\gamma})$ to the
$\varphi^{\prime}_{\rm geo}$ bin spanned by $[\varphi^{\prime}_{\rm
geo,min},\varphi^{\prime}_{\rm geo,max}]$, where the integrals are computed
using

$\begin{split}\int_{\varphi^{\prime}_{\rm geo,min}}^{\varphi^{\prime}_{\rm
geo,max}}\sigma_{\rm KN}(\varphi^{\prime\prime}_{\rm
geo},E_{\gamma})\,d\varphi^{\prime\prime}_{\rm geo}=&\int_{0}^{E_{1}^{\rm
up}}\sigma_{\rm KN}(E^{\prime}_{1},E_{\gamma})\,dE^{\prime}_{1}\,-\\\
&\int_{0}^{E_{1}^{\rm low}}\sigma_{\rm
KN}(E^{\prime}_{1},E_{\gamma})\,dE^{\prime}_{1}\end{split}$ (75)

with

$\begin{split}\int_{0}^{E_{1}}\sigma_{\rm
KN}(E^{\prime}_{1},&E_{\gamma})\,dE^{\prime}_{1}=\frac{E_{1}}{E_{\gamma}}\left(\frac{m_{e}c^{2}}{E_{1}}+1\right)^{2}+\\\
&\left(2\left(\frac{m_{e}c^{2}}{E_{\gamma}}\right)^{2}+2\frac{m_{e}c^{2}}{E_{\gamma}}-1\right)\times\log\left(1-\frac{E_{1}}{E_{\gamma}}\right)-\\\
&\frac{1}{2}\left(\frac{E_{1}}{E_{\gamma}}\right)^{2}+\left(\frac{m_{e}c^{2}}{E_{\gamma}}\right)^{2}\frac{1}{1-\frac{E_{1}}{E_{\gamma}}}\end{split}.$
(76)

###  G.3 Module spectral response functions

Figure 22:  Detector response functions $R_{\rm D1}(E_{1}|\hat{E_{1}})$ (left)
and $R_{\rm D2}(E_{2}|\hat{E_{2}})$ (right) for a selected number of energies
that have been chosen to allow for a comparison with Figs. 3c and 4b in Diehl
et al. (1992).

####  G.3.1 D1 spectral response

The spectral response $R_{\rm D1}(E_{1}|\hat{E_{1}})$ provides the probability
for measuring an energy $E_{1}$ when an incident photon with energy
$E_{\gamma}$ was Compton scattered by an angle $\varphi_{\rm geo}$, producing
a true energy deposit of $\hat{E_{1}}$ in the detector module. The D1 response
is implemented by the GCOMD1Response class and the response evaluation is done
using the GCOMD1Response::operator() operator. The spectral D1 response is
computed using a Gaussian

$R_{\rm
D1}(E_{1}|\hat{E_{1}})=\frac{T_{1}(E_{1}|\hat{E_{1}})}{\sigma(\hat{E_{1}})\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{\mu_{\rm
p}(\hat{E_{1}})-E_{1}}{\sigma(\hat{E_{1}})}\right)^{2}\right),$ (77)

where $\mu_{\rm p}(\hat{E_{1}})$ is the position of the Gaussian peak for true
energy deposit of $\hat{E_{1}}$, with $\mu_{\rm
p}(\hat{E_{1}})\approx\hat{E_{1}}$, and $\sigma(\hat{E_{1}})$ is the standard
deviation of the Gaussian (Diehl et al. 1992). $T_{1}(E_{1}|\hat{E_{1}})$ is a
threshold function, defined by

$T_{1}(E_{1}|\hat{E_{1}})=\left\\{\begin{array}[]{l l}\displaystyle
0,&\mbox{if $E_{1}\leq E_{1}^{\rm thres}(\hat{E_{1}})$}\\\ \\\
\displaystyle\frac{E_{1}-E_{1}^{\rm thres}(\hat{E_{1}})}{\Delta
E_{1}(\hat{E_{1}})},&\mbox{if $0<E_{1}-E_{1}^{\rm
thres}(\hat{E_{1}})\leq\Delta E_{1}(\hat{E_{1}})$}\\\ \\\ \displaystyle
0,&\mbox{if $E_{1}>E_{1}^{\rm max}(\hat{E_{1}})$}\\\ \\\ \displaystyle
1,&\mbox{otherwise}\\\ \end{array}\right.,$ (78)

where $E_{1}^{\rm thres}(\hat{E_{1}})=E_{1}^{\rm
min}(\hat{E_{1}})-\frac{1}{2}\Delta E_{1}(\hat{E_{1}})$. $\mu_{\rm
p}(\hat{E_{1}})$ and $\sigma(\hat{E_{1}})$ as well as $E_{1}^{\rm
min}(\hat{E_{1}})$, $E_{1}^{\rm max}(\hat{E_{1}})$ and $\Delta
E_{1}(\hat{E_{1}})$ are tabulated as a function of $\hat{E_{1}}$ in a
calibration file of type SDA that is included in GammaLib. Values for a given
energy $\hat{E_{1}}$ are obtained by linear interpolation of the tabulated
values.

The spectral response $R_{\rm D1}(E_{1}|\hat{E_{1}})$ is illustrated in the
left panel of Fig. 22 for a selected number of input energies $\hat{E_{1}}$.
The figure can be compared with Fig. 3c in Diehl et al. (1992), which presents
the response functions as implemented in COMPASS at the time of the CGRO
launch for comparable energies.

####  G.3.2 D2 spectral response

The spectral response $R_{\rm D2}(E_{2}|\hat{E_{2}})$ provides the probability
for measuring an energy $E_{2}$ when an photon with energy $\hat{E_{2}}$ hit a
D2 detector module. The D2 response is implemented by the GCOMD2Response class
and the response evaluation is done using the GCOMD2Response::operator()
operator. The spectral D2 response is computed using

$\begin{split}R_{\rm D2}(E_{2}|\hat{E_{2}})=&(P_{\rm
pp}(E_{2}|\hat{E_{2}})+P_{\rm se}(E_{2}|\hat{E_{2}})+P_{\rm
de}(E_{2}|\hat{E_{2}})\,+\\\ &C_{\rm com}(E_{2}|\hat{E_{2}})+C_{\rm
bkg}(E_{2}|\hat{E_{2}}))\times T_{2}(E_{2}|\hat{E_{2}})\end{split},$ (79)

which is a combination of three Gaussian and two continuum components
multiplied by a threshold function $T_{2}(E_{2}|\hat{E_{2}})$ (Diehl et al.
1992). Here, $\hat{E_{2}}$ is the energy of the gamma-ray photon incident on a
D2 module, and $E_{2}$ is the measured energy deposit.

The first Gaussian,

$P_{\rm pp}(E_{2}|\hat{E_{2}})=\frac{A_{\rm
pp}(\hat{E_{2}})}{\sigma(\hat{E_{2}})\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{\mu_{\rm
pp}(\hat{E_{2}})-E_{2}}{\sigma(\hat{E_{2}})}\right)^{2}\right),$ (80)

describes the so-called photo-peak which represents photons that were fully
absorbed in the D2 module. We note that the position $\mu_{\rm
pp}(\hat{E_{2}})$ of the photo-peak is not necessarily identical to the
incident energy, although below $\sim 10$ MeV $\mu_{\rm
pp}(\hat{E_{2}})\approx\hat{E_{2}}$ is well satisfied.

The two subsequent Gaussians describe the single and double escape peak,
respectively, which are generated by photons above 1.022 MeV that undergo pair
creation resulting in the production of two 511 keV gamma rays. The escape of
one or both of these 511 keV photons from the D2 module leads to
characteristic peaks at energies 0.511 MeV and 1.022 MeV below the photo-peak
energy. The escape peaks are modelled using

$P_{\rm se}(E_{2}|\hat{E_{2}})=\frac{A_{\rm se}(\hat{E_{2}})}{\sigma(\mu_{\rm
se}(\hat{E_{2}}))\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{\mu_{\rm
se}(\hat{E_{2}})-E_{2}}{\sigma(\mu_{\rm se}(\hat{E_{2}}))}\right)^{2}\right)$
(81)

and

$P_{\rm de}(E_{2}|\hat{E_{2}})=\frac{A_{\rm de}(\hat{E_{2}})}{\sigma(\mu_{\rm
de}(\hat{E_{2}}))\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{\mu_{\rm
de}(\hat{E_{2}})-E_{2}}{\sigma(\mu_{\rm de}(\hat{E_{2}}))}\right)^{2}\right),$
(82)

with $\mu_{\rm se}(\hat{E_{2}})=\mu_{\rm pp}(\hat{E_{2}})-m_{e}c^{2}$ and
$\mu_{\rm de}(\hat{E_{2}})=\mu_{\rm pp}(\hat{E_{2}})-2m_{e}c^{2}$, where
$m_{e}c^{2}\approx 511$ keV is the rest energy of the electron.

The first continuum component,

$\begin{split}C_{\rm com}(E_{2}|&\hat{E_{2}})=A_{\rm
com}(\hat{E_{2}})\,\times\\\
&\int_{E_{2}-3\sigma(E_{2})}^{\max(E_{2}+3\sigma(E_{2}),E_{c})}\sigma_{\rm
KN}(E|\hat{E_{2}})\,f(E|E_{2},\hat{E_{2}})\,K(E|E_{2})\,dE\end{split},$ (83)

describes the so-called Compton tail which represents photons that underwent a
Compton scattering before escaping the D2 module. The component is the product
of the Klein-Nishina cross-section for Compton scattering,

$\sigma_{\rm
KN}(E|\hat{E_{2}})=\left(\frac{E/\hat{E_{2}}}{1-E/\hat{E_{2}}}\frac{m_{e}c^{2}}{\hat{E_{2}}}\right)^{2}-\frac{E}{\hat{E_{2}}}+\frac{1}{1-E/\hat{E_{2}}},$
(84)

and the probability that the Compton scattered photon does not undergo a
second Compton scattering before escaping, empirically modelled using

$f(E|\hat{E_{2}})=\left\\{\begin{array}[]{l
l}\displaystyle\exp\left(-\mu(E|\hat{E_{2}})\,l(\hat{E_{2}})\right),&\mbox{if
$\hat{E_{2}}>12.14$ MeV}\\\ \displaystyle 0,&\mbox{otherwise}\\\
\end{array}\right.,$ (85)

where

$\mu(E|\hat{E_{2}})=0.72\,e^{-1.28(\hat{E_{2}}-E)^{0.35}}+0.01\,(\hat{E_{2}}-E)+0.014\,(\hat{E_{2}}-E)^{-2.5}$
(86)

is the total linear attenuation coefficient in NaI, which is the scintillator
material of the D2 modules and

$l(\hat{E_{2}})=2.9\ln(\hat{E_{2}}-11.14)$ (87)

is an empirical path length in the D2 module, where $\hat{E_{2}}$ is in units
of MeV. The product is convolved with a Gaussian kernel,

$K(E|E_{2})=\frac{1}{\sigma(E_{2})\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{E-E_{2}}{\sigma(E_{2})}\right)^{2}\right),$
(88)

to take the energy resolution of the D2 module into account, where
$\sigma(E_{2})$ is the standard deviation of the detector response at the
reconstructed energy $E_{2}$.

The second continuum component,

$C_{\rm bkg}(E_{2}|\hat{E_{2}})=A_{\rm
bkg}(\hat{E_{2}})\int_{E_{2}-3\sigma(E_{2})}^{\max(E_{2}+3\sigma(E_{2}),\hat{E_{2}})}K(E|E_{2})\,dE,$
(89)

is a flat background component that is convolved by a Gaussian kernel and that
takes into account any higher-order scatterings in the D2 modules.

The threshold function is defined by

$T_{2}(E_{2}|\hat{E_{2}})=\left\\{\begin{array}[]{l
l}\displaystyle\exp{\left(-\frac{1}{2}\left(\frac{E_{2}^{\rm
thres}(\hat{E_{2}})-E_{2}}{\sigma_{\rm
thres}(\hat{E_{2}})}\right)^{2}\right)},&\mbox{if $E_{2}<E_{2}^{\rm
thres}(\hat{E_{2}})$}\\\ \\\ \displaystyle 0,&\mbox{if $E_{2}>E_{2}^{\rm
max}(\hat{E_{2}})$}\\\ \\\ \displaystyle 1,&\mbox{otherwise}\\\
\end{array}\right.,$ (90)

where $E_{2}^{\rm thres}(\hat{E_{2}})=E_{2}^{\rm
min}(\hat{E_{2}})+\frac{1}{2}\Delta E_{2}(\hat{E_{2}})$ and $\sigma_{\rm
thres}(\hat{E_{2}})=\Delta E_{2}(\hat{E_{2}})/2\sqrt{2\log 2}$

All five response components have amplitude parameters ($A_{\rm pp}$, $A_{\rm
se}$, $A_{\rm de}$, $A_{\rm com}$ and $A_{\rm bkg}$) that are tabulated as a
function of incident energy $\hat{E_{2}}$ in a calibration file of type SDB.
In addition, this file contains also tabulated values for the photo-peak
energy $\mu_{\rm pp}(\hat{E_{2}})$, the standard deviation
$\sigma(\hat{E_{2}})$ and the minimum and maximum threshold energies
$E_{2}^{\rm min}(\hat{E_{2}})$ and $E_{2}^{\rm max}(\hat{E_{2}})$ and width
$\Delta E_{2}(\hat{E_{2}})$. Values for a given energy $\hat{E_{2}}$ are
obtained by linear interpolation of the tabulated values.

The spectral response $R_{\rm D2}(E_{2}|\hat{E_{2}})$ is illustrated in the
right panel of Fig. 22 for a selected number of input energies $\hat{E_{2}}$.
The figure can be compared with Fig. 4b in Diehl et al. (1992), which presents
the response functions as implemented in COMPASS at the time of the CGRO
launch for comparable energies.

###  G.4 Response convolution

The response is convolved with the Gaussian kernel

$K(\varphi^{\prime}_{\rm geo}|\varphi_{\rm
geo},\theta)=\frac{1}{\sigma(\varphi^{\prime}_{\rm
geo},\theta)\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{\varphi_{\rm
geo}-\varphi^{\prime}_{\rm geo}}{\sigma(\varphi^{\prime}_{\rm
geo},\theta)}\right)^{2}\right)$ (91)

to take into account the location uncertainties of the events in the D1 and D2
modules. The width $\sigma(\varphi^{\prime}_{\rm geo},\theta)$ of the Gaussian
kernel is a function of the scatter angle $\varphi^{\prime}_{\rm geo}$ and the
zenith angle $\theta$ of the incoming photon, measured with respect to the
COMPTEL pointing axis. The width

$\sigma(\varphi^{\prime}_{\rm geo}\theta)=\sqrt{\sigma_{\rm
xy}^{2}(\varphi^{\prime}_{\rm geo},\theta)+\sigma_{\rm
z}^{2}(\varphi^{\prime}_{\rm geo},\theta)}$ (92)

is the squared mean of the uncertainty

$\sigma_{\rm xy}^{2}(\varphi^{\prime}_{\rm
geo},\theta)=\frac{\left(\sigma_{\rm x}^{2}+\sigma_{\rm
y}^{2}\right)}{h^{2}}\,f_{\rm xy}(\varphi^{\prime}_{\rm geo},\theta)$ (93)

in the horizontal direction and the uncertainty

$\sigma_{\rm z}^{2}(\varphi^{\prime}_{\rm geo},\theta)=\frac{\left(\sigma_{\rm
z1}^{2}+\sigma_{\rm z2}^{2}\right)}{h^{2}}\,f_{\rm z}(\varphi^{\prime}_{\rm
geo},\theta)$ (94)

in the vertical direction, where $\sigma_{\rm x}=2.3$ cm and $\sigma_{\rm
y}=1.96$ cm is the average horizontal location spread in a D1 module with
respect to the D2 module in $x$ and $y$ direction, respectively, $\sigma_{\rm
z1}=2.45$ cm and $\sigma_{\rm z2}=2.17$ cm is the vertical location spread in
a D1 and D2 module, respectively, assuming a homogenous event distribution in
the z-direction, and $h=158$ cm is the distance between the D1 and D2 modules.
The functions

$\begin{split}f_{\rm xy}(\varphi^{\prime}_{\rm
geo},\theta)=&\cos^{2}\theta\cos^{2}\varphi^{\prime}_{\rm
geo}(4\sin^{2}\theta\sin^{2}\varphi^{\prime}_{\rm
geo}+\cos^{2}\varphi^{\prime}_{\rm geo}-\sin^{2}\theta)\,+\\\
&\frac{1}{2}\sin^{2}\theta\,\times\\\
&\left(\cos^{2}\theta\cos^{2}\varphi^{\prime}_{\rm
geo}+\sin^{2}\theta\sin^{2}\varphi^{\prime}_{\rm
geo}\left(1-\frac{3}{4}\cos^{2}\varphi^{\prime}_{\rm
geo}\right)\right)\end{split}$ (95)

and

$\begin{split}f_{\rm z}(\varphi^{\prime}_{\rm
geo},\theta)=&\cos^{4}\theta\sin^{2}\varphi^{\prime}_{\rm
geo}\cos^{2}\varphi^{\prime}_{\rm geo}+\\\
&\cos^{2}\theta\sin^{2}\theta\left(\frac{1}{2}-3\sin^{2}\varphi^{\prime}_{\rm
geo}\cos^{2}\varphi^{\prime}_{\rm geo}\right)+\\\
&\frac{3}{8}\sin^{4}\theta\sin^{2}\varphi^{\prime}_{\rm
geo}\cos^{2}\varphi^{\prime}_{\rm geo}\end{split}$ (96)

are the geometrical relations that transform from the uncertainties within the
modules to uncertainties in $\varphi^{\prime}_{\rm geo}$ for a given zenith
angle $\theta$.

Since COMPTEL response functions have no explicit zenith angle dependence, the
computations are performed for an average zenith angle of $\theta=25^{\circ}$.
The uncertainty introduced by this approximation is illustrated in Fig. 23,
and amounts to less than $10\%$ for zenith angle $\theta<35^{\circ}$.

Figure 23:  Width $\sigma(\varphi^{\prime}_{\rm geo},\theta)$ of the Gaussian
smoothing kernel as a function of $\varphi^{\prime}_{\rm geo}$ for zenith
angles $\theta$ of $0^{\circ}$ (solid blue), $25^{\circ}$ (dashed orange),
$35^{\circ}$ (dashed-dotted green), and $50^{\circ}$ (dotted purple).

##  Appendix H Carbon footprint of this research

Table 12: Carbon footprint estimate by emission source of the research work
behind this paper.

Source | Emissions kgCO2e | Comments
---|---|---
Electricity to power office building | $525\pm 53$ |
Heating of office building | $422\pm 118$ |
Water usage in office building | $8\pm 2$ |
Air conditioning in office building | $80\pm 23$ |
Waste management | $209\pm 106$ |
Computing infrastructure | $400\pm 200$ | Evaluated using csfootprint
Data flow | $8\pm 8$ | Includes videoconferencing
Lunch meals | $171\pm 103$ | 10% classical, 10% flexitarian, 80% vegetarian
Home-to-office commuting | $89\pm 36$ |  $90\%$ cycling, $5\%$ public transport, $5\%$ car
Total | $1911\pm 284$ |

171717The footprint is based on a total work time estimate of one FTE or 220
working days.

Given the repeated alarms of the scientific community on global warming (IPCC
2021, 2022), biodiversity loss (IPBES 2019) and air, water and soil pollution
(Landrigan et al. 2017), the environmental impact of any human activity,
including obviously research, can no longer be ignored. Here we assess the GHG
emissions related to our research work, including contributions from office
use, research activities and research environment. We based our assessment on
the carbon footprint estimate of the Institut de Recherche en astrophysique et
planétologie (IRAP; Martin et al. 2022) where most of the work for this
research was done. To attribute a share of the total footprint to our work, we
estimated that roughly one full time equivalent (FTE) was necessary to
accomplish this research, corresponding to 220 working days. Consequently, we
use the annual emissions per source from Martin et al. (2022) divided by the
IRAP staff of 263 persons to assess the contributions from the various sources
of GHG emissions.

The results of our assessment are summarised in Table 12. We estimate the
total carbon footprint of our research work to be $1.9\pm 0.3$ tCO2e, composed
of $1.2\pm 0.2$ tCO2e (65%) for running the office building in which the work
was done, $0.4\pm 0.2$ tCO2e (21%) for the computing infrastructure and data
flow, and $0.3\pm 0.1$ tCO2e (14%) for lunch meals and home-to-office
commuting.

We based our assessment for computing on the carbon footprint estimation
implemented in ctools, yet when we started this research the relevant
functionality was not yet available. We note, however, that the carbon
footprint estimate for the final analyses for this publication amounted to 60
kgCO2e, of which 20 kgCO2e were attributed to power consumption and 40 kgCO2e
to the computing infrastructure. Making the conservative estimate that ten
analysis iterations were necessary for this research work we obtain a total
carbon footprint due to computing of 600 kgCO2e, of which 200 kgCO2e were due
to power consumption and 400 kgCO2e due to the computing infrastructure. To
avoid double counting, we disregarded the contribution of power consumption
since it is already accounted for in the electricity to power the office
building.

For emission from lunch meals and home-to-office commuting we took the
specific habits of the co-authors weighted by their contribution to the
research into account. We note that our research work did not require any
professional travelling. The footprint for a couple of video conferences that
we organised for this research is included under the ‘Data flow’ source.


# Oseledets Decomposition on Sub semiflows

Marek Kryspin  Faculty of Pure and Applied Mathematics, Wrocław University of
Science and Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland.
<EMAIL_ADDRESS>
###### Abstract.

The existence of the Oseledets decomposition on continuously embedded
subspaces of Banach spaces is proved in this paper. Natural assumptions
facilitating such transfer of the Oseledets decomposition are presented,
notably conditions often met by dynamical systems generated by differential
equations.

###### Key words and phrases:

Oseledets decomposition, random dynamical systems, random delay differential
systems

###### 2020 Mathematics Subject Classification:

Primary: 37H15. Secondary: 34K06

The author is supported by the National Science Centre, Poland (NCN) under the
grant Sonata Bis with a number NCN 2020/38/E/ST1/00153

## Introduction

The primary objective of this work is to delineate natural assumptions
facilitating the transfer of the Oseledets decomposition from a Banach space
into another Banach space continuously embedded in the former. In general, the
Oseledets-type decomposition implies the partitioning/splitting (in the form
of a direct sum) of the fiber space, within which a dynamical system operates,
into finite-dimensional subspaces, potentially an infinite many of them, each
corresponding to a specific Lyapunov exponent. This process gives rise to a
hierarchy of subspaces often referred to as the Oseledets filtration or flags.
Lyapunov exponents hold paramount significance in the realm of systems
dynamics, as they dictate the exponential asymptotic growth rate along
trajectories. For results, see, e.g., [7, 9, 10, 11, 12, 16, 17, 25].

There are numerous papers that concentrate on the subject of dynamical systems
generated by differential equations and various phase space decompositions;
for instance [1, 4, 5, 6, 7, 8, 14, 20, 21, 22, 18, 19]. In many cases of
differential equations there is no “natural” phase space. Nonetheless, it
appears that there is a scarcity of research papers addressing the topics of
regularization and the transfer of Oseledets-type decomposition. As examples,
we can mention here, first, ordinary or partial differential equations with
delay, as considered in [15, 23, 24], and, second, advection diffusion
equations and others, as investigated in [2]. The paper [13] is also worth
mentioning, where the authors discuss the possibility of transferring the
Oseledets decomposition to dense subspaces of the fiber space with separable
dual.

The presuppositions of this paper, meaning the requirements set for the
solving operators and the fiber space, are often considerably weaker than the
specific properties of these (for instance, compactness or separable dual are
not necessarily required). On the other hand, such relaxation of assumptions
may lead to new results. To be precise, decompositions in subspaces (in
particular, subspaces with a finer/stronger topology are of particular
interest here) of Banach spaces (in terms of scales and continuous embeddings)
frequently encountered in real-world problems related to differential
equations, may arise from this approach. To the author’s knowledge, there are
no known theorems allowing the transfer of Oseledets decompositions to
subspaces without assuming the previously mentioned separability of the fiber
space and/or its dual.

To give a flavor of our results, we formulate now some specializations of our
main results. It is well-known that a linear ordinary differential equation
with delay generates a dynamical system that admits the Oseledets splitting in
the fiber space $L_{p}([-1,0],\mathbb{R}^{N})\oplus\mathbb{R}^{N}$. This
follows from nice properties of fiber spaces, such as separability and
reflexivity. It is also known that in practice, such equations possess a
regularisation property that leads to continuous solutions. Pullback technique
allows us to transfer the Oseledets decomposition to more regular spaces (with
finer topology).

##  1\. Preliminaries and definitions

In this section, we will present definitions of measurable dynamical systems,
measurable linear skew-product semidynamical systems, and the Oseledets
decomposition.

###  1.1. Measurable dynamical systems

We write $\mathbb{R}^{+}$ for $[0,\infty)$. For a metric space $S$ by
$\mathfrak{B}(S)$ we denote the $\sigma$-algebra of Borel subsets of $S$. A
probability space is a triple $(\Omega,\mathfrak{F},\mathbb{P})$, where
$\Omega$ is a set, $\mathfrak{F}$ is a $\sigma$-algebra of subsets of
$\Omega$, and $\mathbb{P}$ is a probability measure defined for all
$F\in\mathfrak{F}$. We always assume that the measure $\mathbb{P}$ is
complete.

A measurable dynamical system on the probability space
$(\Omega,\mathfrak{F},\mathbb{P})$ is a
$(\mathfrak{B}(\mathbb{R})\otimes\mathfrak{F},\mathfrak{F})$-measurable
mapping $\theta\colon\mathbb{R}\times\Omega\to\Omega$ such that

  * $\bullet$

$\theta(0,\omega)=\omega$ for any $\omega\in\Omega$,

  * $\bullet$

$\theta(t+s,w)=\theta(t,\theta(s,\omega))$ for any $\omega\in\Omega$ and
$t,\,s\in\mathbb{R}^{+}$.

We write $\theta(t,\omega)$ as $\theta_{t}\omega$. Also, we usually denote
measurable dynamical systems by
$((\Omega,\mathfrak{F},\mathbb{P}),(\theta_{t})_{t\in\mathbb{R}})$ or simply
by $(\theta_{t})_{t\in\mathbb{R}}$.

A metric dynamical system is a measurable dynamical system
$((\Omega,\mathfrak{F},\mathbb{P}),(\theta_{t})_{t\in\mathbb{R}})$ such that
for each $t\in\mathbb{R}$ the mapping $\theta_{t}\colon\Omega\to\Omega$ is
$\mathbb{P}$-preserving (i.e., $\mathbb{P}(\theta_{t}^{-1}(F))=\mathbb{P}(F)$
for any $F\in\mathfrak{F}$ and $t\in\mathbb{R}$). A subset
$\Omega^{\prime}\subset\Omega$ is _invariant_ if
$\theta_{t}(\Omega^{\prime})=\Omega^{\prime}$ for all $t\in\mathbb{R}$, and
the metric dynamical system is said to be _ergodic_ if for any invariant
subset $F\in\mathfrak{F}$, either $\mathbb{P}(F)=1$ or $\mathbb{P}(F)=0$.
Throughout the paper we will assume that $\mathbb{P}$ is ergodic.

###  1.2. Measurable linear skew-product semidynamical systems

By a measurable linear skew-product semidynamical system or semiflow,
$\Phi=\allowbreak((U_{\omega}(t))_{\omega\in\Omega,t\in\mathbb{R}^{+}},\allowbreak(\theta_{t})_{t\in\mathbb{R}})$
on $X$ covering a metric dynamical system $(\theta_{t})_{t\in\mathbb{R}}$ we
understand a
$(\mathfrak{B}(\mathbb{R}^{+})\otimes\mathfrak{F}\otimes\mathfrak{B}(X),\mathfrak{B}(X))$-measurable
mapping

$[\,\mathbb{R}^{+}\times\Omega\times X\ni(t,\omega,u)\mapsto
U_{\omega}(t)\,u\in X\,]$

satisfying

$\displaystyle U_{\omega}(0)=\mathrm{Id}_{X}\quad$ $\displaystyle\textrm{for
each }\,\omega\in\Omega,$ $\displaystyle U_{\theta_{s}\omega}(t)\circ
U_{\omega}(s)=U_{\omega}(t+s)\qquad$ $\displaystyle\textrm{for each
}\,\omega\in\Omega\textrm{ and }t,\,s\in\mathbb{R}^{+},$ (1.1)
$\displaystyle[\,X\ni u\mapsto U_{\omega}(t)u\in X\,]\in\mathcal{L}(X)$
$\displaystyle\textrm{for each }\,\omega\in\Omega\textrm{ and
}t\in\mathbb{R}^{+}.$

Equation (1.1) is called the cocycle property.

By the positive semiorbit passing through $(\omega,u)\in\Omega\times X$ we
understand a
$(\mathfrak{B}([0,\infty)),\allowbreak\mathfrak{B}(X))$-measurable mapping

$\big{[}[0,\infty)\ni t\mapsto U_{\omega}(t)u\in X\big{]}.$

A negative semiorbit passing through $(\omega,u)\in\Omega\times X$ is a
$(\mathfrak{B}((-\infty,0]),\mathfrak{B}(X))$-measurable mapping
$\tilde{u}\colon(-\infty,0]\to X$ such that

  * $\bullet$

$\tilde{u}(0)=u$;

  * $\bullet$

$\tilde{u}(s+t)=U_{\theta_{s}\omega}(t)\tilde{u}(s)$ for each $s\leq 0$,
$t\geq 0$ such that $s+t\leq 0$.

By a full or entire orbit passing through $(\omega,u)\in\Omega\times X$ we
understand a
$(\mathfrak{B}(\mathbb{R}),\allowbreak\mathfrak{B}(X))$-measurable mapping
${u}\colon\mathbb{R}\to X$ such that

  * $\bullet$

$\tilde{u}(0)=u$;

  * $\bullet$

$\tilde{u}(s+t)=U_{\theta_{s}\omega}\tilde{u}(s)$ for each $s\in\mathbb{R}$
and $t\geq 0$.

From now on, we will focus on separable Banach spaces. Furthermore, based on
the results from [17, Lemma 5.6 and Corollary 7.3], we will refrain from
discussing measurability in the Grassmanian sense in favor of an equivalent
definition of a measurable basis. Let $\Omega_{0}\in\mathfrak{F}$. A family
$\\{E(\omega)\\}_{\omega\in\Omega_{0}}$ of $l$-dimensional vector subspaces of
$X$ is measurable if there are $(\mathfrak{F},\mathfrak{B}(X))$-measurable
functions $v_{1},\dots,v_{l}\colon\Omega_{0}\to X$ such that
$(v_{1}(\omega),\dots,v_{l}(\omega))$ forms a basis of $E(\omega)$ for each
$\omega\in\Omega_{0}$.

Let $\\{E(\omega)\\}_{\omega\in\Omega_{0}}$ be a family of $l$-dimensional
vector subspaces of $X$, and let $\\{F(\omega)\\}_{\omega\in\Omega_{0}}$ be a
family of $l$-codimensional closed vector subspaces of $X$, such that
$E(\omega)\oplus F(\omega)=X$ for all $\omega\in\Omega_{0}$. We define the
family of projections associated with the decomposition $E(\omega)\oplus
F(\omega)=X$ as $\\{P(\omega)\\}_{\omega\in\Omega_{0}}$, where $P(\omega)$ is
the linear projection of $X$ onto $F(\omega)$ along $E(\omega)$, for each
$\omega\in\Omega_{0}$.

The family of projections associated with the decomposition $E(\omega)\oplus
F(\omega)=X$ is called strongly measurable if for each $u\in X$ the mapping
$[\,\Omega_{0}\ni\omega\mapsto P(\omega)u\in X\,]$ is
$(\mathfrak{F},\mathfrak{B}(X))$-measurable.

We say that the decomposition $E(\omega)\oplus F(\omega)=X$, with
$\\{E(\omega)\\}_{\omega\in\Omega_{0}}$ finite-dimensional, is invariant if
$\Omega_{0}$ is invariant, $U_{\omega}(t)E(\omega)=E(\theta_{t}\omega)$ and
$U_{\omega}(t)F(\omega)\subset F(\theta_{t}\omega)$, for each
$t\in\mathbb{R}^{+}$.

A strongly measurable family of projections associated with the invariant
decomposition $E(\omega)\oplus F(\omega)=X$ is referred to as tempered if

$\lim\limits_{t\to\pm\infty}\frac{\ln{\lVert
P(\theta_{t}\omega)\rVert}}{t}=0\qquad\mathbb{P}\text{-a.e. on }\Omega_{0}.$

###  1.3. Oseledets decomposition

From now on we assume that for a given semiflow

  1. (E1)

the functions

     * $\bigl{[}\,\Omega\ni\omega\mapsto\sup\limits_{0\leq s\leq 1}{\operatorname{ln^{+}}{\lVert U_{\omega}(s)\rVert}}\in\mathbb{R}^{+}\,\bigr{]}\in L_{1}(\Omega,\mathfrak{F},\mathbb{P})$,

     * $\bigl{[}\,\Omega\ni\omega\mapsto\sup\limits_{0\leq s\leq 1}{\operatorname{ln^{+}}{\lVert U_{\theta_{s}\omega}(1-s)\rVert}}\in\mathbb{R}^{+}\,\bigr{]}\in L_{1}(\Omega,\mathfrak{F},\mathbb{P})$.

Then it follows from the Kingman subadditive ergodic theorem that there exists
$\lambda_{\mathrm{top}}\in[-\infty,\infty)$ such that

$\lim\limits_{t\to\infty}\frac{\ln{\lVert
U_{\omega}(t)\rVert}}{t}=\lambda_{\mathrm{top}}$

for $\mathbb{P}$-a.e. $\omega\in\Omega$, which is referred to as the top
Lyapunov exponent of $\Phi$.

  1. (E2)

$\lambda_{\mathrm{top}}>-\infty$.

######  Definition 1.1 (Oseledets decomposition).

$\Phi$ admits an Oseledets decomposition if there exists an invariant subset
$\Omega_{0}\subset\Omega$, $\mathbb{P}(\Omega_{0})=1$, with the property that
one of the following mutually exclusive cases, (O1) or (O2), holds:

  1. (O1)

There are $k$ real numbers
$\lambda_{1}=\lambda_{\mathrm{top}}>\cdots>\lambda_{k}$, called the Lyapunov
exponents for $\Phi$, $k$ measurable families
$\\{E_{1}(\omega)\\}_{\omega\in\Omega_{0}}$, …,
$\\{E_{k}(\omega)\\}_{\omega\in\Omega_{0}}$ of finite dimensional vector
subspaces, and a family $\\{F_{\infty}(\omega)\\}_{\omega\in\Omega_{0}}$ of
closed vector subspaces of finite codimension such that

    1. (i)

for $j=1,\dots,k$, any $\omega\in\Omega_{0}$ and $t\geq 0$

$U_{\omega}(t)E_{j}(\omega)=E_{j}(\theta_{t}\omega)\quad\&\quad
U_{\omega}(t)F_{\infty}(\omega)\subset F_{\infty}(\theta_{t}\omega);$

    2. (ii)

$E_{1}(\omega)\oplus\ldots\oplus E_{k}(\omega)\oplus F_{\infty}(\omega)=X$ for
any $\omega\in\Omega_{0}$; we write

$F_{j}(\omega)\vcentcolon=\bigoplus\limits_{m=j+1}^{k}E_{m}(\omega)\oplus
F_{\infty}(\omega)\text{ for }j=0,\dots,k.$

In particular, $F_{j}(\omega)=E_{j+1}(\omega)\oplus F_{j+1}(\omega)$ for
$j=0,1,\dots,k-2\,$;

    3. (iii)

for $j=1,\dots,k$, the families of projections associated with the
decomposition

$\Bigl{(}\bigoplus\limits_{n=1}^{j}E_{n}(\omega)\Bigr{)}\oplus
F_{j}(\omega)=X$

is strongly measurable and tempered;

    4. (iv)

for $j=1,\dots,k\,$, any $\omega\in\Omega_{0}$ and any nonzero $u\in
E_{j}(\omega)$

$\lim_{t\to\infty}\frac{\ln{\lVert U_{\omega}(t)\,u\rVert}}{t}=\lambda_{j};$

    5. (v)

for $j=1,\dots,k$ and any $\omega\in\Omega_{0}$, a nonzero $u\in
F_{j-1}(\omega)$ belongs to $E_{j}(\omega)$ if and only if there exists a
negative semiorbit $\tilde{u}\colon(-\infty,0]\to X$ passing through
$(\omega,u)$ such that

$\lim\limits_{s\to-\infty}\frac{\ln{\lVert\tilde{u}(s)\rVert}}{s}=\lambda_{j};$

    6. (vi)

for any $\omega\in\Omega_{0}$

$\lim\limits_{t\to\infty}\frac{\ln{\lVert
U_{\omega}(t){\restriction}_{F_{\infty}(\omega)}\rVert}}{t}=-\infty.$

In this case,
$\\{F_{1}(\omega)\\}_{\omega\in\Omega_{0}},\ldots,\\{F_{k-1}(\omega)\\}_{\omega\in\Omega_{0}},\\{F_{\infty}(\omega)\\}_{\omega\in\Omega_{0}}$
is called the Oseledets filtration for $\Phi$.

  2. (O2)

There is a decreasing sequence of real numbers
$\lambda_{1}=\lambda_{\mathrm{top}}>\cdots>\lambda_{j}>\lambda_{j+1}>\cdots{}$
with limit $-\infty$, called the Lyapunov exponents for $\Phi$, countably many
measurable families $\\{E_{j}(\omega)\\}_{\omega\in\Omega_{0}}$,
$j\in\mathbb{N}$, of finite dimensional vector subspaces, and countably many
families $\\{F_{j}(\omega)\\}_{\omega\in\Omega_{0}}$, $j\in\mathbb{N}$, of
closed vector subspaces of finite codimensions, called the Oseledets
filtration for $\Phi$, such that

    1. (i)

for $j\in\mathbb{N}$, any $\omega\in\Omega_{0}$ and $t\geq 0$

$U_{\omega}(t)E_{j}(\omega)=E_{j}(\theta_{t}\omega)\quad\&\quad
U_{\omega}(t)F_{j}(\omega)\subset F_{j}(\theta_{t}\omega);$

    2. (ii)

for $j\in\mathbb{N}$ and any $\omega\in\Omega_{0}$

$E_{1}(\omega)\oplus\ldots\oplus E_{j}(\omega)\oplus
F_{j}(\omega)=X\quad\&\quad F_{j}(\omega)=E_{j+1}(\omega)\oplus
F_{j+1}(\omega);$

    3. (iii)

for $j\in\mathbb{N}$, the families of projections associated with the
decompositions

$\Bigl{(}\bigoplus\limits_{n=1}^{j}E_{n}(\omega)\Bigr{)}\oplus
F_{j}(\omega)=X$

are strongly measurable and tempered;

    4. (iv)

for $j\in\mathbb{N}$, any $\omega\in\Omega_{0}$ and any nonzero $u\in
E_{j}(\omega)$

$\lim_{t\to\infty}\frac{\ln{\lVert U_{\omega}(t)\,u\rVert}}{t}=\lambda_{j};$

    5. (v)

for $j\in\mathbb{N}$ and any $\omega\in\Omega_{0}$, a nonzero $u\in
F_{j-1}(\omega)$ belongs to $E_{j}(\omega)$ if and only if there exists a
negative semiorbit $\tilde{u}\colon(-\infty,0]\to X$ passing through
$(\omega,u)$ such that

$\lim\limits_{s\to-\infty}\frac{\ln{\lVert\tilde{u}(s)\rVert}}{s}=\lambda_{j};$

    6. (vi)

for $j\in\mathbb{N}$ and any $\omega\in\Omega_{0}$

$\lim\limits_{t\to\infty}\frac{\ln{\lVert
U_{\omega}(t){\restriction}_{F_{j}(\omega)}\rVert}}{t}=\lambda_{j+1};$

##  2\. Sub-semiflows

Let
$\Phi^{(1)}=\allowbreak((U^{(1)}_{\omega}(t))_{\omega\in\Omega,t\in\mathbb{R}^{+}},(\theta_{t})_{t\in\mathbb{R}})$
and
$\Phi^{(2)}=\allowbreak((U^{(2)}_{\omega}(t))_{\omega\in\Omega,t\in\mathbb{R}^{+}},\allowbreak(\theta_{t})_{t\in\mathbb{R}})$
be measurable linear skew-product semidynamical systems on Banach spaces
$(X_{1},\lVert\cdot\rVert_{1})$ and $(X_{2},\lVert\cdot\rVert_{2})$
respectively. The operator norm on $\mathcal{L}(X_{m},X_{n})$, $m,n=1,2$, will
be denoted by $\lVert\cdot\rVert_{m,n}$. By
$\mathcal{L}_{\mathrm{s}}(X_{m},X_{n})$ we understood the space of linear
bounded operators equipped with strong operator topology.

We assume:

  1. (A1)

There is an injective bounded linear map $i\vcentcolon X_{2}\to X_{1}$.

  2. (A2)

For any $\omega\in\Omega$ and $t\geq 0$ the equality $U^{(1)}_{\omega}(t)\circ
i=i\circ U^{(2)}_{\omega}(t)$ holds.

  3. (A3)

For any $\omega\in\Omega$ there is a map
$U^{(1,2)}_{\omega}(1)\in\mathcal{L}(X_{1},X_{2})$ such that
$U^{(1)}_{\omega}(1)=i\circ U^{(1,2)}_{\omega}(1)$ and
$U^{(2)}_{\omega}(1)=U^{(1,2)}_{\omega}(1)\circ i$. Moreover, the operator
norm bound $\lVert U^{(1,2)}_{\omega}(1)\rVert_{1,2}$ is uniform on $\omega$,
i.e., there exists $M>0$ such that for all $\omega\in\Omega$ the inequality
$\lVert U^{(1,2)}_{\omega}(1)\rVert_{1,2}\leq M$ holds.

  4. (A4)

The operator is $U^{(1,2)}_{\omega}(1)$ is
$(\mathfrak{F},\mathfrak{B}(\mathcal{L}_{\mathrm{s}}(X_{1},X_{2})))$-measurable
i.e., for all $u\in X_{1}$ the mapping

$\big{[}\,\Omega\ni\omega\mapsto U^{(1,2)}_{\omega}(1)u\in
X_{2}\,\big{]}\text{ is
}(\mathfrak{F},\mathfrak{B}(X_{2}))\text{-measurable}.$

  5. (A5)

$X_{1}$ and $X_{2}$ are separable.

  6. (A6)

For any $A\in\mathfrak{B}(X_{2})$ there exists $B\in\mathfrak{B}(X_{1})$ such
that $A=i^{-1}(B)$.

######  Lemma 2.1.

For a separable Banach space $X_{1}$ the mapping

$\big{[}\mathcal{L}_{\mathrm{s}}(X_{1},X_{2})\times X_{1}\ni(T,x)\mapsto Tx\in
X_{2}\big{]}$

$(\mathfrak{B}(\mathcal{L}_{\mathrm{s}}(X_{1},X_{2}))\otimes\mathfrak{B}(X_{1}),\mathfrak{B}(X_{2}))$-measurable.

###### Proof.

Compare with [11, Lemma A.6 (2)] and [3, Lemma. 6.4.2(i)]. ∎

######  Lemma 2.2.

Assume ($\mathrm{A1}$), ($\mathrm{A2}$) and ($\mathrm{A3}$). For any
$\omega\in\Omega$ and $t\geq 1$ the equality

$U^{(1,2)}_{\theta_{t-1}\omega}(1)\circ
U^{(1)}_{\omega}(t-1)=U^{(2)}_{\theta_{1}\omega}(t-1)\circ
U^{(1,2)}_{\omega}(1)$

holds.

###### Proof.

Fix $\omega$ and $t$. We are going to use the assumptions ($\mathrm{A2}$) and
($\mathrm{A3}$). Moreover, since $i$ is injective it suffices to observe that

$\displaystyle i\circ U^{(1,2)}_{\theta_{t-1}\omega}(1)\circ
U^{(1)}_{\omega}(t-1)$ $\displaystyle=U^{(1)}_{\theta_{t-1}\omega}(1)\circ
U^{(1)}_{\omega}(t-1)$ $\displaystyle=U^{(1)}_{\theta_{1}\omega}(t-1)\circ
U^{(1)}_{\omega}(1)$ $\displaystyle=U^{(1)}_{\theta_{1}\omega}(t-1)\circ
i\circ U^{(1,2)}_{\omega}(1)$ $\displaystyle=i\circ
U^{(2)}_{\theta_{1}\omega}(t-1)\circ U^{(1,2)}_{\omega}(1).$

∎

The above observation allows us to extend the definition of the operator
$U^{(1,2)}_{\omega}(1)$ to all $t\geq 1$ as

$U^{(1,2)}_{\omega}(t)\vcentcolon=U^{(1,2)}_{\theta_{t-1}\omega}(1)\circ
U^{(1)}_{\omega}(t-1)=U^{(2)}_{\theta_{1}\omega}(t-1)\circ
U^{(1,2)}_{\omega}(1)\in\mathcal{L}(X_{2},X_{1}).$

Before we investigate how to transfer Oseledets decomposition from one
measurable linear skew-product semidynamical systems $\Phi^{(1)}$ onto another
semiflow $\Phi^{(2)}$, we will introduce a useful Claim.

######  Claim 1.

Assume ($\mathrm{A1}$), ($\mathrm{A2}$), ($\mathrm{A3}$) and let
$\Omega_{0}\in\mathfrak{F}$ be such that $\mathbb{P}(\Omega_{0})=1$. For any
family of subspaces $\\{W(\omega)\\}_{\omega\in\Omega_{0}}$ of $X_{1}$ such
that the equality $U^{(1)}_{\omega}(t)W(\omega)=W(\theta_{t}\omega)$ holds for
all $t\geq 0$ and all $\omega\in\Omega_{0}$ there exists a family of subspaces
$\\{V(\omega)\\}_{\omega\in\Omega_{0}}$ of $X_{2}$ such

  1. (i)

$iV(\omega)=W(\omega)$ for any $\omega\in\Omega_{0}$,

  2. (ii)

$U^{(2)}_{\omega}(t)V(\omega)=V(\theta_{t}\omega)$ for all $t\geq 0$ and
$\omega\in\Omega_{0}$.

###### Proof.

For $\omega\in\Omega_{0}$ let
$V(\omega)\vcentcolon=U^{(1,2)}_{\theta_{-1}\omega}(1)W(\theta_{-1}\omega)$.
Therefore,

$\displaystyle iV(\omega)=i\circ
U^{(1,2)}_{\theta_{-1}\omega}(1)W(\theta_{-1}\omega)$
$\displaystyle=U^{(1)}_{\theta_{-1}\omega}(1)W(\theta_{-1}\omega)=W(\omega),$

and the part (i) is done. Furthermore, for fixed $t\geq 0$ we have

$\displaystyle i\circ U^{(2)}_{\omega}(t)V(\omega)$ $\displaystyle=i\circ
U^{(2)}_{\omega}(t)\circ U^{(1,2)}_{\theta_{-1}\omega}(1)W(\theta_{-1}\omega)$
$\displaystyle=U^{(1)}_{\omega}(t)\circ i\circ
U^{(1,2)}_{\theta_{-1}\omega}(1)W(\theta_{-1}\omega)$
$\displaystyle=U^{(1)}_{\omega}(t)\circ
U^{(1)}_{\theta_{-1}\omega}(1)W(\theta_{-1}\omega)$
$\displaystyle=U^{(1)}_{\theta_{t-1}\omega}(1)\circ
U^{(1)}_{\theta_{-1}\omega}(t)W(\theta_{-1}\omega)$
$\displaystyle=U^{(1)}_{\theta_{t-1}\omega}(1)W(\theta_{t-1}\omega)$
$\displaystyle=i\circ U^{(1,2)}_{\theta_{t-1}\omega}(1)W(\theta_{t-1}\omega)$
$\displaystyle=i\circ
U^{(1,2)}_{\theta_{-1}\theta_{t}\omega}(1)W(\theta_{-1}\theta_{t}\omega)$
$\displaystyle=iV(\theta_{t}\omega),$

which concludes the proof. ∎

It should be noted that $V(\omega)$ can be defined as $i^{-1}(W(\omega))$. In
such a case, firstly, we can observe that by invariance of $W(\omega)$ and the
assumption ($\mathrm{A3}$) we have

$W(\theta_{t}\omega)=U^{(1)}_{\theta_{t-1}\omega}(1)W(\theta_{t-1}\omega)=i\circ
U^{(1,2)}_{\omega}(1)W(\theta_{t-1}\omega)\subset iX_{2}.$

Hence, by proceeding as in the proof of Claim 1 we can show invariance of
$V(\omega)$. Indeed,

$\displaystyle i\circ U^{(2)}_{\omega}(t)V(\omega)$ $\displaystyle=i\circ
U^{(2)}_{\omega}(t)i^{-1}(W(\omega))$ $\displaystyle=U^{(1)}_{\omega}(t)\circ
i(i^{-1}(W(\omega)))$ $\displaystyle=U^{(1)}_{\omega}(t)(W(\omega)\cap
iX_{2})$ $\displaystyle=U^{(1)}_{\omega}(t)W(\omega)$
$\displaystyle=W(\theta_{t}\omega)$ $\displaystyle=W(\theta_{t}\omega)\cap
iX_{2}$ $\displaystyle=i(i^{-1}W(\theta_{t}\omega)).$

Assume that $\Phi^{(1)}$ admits an Oseledets decomposition. By taking in Claim
1 $\\{E^{(1)}_{j}(\omega)\\}_{\omega\in\Omega_{0},j=1,\dots,k}$ or
$\\{E^{(1)}_{j}(\omega)\\}_{\omega\in\Omega_{0},j\in\mathbb{N}}$ for
$\\{W(\omega)\\}$ we obtain families
$\\{E^{(2)}_{j}(\omega)\\}_{\omega\in\Omega_{0},j=1,\dots,k}$ or
$\\{E^{(2)}_{j}(\omega)\\}_{\omega\in\Omega_{0},j\in\mathbb{N}}$ of invariant,
finite-dimensional vector subspaces of $X_{2}$.

Fix $j$. Let $l$ be the dimension of $E^{(1)}_{j}$. For $\omega\in\Omega_{0}$
denote by $G(\omega)$ the linear isomorphism from $E^{(1)}_{j}(\omega)$ onto
$\mathbb{R}^{l}$ given by
$G(\omega)u\vcentcolon=(\alpha_{1},\ldots,\alpha_{l})$ where
$u=\alpha_{1}\,v_{1}(\omega)+\ldots+\alpha_{l}\,v_{l}(\omega)$ is written in
the basis. The family $\\{\widehat{U}_{\omega}(1)\\}_{\omega\in\Omega_{0}}$ of
linear automorphisms of $\mathbb{R}^{l}$ defined by

$\widehat{U}_{\omega}(1)\vcentcolon=G(\theta_{1}\omega)\circ
U^{(1)}_{\omega}(1){\restriction}_{E^{(1)}_{j}(\omega)}\circ
G(\omega)^{-1},\quad\omega\in\Omega_{0},$

generates a two-sided discrete-time linear skew-product dynamical system
$\widehat{\Phi}=((\widehat{U}_{\omega}(n)),(\theta_{n}))$ on
$\Omega_{0}\times\mathbb{R}^{l}$, with

$\widehat{U}_{\omega}(n)\vcentcolon=G(\theta_{n}\omega)\circ
U^{(1)}_{\omega}(n){\restriction}_{E^{(1)}_{j}(\omega)}\circ
G(\omega)^{-1},\quad\omega\in\Omega_{0},\ n\in\mathbb{N}.$

This allows us to use results in [26, Subsection 4.2.4]. Indeed, those results
are formulated for discrete time, but assumption ($\mathrm{E1}$) allows us to
extend them to the continuous time case. As a consequence of [26, Proposition
4.11],

$\lim\limits_{t\to\infty}\frac{1}{t}\ln{\lVert
U^{(1)}_{\omega}(t){\restriction}_{E^{(1)}_{j}(\omega)}\rVert_{1,1}}=\lambda_{j}\text{
and
}\lim\limits_{t\to\infty}\frac{1}{t}\ln{\lVert(U^{(1)}_{\omega}(t){\restriction}_{E^{(1)}_{j}(\omega)})^{-1}\rVert^{-1}_{1,1}}=\lambda_{j}.$
(2.1)

Based on the above, we can introduce the following lemma, which will prove to
be useful in the subsequent analysis.

######  Lemma 2.3.

For $\mathbb{P}$-a.e. $\omega\in\Omega$ and each $j$, there exists a function
$c:(1,\infty)\to(0,\infty)$ such

  1. (i)

for each $t\geq 1$ and $u\in E^{(2)}_{j}(\theta_{t}\omega)$ we have
$c(t)\lVert u\rVert_{2}\leq\lVert iu\rVert_{1}$,

  2. (ii)

$c$ is sub-exponential i.e.

$\lim_{t\to\infty}\frac{\ln c(t)}{t}=0.$

###### Proof.

Let,

$c(t)=\inf\Big{\\{}\frac{\lVert iu^{\prime}\rVert_{1}}{\lVert
u^{\prime}\rVert_{2}}:u^{\prime}\in
E^{(2)}_{j}(\theta_{t}\omega)\setminus\\{0\\}\Big{\\}}$

Hence, part (i) is trivially satisfied. For part (ii) observe that the upper
bound for $c$ is obviously $\lVert i\rVert$. Therefore, we are concentrating
on the sub-exponential lower limit. It is claimed that

$\frac{\lVert(U^{(1)}_{\omega}(t){\restriction}_{E^{(1)}_{j}(\omega)})^{-1}\rVert^{-1}_{1,1}}{\lVert
U^{(1,2)}_{\theta_{t-1}\omega}(1)\rVert_{1,2}\,\lVert
U^{(1)}_{\omega}(t-1){\restriction}_{E^{(1)}_{j}(\omega)}\rVert_{1,1}}\leq
c(t),\quad\text{for }t\geq 1.$

Indeed, fix $t\geq 1$ and $u^{\prime}\in
E_{j}^{(2)}(\theta_{t}\omega)\setminus\\{0\\}$. Let $u\in E_{j}^{(1)}(\omega)$
be such that $u^{\prime}=U^{(1,2)}_{\omega}(t)u$. Since

$\displaystyle\lVert u^{\prime}\rVert_{2}=\lVert
U^{(1,2)}_{\theta_{t-1}\omega}(1)U^{(1)}_{\omega}(t-1)u\rVert_{2}\leq\lVert
U^{(1,2)}_{\theta_{t-1}\omega}(1)\rVert_{1,2}\,\lVert
U^{(1)}_{\omega}(t-1){\restriction}_{E^{(1)}_{j}(\omega)}\rVert_{1,1}\,\lVert
u\rVert_{1},$ $\displaystyle\lVert iu^{\prime}\rVert_{1}=\lVert
U^{(1)}_{\omega}(t)u\rVert_{1,1}\geq\lVert(U^{(1)}_{\omega}(t){\restriction}_{E^{(1)}_{j}(\omega)})^{-1}\rVert_{1,1}^{-1}\,\lVert
u\rVert_{1}.$

We thus have further

$\frac{\lVert(U^{(1)}_{\omega}(t){\restriction}_{E^{(1)}_{j}(\omega)})^{-1}\rVert^{-1}_{1,1}}{\lVert
U^{(1,2)}_{\theta_{t-1}\omega}(1)\rVert_{1,2}\,\lVert
U^{(1)}_{\omega}(t-1){\restriction}_{E^{(1)}_{j}(\omega)}\rVert_{1,1}}\leq\frac{\lVert
iu^{\prime}\rVert_{1}}{\lVert u^{\prime}\rVert_{2}}.$

From this the preliminary claim is obtained, which makes it possible to write
the following

$\displaystyle\frac{1}{t}\ln\lVert(U^{(1)}_{\omega}(t){\restriction}_{E^{(1)}_{j}(\omega)})^{-1}\rVert^{-1}_{1,1}-\frac{1}{t}\ln\lVert
U^{(1)}_{\omega}(t-1){\restriction}_{E^{(1)}_{j}(\omega)}\rVert_{1,1}-\frac{1}{t}\ln\lVert
U^{(1,2)}_{\theta_{t-1}\omega}(1)\rVert_{1,2}$ $\displaystyle\leq\frac{\ln
c(t)}{t}\leq\frac{\ln\lVert i\rVert_{2,1}}{t}$

Equation (2.1) concludes the proof. ∎

######  Theorem 2.1.

Assume ($\mathrm{A1}$)-($\mathrm{A6}$) and moreover, that $\Phi^{(1)}$ admits
an Oseledets decomposition in case (O1) (or (O2)). Then $\Phi^{(2)}$ admits an
Oseledets decomposition in case (O1) (or (O2) respectively).

###### Proof.

We are going to show that if $\Phi^{(1)}$ admits an Oseledets decomposition in
case (O1) with parameters

$(\Omega_{0},(\lambda_{j})_{j=1,\dots,k},\\{E_{j}(\omega)\\}_{\omega\in\Omega_{0},j=1,\dots,k},\\{F_{\infty}(\omega)\\}_{\omega\in\Omega_{0}})$

then $\Phi^{(2)}$ admits an Oseledets decomposition in case (O1) with
parameters

$(\Omega_{0},(\lambda_{j})_{j=1,\dots,k},\\{U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)\\}_{\omega\in\Omega_{0},j=1,\dots,k},\\{i^{-1}(F_{\infty}(\omega))\\}_{\omega\in\Omega_{0}}).$

Before we start with cases (O1i)-(O1vi). Note that
$i^{-1}(F_{\infty}(\omega))$ is a closed subspace as a continuous preimage of
a closed subspace. Moreover, for each $j=1,\dots,k$ the family
$\\{U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)\\}_{\omega\in\Omega_{0}}$
is measurable. Indeed, the mapping

$\big{[}\,E_{j}(\omega)\ni u\mapsto U^{(1,2)}_{\theta_{-1}\omega}(1)u\in
U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)\,\big{]}$

is a bijection. It follows from the observation that $U^{(1)}_{\omega}(1)$ is
a bijection since it preserves finite dimension i.e., $\dim
U^{(1)}_{\omega}(1)E_{j}(\omega)=\dim E_{j}(\theta_{1}\omega)$ (see
(O1)(O1i)). Therefore, from: the injectivity of $i$, the bijectivity of
$U^{(1)}_{\omega}(1)$ and $i\circ U^{(1,2)}_{\omega}(1)=U^{(1)}_{\omega}(1)$
we can deduce that $u=v$ whenever, $i\circ
U^{(1)}_{\theta_{-1}\omega}(1)u=i\circ U^{(1)}_{\theta_{-1}\omega}(1)v$.
Moreover, we have

$\displaystyle U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)$
$\displaystyle=\operatorname{span}\\{U^{(1,2)}_{\theta_{-1}\omega}(1)v_{1}(\theta_{-1}\omega),\dots,U^{(1,2)}_{\theta_{-1}\omega}(1)v_{l}(\theta_{-1}\omega)\\},$

where $v_{1},\dots,v_{l}\colon\Omega_{0}\to X$ are
$(\mathfrak{F},\mathfrak{B}(X))$-measurable functions, such that
$(v_{1}(\omega),\allowbreak\dots,v_{l}(\omega))$ forms a basis of
$E_{j}(\omega)$ for each $\omega\in\Omega_{0}$. Hence, it remains to show the
$(\mathfrak{F},\mathfrak{B}(X_{2}))$-measurability of the maps

$\big{[}\,\Omega_{0}\ni\omega\mapsto
U^{(1,2)}_{\theta_{-1}\omega}(1)v_{j}(\theta_{-1}\omega)\in X_{2}\,\big{]},$

for $j=1,\dots,l$. It can be done by looking at the composition

$\omega\mapsto(\omega,\omega)\mapsto(U_{\theta_{-1}\omega}^{(1,2)}(1),v_{j}(\theta_{-1}\omega))\mapsto
U_{\theta_{-1}\omega}^{(1,2)}(1)v_{j}(\theta_{-1}\omega).$

The second mapping is
$(\mathfrak{F}\otimes\mathfrak{F},\mathfrak{B}(\mathcal{L}_{\mathrm{s}}(X_{1},X_{2}))\otimes\mathfrak{B}(X_{1}))$-measurable.
By the assumption ($\mathrm{A5}$) and Lemma 2.1, the pairing is
$(\mathfrak{B}(\mathcal{L}_{\mathrm{s}}(X_{1},X_{2}))\otimes\mathfrak{B}(X_{1}),\allowbreak\mathfrak{B}(X_{2}))$-measurable.

(O1)(O1i) can be done via Claim 1, indeed

$U^{(2)}_{\omega}(t)U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)=U^{(1,2)}_{\theta_{t-1}\omega}(1)E_{j}(\theta_{t-1}\omega).$

It remains to show $U^{(2)}_{\omega}(t)i^{-1}(F_{\infty}(\omega))\subset
i^{-1}(F_{\infty}(\theta_{t}\omega))$, equivalently

$\displaystyle i\circ U^{(2)}_{\omega}(t)(i^{-1}(F_{\infty}(\omega)))$
$\displaystyle=U^{(1)}_{\omega}(t)\circ i(i^{-1}(F_{\infty}(\omega)))$
$\displaystyle=U^{(1)}_{\omega}(t)(F_{\infty}(\omega)\cap iX_{2})$
$\displaystyle\subset U^{(1)}_{\omega}(t)F_{\infty}(\omega)\cap
U^{(1)}_{\omega}(t)iX_{2}$
$\displaystyle=U^{(1)}_{\omega}(t)F_{\infty}(\omega)\cap
iU^{(2)}_{\omega}(t)X_{2}$ $\displaystyle\subset
F_{\infty}(\theta_{t}\omega)\cap iX_{2}$
$\displaystyle=i(i^{-1}(F_{\infty}(\theta_{t}\omega))).$

Hence, the decomposition is invariant.

(O1)(O1ii) in order to show

$\bigoplus\limits_{j=1}^{k}U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)\oplus
i^{-1}(F_{\infty}(\omega))=X_{2}$

it suffices to show that the spaces
$U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)$ ($j=1,\dots,k$) and
$i^{-1}(F_{\infty}(\omega))$ are linearly independent and

$\sum\limits_{j=1}^{k}U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)+i^{-1}(F_{\infty}(\omega))=X_{2}.$

Therefore, fix $e^{(2)}_{j}\in
U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)$ for each
$j=1,\dots,k$ and $f^{(2)}\in i^{-1}(F_{\infty}(\omega))$ such that
$e^{(2)}_{1}+e^{(2)}_{2}+\dots+e^{(2)}_{k}+f^{(2)}=0$. Since

$0=i\Big{(}\sum\limits_{j=1}^{k}e^{(2)}_{j}+f^{(2)}\Big{)}=\sum\limits_{j=1}^{k}ie^{(2)}_{j}+if^{(2)},$

and by linear independence of the subspaces $E_{j}(\omega)$ ($j=1,\dots,k$)
and $F_{\infty}(\omega)$ we have $ie^{(2)}_{1}=\dots=ie^{(2)}_{k}=if^{(2)}=0$,
so $e^{(2)}_{1}=\dots=e^{(2)}_{k}=f^{(2)}=0$. It remains to show the sets
equality. The inclusion $(\subset)$ is obvious. Therefore, fix $x\in X_{2}$.
Hence, $ix\in X_{1}$ there exists a (unique) representation
$ix=e^{(1)}_{1}+e^{(1)}_{2}+\dots+e^{(1)}_{k}+f^{(1)}$ for some
$e^{(1)}_{j}\in E_{j}(\omega)$ and $f^{(1)}\in F_{\infty}(\omega)$. However,
from the first part of the Claim 1(i) we can conclude that for any
$e^{(1)}_{j}\in E_{j}(\omega)$ there exists $e^{(2)}_{j}\in
U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)$ such that
$e^{(1)}_{j}=ie^{(2)}_{j}$. We can find $f^{(2)}\in F_{\infty}(\omega)$ such
that $if^{(2)}=f^{(1)}$, namely $f^{(2)}=x-e^{(2)}_{1}-\dots-e^{(2)}_{k}$. So,
the inclusion $(\supset)$ is done.

(O1)(O1iii) Fix $l=1,\dots,k$ and $u\in X_{2}$. In order to show that the
family of projections associated with the decomposition

$\Bigl{(}\bigoplus\limits_{j=1}^{l}U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)\Bigr{)}\oplus\Bigl{(}\bigoplus\limits_{j=l+1}^{k}U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)\oplus
i^{-1}(F_{\infty}(\omega))\Bigr{)}=X_{2}$ (2.2)

along the first component is strongly measurable it suffices to see that for
fixed set $A\in\mathfrak{B}(X_{2})$ we have

$\displaystyle(P^{(2)}(\cdot)u)^{-1}(A)$
$\displaystyle=(P^{(2)}(\cdot)u)^{-1}(i^{-1}(B))$ $\displaystyle=(i\circ
P^{(2)}(\cdot)u)^{-1}(B)$ $\displaystyle=(P^{(1)}(\cdot)iu)^{-1}(B)$
$\displaystyle\in\mathfrak{F},$

where $P^{(2)}(\omega)$ is the projection along
$U^{(1,2)}_{\theta_{-1}\omega}(1)E_{1}(\theta_{-1}\omega)\oplus\dots\oplus
U^{(1,2)}_{\theta_{-1}\omega}(1)E_{l}(\theta_{-1}\omega)$ and set
$B\in\mathfrak{B}(X_{1})$ is the set from the assumption ($\mathrm{A6}$) such
that $A=i^{-1}(B)$. It remains to show that the family of projections
associated with the decomposition (2.2) is tempered. The definition refers to
projections onto the finite codimensional closed vector subspaces $F(\omega)$.
However, it is a well-known fact that it is sufficient to equivalently
demonstrate the temperedness of projections onto finite-dimensional vector
subspaces (i.e. first component of (2.2), see [15, Remark 2.1]). We start by
demonstrating that the projections onto
$U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)$ for any
$j=1,\dots,k$ are tempered. Let us fix $j$. Subsequently, in the inductive
process, we will show that temperedness holds universally. Therefore, let
$\widetilde{P}_{j}^{(2)}(\omega)$ denote the complementary part of
$P_{j}^{(2)}(\omega)$, namely the projection onto
$U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)$. Using the results
from Lemma 2.3, we can conclude that for any $u\in X_{2}$ we have

$c(t)\lVert\widetilde{P}_{j}^{(2)}(\theta_{t}\omega)u\rVert_{2}\leq\lVert
i\widetilde{P}^{(2)}(\theta_{t}\omega)u\rVert_{1}\leq\lVert
i\rVert_{2,1}\,\lVert\widetilde{P}_{j}^{(2)}(\theta_{t}\omega)u\rVert_{2}.$

In fact, we have

$\displaystyle
1\leq\lVert\widetilde{P}_{j}^{(2)}(\theta_{t}\omega)\rVert_{2,2}$
$\displaystyle=\sup\Big{\\{}\frac{\lVert\widetilde{P}_{j}^{(2)}(\theta_{t}\omega)u\rVert_{2}}{\lVert
u\rVert_{2}}:u\in X_{2}\setminus\\{0\\}\Big{\\}}$
$\displaystyle\leq\sup\Big{\\{}\frac{\lVert\widetilde{P}_{j}^{(1)}(\theta_{t}\omega)iu\rVert_{1}}{c(t)\lVert
iu\rVert_{1}}\cdot\frac{\lVert iu\rVert_{1}}{\lVert u\rVert_{2}}:u\in
X_{2}\setminus\\{0\\}\Big{\\}}$ $\displaystyle\leq\frac{\lVert
i\rVert_{2,1}}{c(t)}\lVert\widetilde{P}_{j}^{(1)}(\theta_{t}\omega)\rVert_{1,1}.$

Hence, $\ln\lVert\widetilde{P}^{(2)}(\theta_{t}\omega)\rVert_{2,2}/t\to 0$ as
$t\to\infty$. Furthermore, in the spirit of Tanny’s theorem, when
$t\to-\infty$, we can obtain
$\ln\lVert\widetilde{P}^{(2)}(\theta_{t}\omega)\rVert_{2,2}/t\to 0$ as well,
see [11, Lemma C2]. In conclusion, let us observe that generally for the
projection $\widetilde{P}_{1,\dots,l}^{(2)}(\theta_{t}\omega)$ onto the first
component of (2.2) we have

$0\leq\frac{\ln\lVert\widetilde{P}_{1,\dots,l}^{(2)}(\theta_{t}\omega)\rVert_{2,2}}{t}\leq\frac{\ln
l+\max_{j}\ln\lVert\widetilde{P}_{j}^{(2)}(\theta_{t}\omega)\rVert_{2,2}}{t}.$

Hence,
$\ln\lVert\widetilde{P}_{1,\dots,l}^{(2)}(\theta_{t}\omega)\rVert_{2,2}/t\to
0$ as $t\to\infty$ and again by [11, Lemma C2] we can obtain this result for
$t\to-\infty$.

(O1)(O1iv) fix $j=1\dots k$, $\omega\in\Omega_{0}$ and $u\in
U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)$. Hence, from Claim
1(i) $iu\in E_{j}(\omega)$. Therefore, by assumption (O1)(O1iv) for
$\Phi^{(1)}$, $\ln\lVert
U^{(1)}_{\omega}(t)iu\rVert_{1}/t\allowbreak\to\lambda_{j}$. Moreover, we have
two inequalities, first one from the assumption ($\mathrm{A2}$): $\lVert
U^{(1)}_{\omega}(t)\circ iu\rVert_{1}=\lVert i\circ
U^{(2)}_{\omega}(t)u\rVert_{1}\leq\lVert i\rVert_{2,1}\lVert
U^{(2)}_{\omega}(t)u\rVert_{2}$ and the second one, $\lVert
U^{(2)}_{\omega}(t)u\rVert_{2}\leq\lVert
U^{(1,2)}_{\theta_{t-1}\omega}(1)\rVert_{1,2}\,\lVert
U^{(1)}_{\omega}(t-1)\circ iu\rVert_{1}$, from the fact that
$U^{(2)}_{\omega}(t)u=U^{(1,2)}_{\theta_{t-1}\omega}(1)\circ
U^{(1)}_{\omega}(t-1)\circ iu$. Therefore,

$\frac{1}{t}\ln\lVert U^{(1)}_{\omega}(t)\circ
iu\rVert_{1}-\frac{1}{t}\ln\lVert i\rVert_{2,1}\leq\frac{1}{t}\ln\lVert
U^{(2)}_{\omega}(t)u\rVert_{2}\\\ \leq\frac{1}{t}\ln\lVert
U^{(1,2)}_{\theta_{t-1}\omega}(1)\rVert_{1,2}+\frac{1}{t}\ln\lVert
U^{(1)}_{\omega}(t-1)\circ iu\rVert_{1}.$

Thus, we have proven (O1)(O1iv) for $\Phi^{(2)}$.

(O1)(O1v) Fix $\omega$ and

$u^{(2)}\in\bigoplus\limits_{m=j}^{k}U^{(1,2)}_{\theta_{-1}\omega}(1)E_{m}(\theta_{-1}\omega)\oplus
i^{-1}(F_{\infty}(\omega)).$

To begin with, let us note that Claim 1 allows for demonstrating only the
equivalence between the existence of negative semiorbits (in $X_{1}$ and
$X_{2}$) on which the appropriate Lyapunov exponent is achieved, since the
membership of $u^{(2)}$ in the subspace
$U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)$ is equivalent to
the membership of $iu^{(2)}$ in the subspace $E_{j}(\omega)$.

Assume that there exists a negative semiorbit $\tilde{u}^{(2)}(s)$ in $X_{2}$
passing through $(\omega,u^{(2)})$. We commence by demonstrating that within
the space $X_{1}$, there exists a negative semiorbit passing through
$(\omega,iu^{(2)})$, such that the appropriate Lyapunov exponent is
attainable. Namely, we define $\tilde{u}^{(1)}\colon(-\infty,0]\to X_{1}$ by
$\tilde{u}^{(1)}(s)=i\tilde{u}^{(2)}(s)$; such Lyapunov exponent in $X_{2}$ is
(by assumption) $\lambda_{j}$. Upon examination, it becomes evident that the
thus-defined $\tilde{u}^{(1)}$ indeed is a negative semi-orbit. Indeed
$\tilde{u}^{(1)}(0)=iu^{(2)}$ and

$\tilde{u}^{(1)}(s+t)=i\tilde{u}^{(2)}(s+t)=iU^{(2)}_{\theta_{s}\omega}(t)\tilde{u}^{(2)}(s)=U^{(1)}_{\theta_{s}\omega}(t)i\tilde{u}^{(2)}(s)=U^{(1)}_{\theta_{s}\omega}(t)\tilde{u}^{(1)}(s).$

Moreover, $\tilde{u}^{(1)}$ is
$(\mathfrak{B}((-\infty,0]),\mathfrak{B}(X_{1}))$-measurable as a continuous
composition. Furthermore,
$\ln\lVert\tilde{u}^{(1)}(s)\rVert_{1}/s\to\lambda_{j}$ as $s\to-\infty$ since

$\lVert\tilde{u}^{(2)}(s+1)\rVert_{2}=\lVert
U^{(1,2)}_{\theta_{s}\omega}(1)i\tilde{u}^{(2)}(s)\rVert_{2}\\\ \leq\lVert
U^{(1,2)}_{\theta_{s}\omega}(1)\rVert_{1,2}\,\lVert
i\tilde{u}^{(2)}(s)\rVert_{1}=\lVert
U^{(1,2)}_{\theta_{s}\omega}(1)\rVert_{1,2}\,\lVert\tilde{u}^{(1)}(s)\rVert_{1}\\\
\leq\lVert U^{(1,2)}_{\theta_{s}\omega}(1)\rVert_{1,2}\lVert
i\rVert_{2,1}\lVert\tilde{u}^{(2)}(s)\rVert_{2}.$

Conversely, if $\tilde{u}^{(1)}$ is a negative semiorbit on $X_{1}$, then
$\tilde{u}^{(2)}$ defined as
$\tilde{u}^{(2)}(s)=U^{(1,2)}_{\theta_{s-1}\omega}(1)\tilde{u}^{(1)}(s-1)$
serves as a negative semiorbit on $X_{2}$. Clearly,

$i\tilde{u}^{(2)}(0)=iU^{(1,2)}_{\theta_{-1}\omega}(1)\tilde{u}^{(1)}(-1)=U^{(1)}_{\theta_{-1}\omega}(1)\tilde{u}^{(1)}(-1)=\tilde{u}^{(1)}(0)=u^{(1)}=iu^{(2)},$

and for any $s\leq 0$, $t\geq 0$ such that $s+t\leq 0$ we have

$\begin{split}\tilde{u}^{(2)}(s+t)&=U^{(1,2)}_{\theta_{s-1+t}\omega}(1)\tilde{u}^{(1)}(s-1+t)\\\
&=U^{(1,2)}_{\theta_{s-1+t}\omega}(1)\circ
U^{(1)}_{\theta_{s-1}\omega}(t)\tilde{u}^{(1)}(s-1)\\\
&=U^{(2)}_{\theta_{s}\omega}(t)\circ
U^{(1,2)}_{\theta_{s-1}\omega}(1)\tilde{u}^{(1)}(s-1)\\\
&=U^{(2)}_{\theta_{s}\omega}(t)\tilde{u}^{(2)}(s).\end{split}$

Moreover, $\ln\lVert\tilde{u}^{(2)}(s)\rVert_{2}/s\to\lambda_{j}$ as
$s\to-\infty$ since

$\lVert\tilde{u}^{(1)}(s)\rVert_{1}\leq\lVert
i\rVert_{2,1}\,\lVert\tilde{u}^{(2)}(s)\rVert_{2}\leq\lVert
i\rVert_{2,1}\,\lVert
U^{(1,2)}_{\theta_{s-1}\omega}(1)\rVert_{1,2}\,\lVert\tilde{u}^{(1)}(s-1)\rVert_{1}.$

It remains to show the
$(\mathfrak{B}((-\infty,0]),\mathfrak{B}(X_{2}))$-measurability of the map

$\big{[}\,(-\infty,0]\ni s\mapsto
U^{(1,2)}_{\theta_{s-1}\omega}(1)\tilde{u}^{(1)}(s-1)\in X_{2}\,\big{]}.$

For this purpose, let us observe that it can be rewritten as the measurable
composition

$s\mapsto(\theta_{s-1}\omega,s)\mapsto(U_{\theta_{-1}\omega}^{(1,2)}(1),\tilde{u}^{(1)}(s-1))\mapsto
U^{(1,2)}_{\theta_{s-1}\omega}(1)\tilde{u}^{(1)}(s-1).$

The first mapping is
$(\mathfrak{B}((-\infty,0]),\mathfrak{F}\otimes\mathfrak{B}((-\infty,0]))$-measurable,
the second mapping is
$(\mathfrak{F}\otimes\mathfrak{B}((-\infty,0]),\mathfrak{B}(\mathcal{L}_{\mathrm{s}}(X_{1},X_{2}))\otimes\mathfrak{B}(X_{1}))$-measurable.
By the assumption ($\mathrm{A5}$) and Lemma 2.1, the pairing is
$(\mathfrak{B}(\mathcal{L}_{\mathrm{s}}(X_{1},X_{2}))\otimes\mathfrak{B}(X_{1}),\allowbreak\mathfrak{B}(X_{2}))$-measurable.

(O1)(O1vi) Firstly, we claim that

$\displaystyle U^{(2)}_{\omega}(t)=U^{(1,2)}_{\theta_{t-1}\omega}(1)\circ
U^{(1)}_{\omega}(t-1)\circ i.$

Together with $i(i^{-1}(F_{\infty}(\omega)))\subset F_{\infty}(\omega)$ we can
see that the inequality

$\displaystyle\lVert
U^{(2)}_{\omega}(t){\restriction}_{i^{-1}(F_{\infty}(\omega))}\rVert_{2,2}$
$\displaystyle\leq\lVert i\rVert_{2,1}\,\lVert
U^{(1,2)}_{\theta_{t-1}\omega}(1)\rVert_{1,2}\,\lVert
U^{(1)}_{\omega}(t-1){\restriction}_{i(i^{-1}(F_{\infty}(\omega)))}\rVert_{1,1}$
$\displaystyle\leq\lVert i\rVert_{2,1}\,\lVert
U^{(1,2)}_{\theta_{t-1}\omega}(1)\rVert_{1,2}\,\lVert
U^{(1)}_{\omega}(t-1){\restriction}_{F_{\infty}(\omega)}\rVert_{1,1}$

holds for any $\omega\in\Omega$ and $t\geq 1$. So,

$\frac{1}{t}\ln\lVert
U^{(2)}_{\omega}(t){\restriction}_{i^{-1}(F_{\infty}(\omega))}\rVert_{2,2}\leq\frac{1}{t}\ln\lVert
i\rVert_{2,1}+\frac{1}{t}\ln\lVert
U^{(1,2)}_{\theta_{t-1}\omega}(1)\rVert_{1,2}+\frac{1}{t}\ln\lVert
U^{(1)}_{\omega}(t-1){\restriction}_{F_{\infty}(\omega)}\rVert_{1,1},$

with under assumptions ($\mathrm{A1}$) and ($\mathrm{A3}$) suffices to finish
the proof of this part.

Furthermore, we have concluded the first and, as it turns out, the main part
of the proof. The remaining part (O2) pertains to the scenario in which there
is a countable number of Lyapunov exponents. Nevertheless, it is easy to
notice that practically the entire proof is replicated word for word in this
situation. However, for the sake of completeness, let us note the fact that if
$\Phi^{(1)}$ admits Oseledets decomposition in the case (O2) with parameters

$(\Omega_{0},(\lambda_{j})_{j\in\mathbb{N}},\\{E_{j}(\omega)\\}_{\omega\in\Omega_{0},j\in\mathbb{N}},\\{F_{j}(\omega)\\}_{\omega\in\Omega_{0},j\in\mathbb{N}})$

then $\Phi^{(2)}$ also admits Oseledets decomposition in the case (O2), with
parameters

$(\Omega_{0},(\lambda_{j})_{j\in\mathbb{N}},\\{U^{(1,2)}_{\theta_{-1}\omega}(1)E_{j}(\theta_{-1}\omega)\\}_{\omega\in\Omega_{0},j\in\mathbb{N}},\\{i^{-1}(F_{j}(\omega))\\}_{\omega\in\Omega_{0},j\in\mathbb{N}}).$

∎

## Assumption discussion

In this section, we aim to discuss the scenario where assumption
($\mathrm{A6}$) is met. Indeed, we will show something more, namely that this
assumption is fulfilled when, in particular, $X_{1}=L_{p}([-1,0])$
($1<p<\infty$) and $X_{2}=C([-1,0])$, and that measurability of function
factors follows from it.

######  Lemma 2.4.

Let $i:C([-1,0])\to L_{p}([-1,0])$ be the continuous injection defined by
$i(f)=f$. Moreover, let $w^{(1)}:\Omega\to L_{p}([-1,0])$ be a
$(\mathfrak{F},\mathfrak{B}(L_{p}([-1,0])))$-measurable function with the
factor $w^{(2)}:\Omega\to C([-1,0])$, i.e. $w^{(1)}=i\circ w^{(2)}$. Then
$w^{(2)}:\Omega\to C([-1,0])$ is
$(\mathfrak{F},\mathfrak{B}(C([-1,0])))$-measurable.

###### Proof.

The proof is based on observation that every closed ball, specifically the
unit ball, in $C([-1,0])$ corresponds to a pre-image of $i$ of some
$L_{p}$-closed subset $F\subset L_{p}([-1,0])$, i.e.
$\bar{B}_{C([-1,0])}(0,1)=i^{-1}(F)$ for some $L_{p}$-closed $F$. The set
$F\vcentcolon=\\{i(f)\in L_{p}([-1,0]):-1\leq f(x)\leq 1\text{ for a.e.
}x\in[-1,0]\\}$ proves to be a witness, as $L_{p}([-1,0])\setminus F$ is
$L_{p}$-open. Indeed, for $g\in L_{p}([-1,0])\setminus F$ there exists a
Lebesgue-measurable set $M\subset[-1,0]$ such that its Lebesgue measure
$\lambda(M)>0$ and $\epsilon>0$ such that $g(x)\geq 1+\epsilon$ for all $x\in
M$. Therefore, for $f\in F$ we have
$\|g-f\|_{L_{p}}\geq\epsilon\lambda^{1/p}(M)$ so
$B_{L_{p}(-1,0)}(g,\epsilon\lambda^{1/p}(M))\subset L_{p}([-1,0])\setminus F$.
It remains to show $i^{-1}(F)=\bar{B}_{C([-1,0])}(0,1)$. However, it can be
easily done via definition of pre-image since, conditions $-1\leq f(x)\leq 1$
for a.e. $x$ and $-1\leq f(x)\leq 1$ for all $x$ are equivalent for continuous
functions.

As a final step, we demonstrate that the map $w^{(2)}$ is
$(\mathfrak{F},\mathfrak{B}(C([-1,0])))$-measurable. Let $D$ be an open subset
of $C([-1,0])$ in the sense of the $C([-1,0])$ topology. Since $C([-1,0])$ is
separable, it has a countable base, and $D$ can be expressed as a countable
union of open balls. Moreover, open balls can be written as countable union of
closed ones. As a consequence the set $D$ is a countable union of pre-images
of $L_{p}$-closed sets. The observation that

$(w^{(2)})^{-1}[D]=(w^{(2)})^{-1}\Big{(}\bigcup_{n\in\mathbb{N}}i^{-1}(F_{n})\Big{)}=(i\circ
w^{(2)})^{-1}\Big{(}\bigcup_{n\in\mathbb{N}}F_{n}\Big{)}=(w^{(1)})^{-1}\Big{(}\bigcup_{n\in\mathbb{N}}F_{n}\Big{)}\in\mathfrak{F}$

finishes the proof. ∎

Acknowledgment.

I am grateful to Janusz Mierczyński for inspiring conversations.

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