# Tropes in films: an initial analysis

Rubén Héctor García-Ortega Department of Computer Architecture and Technology,
University of Granada Pablo García-Sánchez Department of Languages and
Computer Systems, University of Granada JJ Merelo Department of Computer
Architecture and Technology, University of Granada

###### Abstract

TVTropes is a wiki that describes tropes and which ones are used in which
artistic work. We are mostly interested in films, so after releasing the
TropeScraper Python module that extracts data from this site, in this report
we use scraped information to describe statistically how tropes and films are
related to each other and how these relations evolve in time. In order to do
so, we generated a dataset through the tool TropeScraper in April 2020. We
have compared it to the latest snapshot of DB Tropes, a dataset covering the
same site and published in July 2016, providing descriptive analysis, studying
the fundamental differences and addressing the evolution of the wiki in terms
of the number of tropes, the number of films and connections. The results show
that the number of tropes and films doubled their value and quadrupled their
relations, and films are, at large, better described in terms of tropes.
However, while the types of films with the most tropes has not changed
significantly in years, the list of most popular tropes has. This outcome can
help on shedding some light on how popular tropes evolve, which ones become
more popular or fade away, and in general how a set of tropes represents a
film and might be a key to its success. The dataset generated, the information
extracted, and the summaries provided are useful resources for any research
involving films and tropes. They can provide proper context and explanations
about the behaviour of models built on top of the dataset, including the
generation of new content or its use in machine learning.

Keywords : Tropes; TvTropes; tropescraper.

##  1 Motivations

TV Tropes [3] is an outstanding source of knowledge because it describes
thousands of tropes and relates them to artistic works, providing examples
with a broad context in terms of the characters, places and actions. These
relations, once extracted and processed, have shown to be very valuable in the
field of Artificial Intelligence, in both content generation and quality
prediction, as in [5].

Since TV Tropes is a non-structured wiki-style database, most researchers have
accessed to its data through a structured n-tuple-based database called DB
Tropes [1], which we enriched and analyzed in [4]. Unfortunately, BDTropes
remains unattended since July 2016, while the prolific community of TV Tropes
has doubled the number of films and their tropes since then. For this reason,
in order to allow the researchers to use the films and tropes from TV Tropes,
we implemented a scraper called TropeScraper [2] and licensed it as free
software, so it becomes part of the Python ecosystem. This new tool, which
takes days to download the whole database, has allowed us, and now you, to
make a more up-to-date analysis of the tropes used in films. The rest of the
paper will be devoted to this.

Although the quantity of the information extracted is promising, doubling the
number of films and tropes in TV Tropes, an exploratory statistical analysis
shows that the data is still incomplete, with distributions that point out to
huge biases due to the popularity of the films and tropes. The detection of
these biases is essential to explain the outcomes of future automatic learning
models build on top of the dataset, for example, according to [5] the number
of tropes in TV Tropes is proportional to the popularity of a movie and
inversely proportional to its age.

In the following report, we evaluate the dataset of films and tropes from TV
Tropes as extracted April 2020, because knowing their absolute numbers,
frequencies and the most relevant ones can be critical in different stages of
future researches in automatic authoring tools, from data wrangling to
technique selection, configuration and outcome interpretation. As a secondary
goal, we compare and analyse how TV Tropes has evolved from July 2016 to April
2020 in order to detect general trends that have emerged which could help on
the prediction of TV Tropes in some years away. Finally, since TV Tropes is an
unstructured wiki and we have to follow different heuristics to crawl the
pages correctly, we also assess the quality of the scraping process in this
report, considering possible inconsistencies and improvements based on the
results.

We divide the rest of the report as follows: Section 2 explains the data
acquisition phase. The analysis in Sections 3 and 4 includes the two
perspectives of the relation between films and tropes; in other words, how
many tropes appear in each movie and vice versa. The conclusion in Section 5
highlights and summarises the outcomes.

##  2 Notes on data Acquisition

It is important to remark that we have improved the original scraper that we
used in [5]. The first stable version of the scraper (v1.0.2) crawled through
the film pages and scraped tropes found in every film. The data seemed to
correlate with DBTropes; however, the analysis of the top tropes showed
missing information because it was not considering tropes that were very
common in 2020. After studying the situation, we discovered that counting the
tropes listed in the film pages was not enough; the trope pages also include
the list of films that they belong to, and this information complements the
previous one, increasing the relations between films and tropes. Finally, some
massively used tropes divide their occurrences into wiki pages that are in the
same level of hierarchy than films and tropes, although semantically they are
at the trope level. For these reasons, we evolved the scraper to find the
relations from the film’s perspective, from the trope’s perspective and also,
considering pagination, providing a more solid dataset that is very aligned to
the data in 2016 and the changes stated inside TV Tropes since then. We
published it as version v1.1.0. As a side effect, its scraping time passed
from a couple of hours to days, because it included ten times more pages to
process.

##  3 Analysis of the number of tropes used in the films

The descriptive analysis of the films regarding their tropes (Table 1) reveals
the growth of the number of films by 99.6%. The average number of tropes per
film has also grown by 279.38%. In other words, the dataset doubles the number
of films (2x) and quadruples the number of tropes by film (4x). It is
interesting to point out that the scraper has been able to find five movies
without tropes. However, in all the cases, we could manually check and confirm
that the films do not have a page in TV Tropes, so they are listed in the
categories by mistake.

The increase of the frequency is visible through the comparative boxplots
(Figure 1), where a reduced interquartile range shows that there are fewer
differences among the number of tropes in 50% of the films, and moved to the
top, that shows how the average number of tropes per film has grown.

| Jul. 2016 | April. 2020
---|---|---
min | 1.000 | 0.000
max | 1075.000 | 3611.000
nobs | 6296.000 | 12567.000
mean | 21.984 | 83.402
kurtosis | 85.792 | 76.165
skewness | 7.046 | 6.750
variance | 2164.167 | 20937.037
Table 1: Comparative Descriptive analysis of trope occurrences by film between
July 2016 and April 2019 Figure 1: Comparative boxplots of trope occurrences
by film between July 2016 and April 2019

If we represent the number of tropes per film concerning the number of movies
(Figure 2), we can observe a clear trend: the number of films with just one
trope is high, and the number of films decreases as the number of tropes
increases, reaching the minimum in 4 tropes. Then, the value reaches a maximum
value close to 20 tropes and decreases again following a long-tail
distribution.

Figure 2: Comparative boxplots of film occurrences by tropes between July 2016
and April 2019

The films with more tropes (Table 2) are adventure/action film franchises,
such as James Bond, The Avengers and Star Wars. If we compare the films in
2020 and 2016, we can observe that, in general, the top ones remain. As a
rule, the films that were previously at the top have multiplied the number of
tropes by up to 10.

| Film (Jul. 2016) | Tropes | Film (April. 2020) | Tropes | Increment
---|---|---|---|---|---
0 | JamesBond | 1075 | JamesBond | 3611 | +235.9%
1 | TheLordOfTheRings | 769 | TheLordOfTheRings | 2413 | +213.8%
2 | TheMatrix | 691 | TheAvengers2012 | 2220 | +558.8%
3 | TheDarkKnight | 613 | TheDarkKnight | 1999 | +226.1%
4 | StarTrek | 562 | ANewHope | 1932 | +732.8%
5 | WhoFramedRogerRabbit | 491 | ReturnOfTheJedi | 1926 | +519.3%
6 | Serenity | 472 | XMenFilmSeries | 1870 | +307.4%
7 | Transformers | 459 | TheMatrix | 1861 | +169.3%
8 | XMen | 459 | StarTrek2009 | 1769 | +214.8%
9 | XMenFirstClass | 446 | AvengersEndgame | 1765 | –
10 | Ghostbusters | 445 | AvengersInfinityWar | 1729 | +28,716.7%
11 | MenInBlack | 427 | TheForceAwakens | 1717 | +9,438.9%
12 | Avatar | 426 | RevengeOfTheSith | 1706 | +635.3%
13 | TheDarkKnightRises | 408 | Thor | 1697 | +366.2%
14 | XMenDaysOfFuturePast | 394 | ThorRagnarok | 1640 | –
15 | KillBill | 381 | AvengersAgeOfUltron | 1585 | –
16 | Thor | 364 | TheEmpireStrikesBack | 1584 | +504.6%
17 | BackToTheFuture | 360 | CaptainAmericaTheWinterSoldier | 1483 | +493.2%
18 | Spaceballs | 359 | CaptainAmericaCivilWar | 1481 | +21,057.1%
19 | PacificRim | 356 | TheDarkKnightRises | 1452 | +255.9%
20 | HarryPotter | 346 | WhoFramedRogerRabbit | 1444 | +194.1%
21 | MontyPythonAndTheHolyGrail | 345 | TheWizardOfOz | 1437 | +380.6%
22 | ThePrincessBride | 342 | XMenDaysOfFuturePast | 1425 | +261.7%
23 | BatmanBegins | 339 | CaptainAmericaTheFirstAvenger | 1420 | +346.5%
24 | TheAvengers | 337 | XMenFirstClass | 1420 | +218.4%
25 | TheThreeStooges | 336 | TheLastJedi | 1397 | –
26 | IronMan | 327 | ThePhantomMenace | 1379 | +529.7%
27 | CaptainAmericaTheFirstAvenger | 318 | GuardiansOfTheGalaxy | 1373 | +406.6%
28 | AlienS | 314 | Serenity | 1355 | +187.1%
29 | ReturnOfTheJedi | 311 | Terminator2JudgmentDay | 1352 | +349.2%
30 | Terminator2JudgmentDay | 301 | Transformers | 1231 | +168.2%
31 | TheWizardOfOz | 299 | Aliens | 1229 | +291.4%
32 | HotFuzz | 296 | KillBill | 1226 | +221.8%
33 | StarTrekIITheWrathOfKhan | 295 | XMenApocalypse | 1226 | +8,073.3%
34 | IndependenceDay | 288 | BackToTheFuture | 1219 | +238.6%
35 | StarTrekIntoDarkness | 283 | StarTrekIITheWrathOfKhan | 1219 | +313.2%
36 | TheFifthElement | 276 | StarTrekIntoDarkness | 1168 | +312.7%
37 | GalaxyQuest | 276 | Avatar | 1167 | +173.9%
38 | ManOfSteel | 274 | PacificRim | 1165 | +227.2%
39 | TronLegacy | 271 | BatmanVSupermanDawnOfJustice | 1164 | +11,540.0%
40 | GuardiansOfTheGalaxy | 271 | Batman1989 | 1161 | +333.2%
41 | AustinPowers | 268 | HarryPotter | 1146 | +231.2%
42 | DieHard | 268 | BlackPanther2018 | 1126 | –
43 | Batman | 268 | ManOfSteel | 1124 | +310.2%
44 | TheGodfather | 267 | TheGodfather | 1122 | +320.2%
45 | Inception | 267 | MontyPythonAndTheHolyGrail | 1117 | +223.8%
46 | IronMan2 | 265 | AttackOfTheClones | 1113 | +525.3%
47 | TheEmpireStrikesBack | 262 | MenInBlack | 1105 | +158.8%
48 | Godzilla2014 | 257 | GuardiansOfTheGalaxyVol2 | 1104 | –
49 | GIJoeTheRiseofCobra | 257 | BatmanBegins | 1103 | +225.4%
Table 2: Comparative top fifty films by the number of tropes between July 2016
and April 2019. *Common elements are marked in blue.

In some cases, the films have changed the name from 2016 to avoid ambigüity
(TheAvengers becomes TheAvengers2012, StarTrek becomes StarTrek2009, AlienS
becomes Aliens, Batman becomes Batman1989, Xmen becomes XmenFilmSeries). As
shown in the column "Increment", most of the films have around 100%-300% more
tropes in 2020 than in 2016. In just some cases, some films have increased the
number of tropes drastically, to cite the ones with an increment above 500%:
TheAvengers, ANewHope, ReturnOfTheJedi, RevengeOfTheSith,
TheEmpireStrikesBack, ThePhantomMenace and AttackOfTheClones. It is
interesting that 6 out of 7 films with such increment are from the Star Wars
franchise, that, in 2016 (with Episode VI just released), only had two films
in the top and, in 2020, with just four films more released by then, has
seven.

Thirty-four films out of the top fifty were already in the top list in 2016.
The remaining set includes films that had just been released at that time and
had a limited number of tropes (CaptainAmericaCivilWar, TheForceAwakens,
XMenApocalyse, BatmanVSupermanDawnOfJustice). Also, films that were released
but not included in TV Tropes yes (AvengersAgeOfUltron) and films not released
at that time (AvengerEndGame, ThorRagnarok, BlackPanther2018,
GuardiansOfTheGalaxyVol2). Finally, TV Tropes also includes films that are not
released yet but are already fulfilling some meta-tropes; this is the case of
AvengersInfinityWar, that already had the tropes LoadsAndLoadsOfCharacters and
GrandFinale in 2016.

##  4 Analysis of the number of films where the tropes appear

The descriptive analysis of the tropes and the films they appear in (Table 3)
reveals a growth of the number of tropes by 110.38%. The average number of
films that every trope appears in also grows by 259.95%. While there are
tropes that only exist in a single film, every trope is used on average by 28
films. The comparative boxplots (Figure 3) shows a more significant
interquartile range, and a shift to the top, that corresponds to stretching
the histogram from 2016 and a longer tail.

| Jul. 2016 | April. 2020
---|---|---
min | 1.000 | 1.000
max | 480.000 | 6591.000
nobs | 17738.000 | 37317.000
mean | 7.803 | 28.087
kurtosis | 274.065 | 1186.002
skewness | 10.043 | 22.339
variance | 117.915 | 7111.168
Table 3: Comparative descriptive analysis of trope occurrences by film between
July 2016 and April 2019 Figure 3: Comparative boxplots of trope occurrences
by films between July 2016 and April 2019

The representation of the number of films vs tropes (Figure 4) shows a
distribution similar to 2016 but stretched and extended. The only main
difference is that the number of films with a single trope is three times as
high now as in 2016. In other words, we have more tropes, and in general,
films have more tropes as well.

Figure 4: Comparative boxplots of trope occurrences by film between July 2016
and April 2019

However, having a distribution that resembles the one in 2016 does not
necessarily mean that TV Tropes has not suffered many changes. Although the
top films have not changed a lot since 2016, the top tropes have, keeping only
three out of fifty (Table 4). The tropes have evolved so much since 2016 that
43 out of the top fifty in 2020 did not even exist in 2016.

| Trope (Jul. 2016) | Films | Trope (April. 2020) | Films | Increment
---|---|---|---|---|---
0 | BoxOfficeBomb | 480 | AmericanFilms | 6591 | –
1 | CultClassic | 283 | ShoutOut | 3603 | –
2 | HeyItsThatGuy | 242 | BigBad | 2538 | –
3 | ArchEnemy | 229 | BoxOfficeBomb | 2193 | +356.9%
4 | BMovie | 205 | ChekhovsGun | 1977 | –
5 | CaliforniaDoubling | 190 | HorrorFilms | 1944 | –
6 | LogoJoke | 153 | Foreshadowing | 1850 | +184,900.0%
7 | HorrorDoesntSettleForSimpleTuesday | 132 | OhCrap | 1787 | –
8 | RetroactiveRecognition | 122 | BittersweetEnding | 1727 | –
9 | SignatureSong | 122 | TitleDrop | 1541 | –
10 | FilmNoir | 119 | DeadpanSnarker | 1456 | –
11 | DisneyVillainDeath | 116 | OneWordTitle | 1429 | +1,329.0%
12 | DuringTheWar | 113 | FilmsOfThe1980s | 1410 | –
13 | ProtagonistTitle | 112 | DownerEnding | 1400 | –
14 | PreMortemOneLiner | 111 | WMG | 1396 | –
15 | CompletelyDifferentTitle | 111 | Jerkass | 1377 | –
16 | RomanticComedy | 111 | NightmareFuel | 1356 | –
17 | BillingDisplacement | 110 | LargeHam | 1353 | –
18 | TheForeignSubtitle | 109 | FilmsOf20102014 | 1349 | –
19 | MissingTrailerScene | 108 | FilmsOf20052009 | 1311 | –
20 | FakeNationality | 108 | TearJerker | 1307 | –
21 | HollywoodActionHero | 106 | RunningGag | 1247 | –
22 | OneWordTitle | 100 | Headscratchers | 1221 | –
23 | EpicMovie | 99 | WhatHappenedToTheMouse | 1217 | –
24 | TheStinger | 98 | FilmsOf20152019 | 1167 | –
25 | DemotedToExtra | 96 | KarmaHoudini | 1163 | –
26 | ImpaledWithExtremePrejudice | 95 | TheCameo | 1162 | +2,012.7%
27 | PrettyInMink | 94 | MeaningfulName | 1146 | –
28 | SpiritualLicensee | 94 | AssholeVictim | 1116 | –
29 | AllStarCast | 94 | DrivenToSuicide | 1111 | –
30 | BestKnownForTheFanservice | 93 | Tagline | 1105 | –
31 | AndStarring | 93 | TooDumbToLive | 1096 | –
32 | CriticalDissonance | 93 | MohsScaleOfViolenceHardness | 1071 | –
33 | CreatorCameo | 93 | ImageSource | 1067 | –
34 | FinalGirl | 93 | ActorAllusion | 1061 | –
35 | SequelHook | 92 | KickTheDog | 1039 | –
36 | BottomlessMagazines | 91 | BookEnds | 1034 | –
37 | SignatureLine | 90 | BerserkButton | 1025 | –
38 | DVDCommentary | 90 | FilmsOf20002004 | 1022 | –
39 | NotEvenBotheringWithTheAccent | 89 | YourCheatingHeart | 1011 | +2,251.2%
40 | TheOner | 87 | RealityEnsues | 1011 | +101,000.0%
41 | Fingore | 85 | PrecisionFStrike | 1002 | –
42 | CarFu | 80 | MoodWhiplash | 997 | –
43 | ThreeDMovie | 79 | CompletelyDifferentTitle | 984 | +786.5%
44 | DeliberatelyMonochrome | 77 | GroinAttack | 983 | –
45 | NeckSnap | 77 | BrickJoke | 979 | –
46 | BlackDudeDiesFirst | 76 | FilmsOfThe1970s | 973 | –
47 | DeathByAdaptation | 76 | EstablishingCharacterMoment | 969 | –
48 | ArcWords | 76 | FilmsDiscussedByMoviebob | 967 | –
49 | ArtisticLicenseGeography | 76 | BritishFilms | 964 | –
Table 4: Comparative top fifty films by the number of tropes between July 2016
and April 2019. *Common elements are marked in blue.

The first question that comes into our minds is if the scraper is working
fine. In order to answer this question, we analyse the top tropes from 2016
and try to trace the reasons why they are not so popular in 2020. Table 5
shows the comparative ranking of the top tropes between 2016 and their
position in 2020. According to it, the vast majority of tropes have
incremented their appearance in films by a 200%-300%. Nevertheless, this is
not enough for them to be in the top fifty, although they mostly keep being in
the top 1,300. There are four cases with a reduced use: TV Tropes avoids
CultClassic now because of its lack of universality, removes HeyItsThatGuy and
replaces it by RetroactiveRecognition, replaces SpiritualLicensee by
SpiritualAdaptation (although the old trope still appears in some films) and
removes ThreeDMovie.

| Trope | Films (Jul. 2016) | Films (April. 2020) | Increment | Moves to
---|---|---|---|---|---
0 | BoxOfficeBomb | 480 | 2193 | +356.9% | +3rd
1 | CultClassic | 283 | 13 | -95.4% | +16,030th
2 | HeyItsThatGuy | 242 | 0 | – | –
3 | ArchEnemy | 229 | 447 | +95.2% | +212th
4 | BMovie | 205 | 577 | +181.5% | +138th
5 | CaliforniaDoubling | 190 | 418 | +120.0% | +238th
6 | LogoJoke | 153 | 420 | +174.5% | +231st
7 | HorrorDoesntSettleForSimpleTuesday | 132 | 179 | +35.6% | +854th
8 | RetroactiveRecognition | 122 | 505 | +313.9% | +170th
9 | SignatureSong | 122 | 319 | +161.5% | +370th
10 | FilmNoir | 119 | 190 | +59.7% | +791st
11 | DisneyVillainDeath | 116 | 400 | +244.8% | +251st
12 | DuringTheWar | 113 | 130 | +15.0% | +1,292nd
13 | ProtagonistTitle | 112 | 204 | +82.1% | +705th
14 | PreMortemOneLiner | 111 | 385 | +246.8% | +266th
15 | CompletelyDifferentTitle | 111 | 984 | +786.5% | +43rd
16 | RomanticComedy | 111 | 314 | +182.9% | +376th
17 | BillingDisplacement | 110 | 241 | +119.1% | +559th
18 | TheForeignSubtitle | 109 | 208 | +90.8% | +689th
19 | MissingTrailerScene | 108 | 266 | +146.3% | +490th
20 | FakeNationality | 108 | 276 | +155.6% | +468th
21 | HollywoodActionHero | 106 | 136 | +28.3% | +1,211th
22 | OneWordTitle | 100 | 1429 | +1,329.0% | +11th
23 | EpicMovie | 99 | 219 | +121.2% | +639th
24 | TheStinger | 98 | 456 | +365.3% | +205th
25 | DemotedToExtra | 96 | 412 | +329.2% | +241st
26 | ImpaledWithExtremePrejudice | 95 | 484 | +409.5% | +188th
27 | PrettyInMink | 94 | 303 | +222.3% | +405th
28 | SpiritualLicensee | 94 | 35 | -62.8% | +6,670th
29 | AllStarCast | 94 | 178 | +89.4% | +861st
30 | BestKnownForTheFanservice | 93 | 226 | +143.0% | +613th
31 | AndStarring | 93 | 437 | +369.9% | +216th
32 | CriticalDissonance | 93 | 272 | +192.5% | +478th
33 | CreatorCameo | 93 | 765 | +722.6% | +77th
34 | FinalGirl | 93 | 373 | +301.1% | +281st
35 | SequelHook | 92 | 517 | +462.0% | +165th
36 | BottomlessMagazines | 91 | 295 | +224.2% | +416th
37 | SignatureLine | 90 | 267 | +196.7% | +489th
38 | DVDCommentary | 90 | 198 | +120.0% | +742nd
39 | NotEvenBotheringWithTheAccent | 89 | 320 | +259.6% | +367th
40 | TheOner | 87 | 340 | +290.8% | +332nd
41 | Fingore | 85 | 363 | +327.1% | +296th
42 | CarFu | 80 | 349 | +336.2% | +321st
43 | ThreeDMovie | 79 | 1 | -98.7% | +35,034th
44 | DeliberatelyMonochrome | 77 | 313 | +306.5% | +381st
45 | NeckSnap | 77 | 254 | +229.9% | +524th
46 | BlackDudeDiesFirst | 76 | 402 | +428.9% | +247th
47 | DeathByAdaptation | 76 | 397 | +422.4% | +252nd
48 | ArcWords | 76 | 639 | +740.8% | +116th
49 | ArtisticLicenseGeography | 76 | 262 | +244.7% | +501st
Table 5: Top fifty films by the number of tropes in July 2016 and their
increment of tropes in April 2020.

##  5 Conclusions

We extract the following outcomes from the previous analysis.

In terms of gross numbers, the number of films and tropes in 2020 doubles the
number in 2016. It goes from 6,296 films to 12,567 and from 17,738 tropes to
37,317. The average number of relations between films and tropes in 2020
quadruples the number in 2016; in other words, films are better described in
terms of tropes and tropes appear in more films. In absolute terms, the number
of connections has increased by 657.23%. Both distribution curves shift right
(less positive skewness). However, while the interquartile range of tropes’
distribution has reduced, the interquartile range of the films’ distribution
has increased.

Regarding the films with more tropes in TV Tropes, only 32% of the films in
the top fifty have changed from 2016 to 2020, that is to say, in general,
popular films in terms of tropes remain. The James Bond franchise is the
artistic work with more tropes, with almost a third more than the following,
The Lord of the rings, that overcomes the next by a few hundreds of tropes,
The Avengers. It is imperative to hint how the franchise Star Wars boosted its
popularity from 2016 to 2020, also. Regarding the most popular tropes,
although the majority of them in 2016 are still relevant in 2020, the new top
list if almost entirely replaced by tropes that did not exist in 2016. The
most popular trope in 2020 is AmericanFilms, followed by ShoutOut and BigBad.

We discovered that TV Tropes adapts the name of the films to disambiguate
second parts or remakes as they appear. They also include films that have not
been premiered and extrapolate possible tropes from the existing information.

We could detect errors in the wiki that lead to parsing inexisting film pages
and incomplete information that made us evolve the scraper several times. The
latest modifications made to tropescraper allows extracting the data
correctly, with results that correlate to the ones from DB Tropes in 2016.
However, due that the wiki is unstructured and maintained manually, it is
crucial to keep a critic eye on the data extracted from it, especially the
outliers, and always consider a data wrangling phase before using the data to
build models in machine learning. It also points to specific, and network-
based, dynamic of tropes and films.

## Acknowledgements

This work has been partially funded by projects DeepBio (TIN2017-85727-C4-2-P)
and TEC2015-68752 and “Ayuda del Programa de Fomento e Impulso de la actividad
Investigadora de la Universidad de Cádiz”.

##  6 Bibliography

## References

  * [1] DBTropes – skipforward.  http://skipforward.opendfki.de/wiki/DBTropes.  (Accessed on 05/31/2020).
  * [2] tropescraper · pypi.  https://pypi.org/project/tropescraper/.  (Accessed on 05/31/2020).
  * [3] TV Tropes.  https://tvtropes.org/.  (Accessed on 06/07/2020).
  * [4] R. H. García-Ortega, J. J. Merelo-Guervós, P. G. Sánchez, and G. Pitaru.  Overview of PicTropes, a film trope dataset.  arXiv preprint arXiv:1809.10959, 2018.
  * [5] R. H. García-Ortega, P. García-Sánchez, and J. J. Merelo-Guervós.  Startroper, a film trope rating optimizer using machine learning and evolutionary algorithms.  Expert Systems, 37(6):e12525, 2020.


# Generalized evolution semigroups and general dichotomies

Nicolae Lupa  N. Lupa, Department of Mathematics, Politehnica University of
Timişoara, Piaţa Victoriei 2, 300006 Timişoara, Romania<EMAIL_ADDRESS>and  Liviu Horia Popescu  L. H. Popescu, Department of Mathematics and
Informatics, Faculty of Sciences, University of Oradea, Universităţii St. 1,
410087 Oradea, Romania<EMAIL_ADDRESS>
###### Abstract.

We introduce a special class of real semiflows, which is used to define a
general type of evolution semigroups, associated to not necessarily
exponentially bounded evolution families. Giving spectral characterizations of
the corresponding generators, our results directly apply to a wide class of
dichotomies, such as those with time-varying rate of change.

###### Key words and phrases:

Real semiflows, evolution families, evolution semigroups, hyperbolicity,
dichotomies

###### 2010 Mathematics Subject Classification:

34D09, 34G10, 47D06

##  1\. Introduction

It is known that if $A$ is a _bounded_ linear operator on a Banach space $X$,
then the asymptotic behavior of solutions $x(t)=e^{tA}x_{0}$ of the autonomous
equation

$\dot{x}=Ax$ (1)

is completely determined by the spectrum of $A$, $\sigma(A)$ (for much more
information we refer the reader to [8, Chapter II] and [10, Section I.3]). For
instance, the differential equation (1) is exponentially stable (i.e., there
exist constants $\nu>0$ and $N\geq 1$ such that $\|e^{tA}\|\leq Ne^{-\nu t}$
for every $t\geq 0$) if and only if the spectrum $\sigma(e^{tA})$ lies in the
open unit disk $\mathbb{D}=\\{\lambda\in\mathbb{C}:\,|\lambda|<1\\}$ or,
equivalently, $\sigma(A)$ is contained in the open left half of the complex
plane (see [10, Theorem I.3.14]). Consequently, the following statements are
equivalent:

  * •

the differential equation (1) admits an exponential dichotomy on $X$, i.e.,
there exists a direct decomposition $X=X_{s}\oplus X_{u}$, into $A$-invariant
closed subspaces $X_{s}$ and $X_{u}$, such that there are constants $\nu>0$,
$N\geq 1$ satisfying

$\|e^{tA}x\|\leq Ne^{-\nu t}\|x\|,\;x\in X_{s},$

and

$\|e^{tA}x\|\geq\frac{1}{N}e^{\,\,\nu t}\|x\|,\;x\in X_{u},$

for all $t\geq 0$;

  * •

the semigroup $\left\\{e^{tA}\right\\}_{t\geq 0}$ is hyperbolic, i.e.,
$\sigma(e^{tA})\cap\mathbb{T}=\emptyset$ for some (and hence all) $t>0$, where
$\mathbb{T}$ denotes the unit circle on $\mathbb{C}$;

  * •

the spectrum $\sigma(A)$ does not intersect the imaginary axis, $\sigma(A)\cap
i\mathbb{R}=\emptyset$.

The key to the proof of the above results lies in the spectral mapping
property

$\sigma(e^{tA})=e^{t\sigma(A)},\;t\geq 0,$

which in this case is a simple consequence of Dunford’s functional calculus
[9, Theorem 2.9].

For a general $C_{0}$-semigroup $\left\\{S_{t}\right\\}_{t\geq 0}$ on $X$,
generated by an _unbounded_ linear operator $A$, the situation is more
complex. We recall that a _$C_{0}$ -semigroup_ on the Banach space $X$ is a
family $\left\\{S_{t}\right\\}_{t\geq 0}$ of bounded linear operators on $X$
such that the following properties are satisfied:

  * •

$S_{0}=\mathrm{Id}$ (the identity on $X$);

  * •

$S_{t}\,S_{\tau}=S_{t+\tau},\text{ for }t,\tau\geq 0;$

  * •

$S_{t}x\to x$ in $X$ as $t\to 0^{+}$ for all $x\in X$.

The _generator_ of a $C_{0}$-semigroup $\left\\{S_{t}\right\\}_{t\geq 0}$ is
the linear operator $A$, with domain $D(A)$, defined by

$D(A)=\left\\{x\in X:\,\lim\limits_{t\to 0^{+}}\frac{S(t)x-x}{t}\text{ exists
}\right\\}$

and

$Ax=\lim\limits_{t\to 0^{+}}\frac{S(t)x-x}{t},\,x\in D(A).$

We refer the reader to the monograph of Engel and Nagel [10] for a brief
history of $C_{0}$-semigroups (see also [27] for a short and very nice
presentation of semigroups and their recent applications).

For any $C_{0}$-semigroup, a spectral inclusion is always valid (see, for
instance, Theorem 2.3 in [32, Chapter 2]):

$\sigma(S_{t})\supseteq e^{t\sigma(A)},\,t\geq 0.$

On the other hand, if we take a $C_{0}$-semigroup that cannot be extended to a
group (e.g., the left translation semigroup on the Banach space of all
continuous functions on $\mathbb{R}_{+}$ vanishing at infinity), then
$0\in\sigma(S_{t})$ for all $t>0$, while evidently $0$ does not belong to
$e^{t\sigma(A)}$. Hence, to obtain a spectral mapping property, we must
exclude $0$ from the spectrum of $S_{t}$. Even so, the _spectral mapping
theorem_

$\sigma(S_{t})\setminus\\{0\\}=e^{t\sigma(A)},\,t\geq 0,$

generally fails (see [7, Section 2.1.5] or [10, pp. 270-275]). However, it is
valid for many classes of $C_{0}$-semigroups, such as eventually norm
continuous semigroups (see [10, Paragraph IV.3.10] or [31, Theorem 2.3.2])
and, consequently (see Diagram (4.26) in [10, p. 119]), eventually compact
semigroups, eventually differentiable semigroups, analytic semigroups,
uniformly continuous semigroups [10, Corollary IV.3.12]. Furthermore, some
spectral mapping formula holds for the point and residual spectrum (see [31,
Theorems 2.1.2, 2.1.3]). Thus, the failure of the spectral mapping theorem is
determined by the approximate point spectrum. For a detailed analysis of the
spectral mapping theorem we refer the reader to [10, Section IV.3] or [31,
Chapter 2].

In order to overcome the failure of the spectral mapping theorem, Latushkin
and Montgomery-Smith [15, 16] proved that the hyperbolicity of
$\left\\{S_{t}\right\\}_{t\geq 0}$ can be completely determined by the
generator $G$ of a $C_{0}$-semigroup $\left\\{T_{t}\right\\}_{t\geq 0}$,
defined by

$(T_{t}u)(s)=S_{t}u(t-s),$

on certain _super-space_ $E(X)$ of functions $u:\mathbb{R}\to X$. This
semigroup is called the _evolution semigroup_ associated to
$\left\\{S_{t}\right\\}_{t\geq 0}$. For instance, if $E(X)$ is one of the
spaces $C_{0}(\mathbb{R},X)$ or $L^{p}(\mathbb{R},X)$, for $1\leq p<\infty$,
then the hyperbolicity of $\left\\{S_{t}\right\\}_{t\geq 0}$ on $X$ is
equivalent to the hyperbolicity of $\left\\{T_{t}\right\\}_{t\geq 0}$ on
$E(X)$ or, equivalently, to the invertibility of $G$ (see [16, Theorem 2.5] or
[7, Theorem 2.39]). Moreover, the evolution semigroup
$\left\\{T_{t}\right\\}_{t\geq 0}$ always satisfies the spectral mapping
theorem in $E(X)$ even if the underlying semigroup
$\left\\{S_{t}\right\\}_{t\geq 0}$ does not have this property on $X$ [7,
Corollary 2.40].

For a non-autonomous differential equation $\dot{x}=A(t)x$ the situation is
much more difficult then the autonomous case. It is known that if the
associated Cauchy problem

$\quad\left\\{\begin{array}[]{ll}\dot{x}(t)=A(t)x(t),\;t\geq{s},\\\
x(s)=x_{s},\end{array}\right.$

is well-posed, then one can define an evolution family
$\left\\{U(t,s)\right\\}_{t\geq s}$ on $X$ such that $x(t)=U(t,s)x(s)$ for
$t\geq s$ (for more details on well-posed non-autonomous Cauchy problems we
refer the reader to [26] and the references therein).

Let now $\mathcal{U}=\left\\{U(t,s)\right\\}_{t\geq s}$ be an evolution family
on $X$ (not necessarily associated to a non-autonomous differential equation)
such that there exist constants $\alpha>0$ and $K\geq 1$ satisfying

$\|U(t,s)\|\leq Ke^{\alpha(t-s)},\text{ for }t\geq s,$ (2)

that is $\mathcal{U}$ is (uniformly) _exponentially bounded_. Then, one can
define a $C_{0}$-semigroup $\left\\{T_{t}\right\\}_{t\geq 0}$ on
$C_{0}(\mathbb{R},X)$ or $L^{p}(\mathbb{R},X)$, for $1\leq p<\infty$, by

$T_{t}u(s)=U(s,s-t)u(s-t),$ (3)

called the _evolution semigroup_ associated to the evolution family
$\mathcal{U}$ (see [10, Lemma VI.9.10] or [7, Proposition 3.11]), and denote
its generator by $(G,D(G))$.

The theory of evolution (semi)groups associated to an evolution family has a
long history going back to Howland [12] and Lovelady [18]. Significant
contributions to this theory are due to Evans [11], Neidhardt [30], Rau [35,
36], Van Minh [22, 23], Latushkin and Montgomery-Smith [15, 16], Latushkin and
Randolph [17], Räbiger and Schnaubelt [34], Räbiger, Rhandi, and Schnaubelt
[33], and many others.

The spectra of the operators $T_{t}$ and the generator $G$ of the evolution
semigroup $\left\\{T_{t}\right\\}_{t\geq 0}$ have some important symmetry
properties: _$\sigma\left(T_{t}\right)$ is rotationally invariant for each
$t>0$_, that is

$\lambda\sigma\left(T_{t}\right)=\sigma\left(T_{t}\right)\text{ for every
$\lambda\in\mathbb{C}$ with $|\lambda|=1$},$

and _the spectrum $\sigma(G)$ is invariant under translations along the
imaginary axis_, respectively, i.e.,

$\sigma(G)=\sigma(G)+i\mathbb{R}.$

Furthermore, _the evolution semigroup is hyperbolic if and only if its
generator is invertible_ (see [16, Theorem 3.1] or [7, Theorem 3.13]). In
particular, _the evolution semigroup satisfies the spectral mapping theorem_.

On the other hand, the hyperbolicity of the evolution semigroup characterizes
the exponential dichotomy of the underlying evolution family (see Theorem 1.1
below), and thus the evolution semigroups method provides a strong tool to
study the exponential dichotomy of evolution families. More precisely, the
evolution family $\mathcal{U}$ admits a (uniform) _exponential dichotomy_ if:

  1. (a)

there exist projections $P(t):X\to X$, $t\in\mathbb{R}$, and write
$Q(t)=\mathrm{Id}-P(t)$, _compatible with $\mathcal{U}$_, that is
$P(t)U(t,s)=U(t,s)P(s)$ and the restriction $U_{Q}(t,s):Q(s)X\to Q(t)X$ of
$U(t,s)$ is invertible, for all $t\geq s$;

  2. (b)

there exist constants $\nu>0$, $N\geq 1$ such that

$\|U(t,s)P(s)\|\leq Ne^{-\nu(t-s)}\text{ and }\|U_{Q}(s,t)Q(t)\|\leq
Ne^{-\nu(t-s)},\;t\geq s,$

where $U_{Q}(s,t)=U_{Q}(t,s)^{-1}$.

The above mentioned result is now stated in the following theorem (see, for
instance, [7, Theorem 3.17 and Theorem 4.25] or [10, Theorem VI.9.18]).

######  Theorem 1.1.

Let $\mathcal{U}=\left\\{U(t,s)\right\\}_{t\geq s}$ be an exponentially
bounded evolution family on a Banach space $X$, let
$\left\\{T_{t}\right\\}_{t\geq 0}$ be the associated evolution semigroup on
$E(X)$ defined by (3), where $E(X)$ is $C_{0}(\mathbb{R},X)$ or
$L^{p}(\mathbb{R},X)$, for $1\leq p<\infty$, and denote $G$ its generator. The
following assertions are equivalent:

  1. (1)

$\mathcal{U}$ admits an exponential dichotomy on $X$;

  2. (2)

$\left\\{T_{t}\right\\}_{t\geq 0}$ is hyperbolic on $E(X)$;

  3. (3)

$\sigma(G)\cap i\mathbb{R}=\emptyset$. In this case, $G$ is invertible and its
inverse is given by

$(G^{-1}f)(t)=-\int_{\mathbb{R}}\Gamma(t,s)f(s)\,ds,\text{ for all }f\in
E(X)\text{ and }t\in\mathbb{R},$ (4)

where

$\Gamma(t,s)=\begin{cases}\;\;\phantom{-}U(t,s)P(s),&t>s,\\\
-U_{Q}(t,s)Q(s),&t<s,\end{cases}$

is the Green function associated to the exponential dichotomous evolution
family $\mathcal{U}$.

To our knowledge, formula (4) was first proved in this context by Latushkin
and Randolph [17].

Räbiger and Schnaubelt extended Theorem 1.1 to a large class of $X$-valued
function spaces, including in particular $C_{0}(\mathbb{R},X)$ and
$L^{p}(\mathbb{R},X)$, for $1\leq p<\infty$ [34]. For an excellent survey on
the theory of evolution semigroups and their applications we refer the reader
to the monograph of Chicone and Latushkin [7], where the authors also
developed a systematic theory of evolution semigroups induced by a cocycle
over a locally compact metric space acting on Banach fibers. More recently,
the evolution semigroups method was extended to nonuniform behavior (see, for
instance, [1, 2, 5, 19, 20]).

For non-autonomous differential equations, the classical concept of
exponential dichotomy may look as too restrictive, therefore it is important
to search for more general behavior, for instance considering dichotomies
given by general growth rates, an approach initiated to the best of our
knowledge by Naulin and Pinto [28, 29]. In particular, the contraction and
expansion can be of the form $e^{\gamma\int_{0}^{t}\rho(\xi)d\xi}$ (see [13,
14, 21, 25]) or $e^{\gamma\mu(t)}$ [3], including the usual exponential
behavior and polynomial behavior [4, 6] as special cases. These types of
asymptotic behavior can occur in the critical situations when all Lyapunov
exponents are infinite ($\pm\infty$) or they are all zero [3]. We also
emphasize that some of these types of dichotomy were connected to the
important problem of linearization of dynamical systems, which is of a great
interest for geometry.

In this paper, we generalize the classical concept of the evolution semigroup
associated to an evolution family, hereby called the _generalized evolution
semigroup_ , replacing the right translation semiflow
$\varphi(t,s)=s-t,\;t\geq 0,\,s\in\mathbb{R},$ in formula (3) with a general
real semiflow. We point out that in our setting the evolution family might not
be exponentially bounded.

The main purpose of generalizing the concept of evolution semigroup is to
study a wide class of dichotomies presented above. We prove that under some
suitable conditions, the evolution semigroup we consider is similar to the
classical evolution semigroup defined by (3). This enables us to apply well
established results to completely characterize a general type of exponential
dichotomy of an evolution family in terms of spectral conditions imposed to
the generator of the corresponding generalized evolution semigroup. We
restrict our study to the case of the space of all continuous functions
vanishing at infinity.

Present paper is organized as follows. In Section 2 we introduce the notion of
a _real semiflow_. We think the main result here is the one in Theorem 2.5,
that completely characterizes the non-degenerated real semiflows. The third
section deals with what we call as _generalized evolution semigroups_ ,
associated to real semiflows. In Theorem 3.3 we prove that the generalized
evolution semigroups are $C_{0}$-semigroups on $C_{0}(\mathbb{R},X)$.
Furthermore, in subsection 3.2 we show that the generalized evolution
semigroups of a certain type are similar to the classical evolution semigroup,
thus satisfying the spectral mapping theorem. In the last section we analyze a
wide class of exponential dichotomies with time-varying rates of change, by
using the generalized evolution semigroups previously constructed.

##  2\. Real semiflows

In this section, we introduce a special class of real semiflows which will be
used to generalize the classical notion of the evolution semigroup associated
to an evolution family. To the best of our knowledge, the results in this
section are completely new.

As usually, $\mathbb{R}$ denotes the field of real numbers and write
$\mathbb{R}_{+}=[0,\infty)$.

######  Definition 2.1.

Let $\Theta$ be a locally compact metric space. A continuous mapping
$\varphi:\mathbb{R}_{+}\times\Theta\to\Theta$ is called a _semiflow_ on
$\Theta$ if the following properties hold:

  1. (i)

$\varphi_{0}(\theta)=\theta$, for every $\theta\in\Theta$,

  2. (ii)

$\varphi_{t}\circ\varphi_{\tau}=\varphi_{t+\tau}$, for all $t,\tau\geq 0$,

where $\varphi_{t}(\theta)=\varphi(t,\theta)$, for $t\geq 0$ and
$\theta\in\Theta$. In this case, the function $\varphi_{t}(\cdot)$ is called
_the transition map_ defined by $t$.

In particular, a semiflow
$\varphi:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$ (that is
$\Theta=\mathbb{R}$) will be called a _real semiflow_ if in addition the
following inequality holds:

$\varphi_{t}(s)\leq s,\text{ for all $t\geq 0$ and $s\in\mathbb{R}$.}$ (5)

It is easy to observe that (5) implies

$\lim\limits_{s\to-\infty}\varphi_{t}(s)=-\infty,\text{ for every }t\geq 0.$
(6)

The orbit of $s\in\mathbb{R}$ is the set
$o\left(s\right)=\left\\{\varphi_{t}\left(s\right):\;t\geq 0\right\\}.$ If
$o\left(s\right)=\left\\{s\right\\}$, we say that the orbit $o\left(s\right)$
is _trivial_.

A classical example of real semiflow is _the right translation semiflow_ ,
defined by

$\varphi(t,s)=s-t,\text{ for }t\geq 0\text{ and }s\in\mathbb{R},$ (7)

with the orbits $o(s)=(-\infty,s]$, $s\in\mathbb{R}$.

######  Remark 2.2.

For each $s\in\mathbb{R}$ the function $\mathbb{R}_{+}\ni
t\mapsto\varphi_{t}\left(s\right)\in\mathbb{R}$ is decreasing, meanwhile for
each $t\geq 0$ the function $\mathbb{R}\ni
s\mapsto\varphi_{t}\left(s\right)\in\mathbb{R}$ is increasing.

###### Proof.

Indeed, for each fixed $s\in\mathbb{R}$, if $t_{1},t_{2}\geq 0$ with
$t_{1}<t_{2}$, one has

$\varphi_{t_{2}}\left(s\right)=\varphi_{t_{2}-t_{1}}\left(\varphi_{t_{1}}\left(s\right)\right)\leq\varphi_{t_{1}}\left(s\right),$

and thus the function $t\mapsto\varphi_{t}\left(s\right)$ is decreasing. For
the second statement, assume that there exist $s_{1},s_{2}\in\mathbb{R}$ with
$s_{1}<s_{2}$ such that
$\varphi_{t_{0}}\left(s_{1}\right)>\varphi_{t_{0}}\left(s_{2}\right)$ for some
$t_{0}>0$. Then,

$\varphi_{t_{0}}(s_{2})<\varphi_{t_{0}}\left(s_{1}\right)\leq
s_{1}<s_{2}=\lim\limits_{t\to 0_{+}}\varphi_{t}\left(s_{2}\right),$

and hence there exists $t_{1}\in(0,t_{0})$ such that
$\varphi_{t_{1}}\left(s_{2}\right)=s_{1}$. This yields

$\varphi_{t_{0}}\left(s_{2}\right)<\varphi_{t_{0}}\left(\varphi_{t_{1}}\left(s_{2}\right)\right)=\varphi_{t_{0}+t_{1}}\left(s_{2}\right),$

which contradicts the first statement above. ∎

For any fixed $s\in\mathbb{R}$, we set

$\omega\left(s\right)=\lim\limits_{t\rightarrow\infty}\varphi_{t}\left(s\right).$

Notice that $\omega(s)$ is a real number or $\omega(s)=-\infty$, and

$\omega(s)\leq\varphi_{t}(s)\leq s,\text{ for all }t\geq 0\text{ and
}s\in\mathbb{R}.$ (8)

Moreover,

$s_{1}<s_{2}\text{ in
}\mathbb{R}\;\Rightarrow\;\omega(s_{1})\leq\omega(s_{2})\text{ in
}\mathbb{R}\cup\\{-\infty\\}.$ (9)

Hence, if $\omega(s)=-\infty$ for some $s\in\mathbb{R}$, then
$\omega(\tau)=-\infty$ for every $\tau\in(-\infty,s]$. On the other hand, if
there exists $s\in\mathbb{R}$ such that $\omega(s)\in\mathbb{R}$, then
$\omega(\tau)\in\mathbb{R}$ for every $\tau\geq s$, thus
$\omega:[s,\infty)\to\mathbb{R}$ is well-defined. Furthermore, _$\omega$ is
right continuous at $s$_. Indeed, since $t\mapsto\varphi_{t}(s)$ is a
decreasing function, we have

$\omega(s)=\inf\limits_{t\geq 0}\varphi_{t}(s).$

It is well-known that the pointwise infimum of an arbitrary collection of
upper semi-continuous functions is upper semi-continuous. Thus,

$\limsup\limits_{\tau\to s^{+}}\,\omega(\tau)\leq\omega(s).$

On the other hand, by (9) we have

$\omega(s)\leq\lim\limits_{\tau\to s^{+}}\omega(\tau)=\limsup\limits_{\tau\to
s^{+}}\,\omega(\tau).$

Therefore, $\lim\limits_{\tau\to s^{+}}\omega(\tau)=\omega(s),$ that is
$\omega$ is right continuous at $s$.

Now, let us observe that letting $\tau\to\infty$ in the relation

$\varphi_{\tau}(\varphi_{t}(s))=\varphi_{\tau+t}(s)=\varphi_{t}(\varphi_{\tau}(s)),$

we get

$\omega(\varphi_{t}(s))=\omega(s),\;t\geq 0,$ (10)

and, in addition, using the continuity of the function $\varphi_{t}(\cdot)$,
we obtain

$\varphi_{t}(\omega(s))=\omega(s),\;t\geq 0.$ (11)

On the other hand, letting $t\to\infty$ in the above relation, it results that

$\omega(\omega(s))=\omega(s).$ (12)

We stress that relations (10)–(12) are valid provided that
$\omega(s)\in\mathbb{R}$. In particular, (10) shows that the restriction of
$\omega$ on the orbit $o(s)$ is constant for each fixed $s\in\mathbb{R}$ with
$\omega(s)\in\mathbb{R}$. In fact, $\omega(\tau)=\omega(s)$ for every $\tau\in
o(s)$.

######  Definition 2.3.

A _non-degenerate semiflow_ is a real semiflow without trivial orbits and, if
contrary, we call it _degenerate_.

For instance, the right translation semiflow, defined in (7), is a non-
degenerate semiflow.

The next result gives some simple equivalent conditions for a real semiflow to
be non-degenerate.

######  Lemma 2.4.

Let $\varphi:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$ be a real semiflow.
The following statements are equivalent:

  1. (i)

$\varphi$ is non-degenerate;

  2. (ii)

$\lim\limits_{t\rightarrow\infty}\varphi_{t}(s)=-\infty,$ for every
$s\in\mathbb{R}$;

  3. (iii)

$\varphi_{t}\left(s\right)<s$, for all $t>0$ and $s\in\mathbb{R}$;

  4. (iv)

For each $s\in\mathbb{R}$ there exists $t_{s}>0$ such that
$\varphi_{t_{s}}(s)\neq s$.

###### Proof.

(i) $\Rightarrow$ (ii) If there exists $s_{0}\in\mathbb{R}$ such that
$\lim\limits_{t\rightarrow\infty}\varphi_{t}(s_{0})$ is finite, which is
equivalent to $\omega(s_{0})\in\mathbb{R},$ then (11) shows that the orbit
$o\left(\omega\left(s_{0}\right)\right)$ is trivial and hence $\varphi$ is
degenerate, which contradicts (i).

(ii) $\Rightarrow$ (iii) Assume that there exists $t_{0}>0$ and
$s_{0}\in\mathbb{R}$ such that $\varphi_{t_{0}}\left(s_{0}\right)=s_{0}.$
Then,

$\varphi_{nt_{0}}\left(s_{0}\right)=s_{0},\text{ for all }n\in\mathbb{N},$

and letting $n\to\infty$ we get a contradiction. Implications (iii)
$\Rightarrow$ (iv) and (iv) $\Rightarrow$ (i) are trivial. ∎

Using the equivalence (i) $\Leftrightarrow$ (iii) in the previous lemma and
the same type of arguments as in Remark 2.2, one can easily prove that _if
$\varphi:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$ is a non-degenerate
semiflow, then the function $t\mapsto\varphi_{t}\left(s\right)$ is strictly
decreasing for each $s\in\mathbb{R}$ and the function
$s\mapsto\varphi_{t}\left(s\right)$ is strictly increasing for each $t\geq
0$_.

The next result is of a significant importance. It states that any non-
degenerate semiflow is generated by a continuous, strictly increasing real
function.

######  Theorem 2.5.

A mapping $\varphi:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$ is a non-
degenerate semiflow if and only if there exists a continuous, strictly
increasing function $\mu:\mathbb{R}\to\mathbb{R}$ with
$\lim\limits_{s\rightarrow-\infty}\mu(s)=-\infty$ and

$\varphi_{t}(s)=\mu^{-1}(\mu(s)-t),\text{ for all }t\geq 0\text{ and
}s\in\mathbb{R}.$ (13)

###### Proof.

Suppose that $\varphi$ is a non-degenerate semiflow. Let $s,\tau\in\mathbb{R}$
with $s>\tau$. According to Lemma 2.4 we have

$-\infty=\lim\limits_{t\rightarrow\infty}\varphi_{t}\left(s\right)<\tau<s=\varphi_{0}(s),$

and thus there exists a unique number $t(s,\tau)>0$ such that
$\varphi_{t(s,\tau)}(s)=\tau$. On the other hand, since $\varphi_{0}(s)=s$, we
can consider $t(s,\tau)=0$ whenever $s=\tau$. Hence, $t(s,\tau)\geq 0$ is the
unique number which satisfies the identity

$\varphi_{t(s,\tau)}(s)=\tau,\text{ for }s\geq\tau.$ (14)

Let us first prove that

$t(s,\tau)+t(\tau,\eta)=t(s,\eta),\text{ for }s\geq\tau\geq\eta.$

Indeed, for $s\geq\tau\geq\eta$ we have

$\displaystyle\varphi_{t(s,\tau)+t(\tau,\eta)}(s)=\varphi_{t(\tau,\eta)}\left(\varphi_{t(s,\tau)}(s)\right)=\varphi_{t(\tau,\eta)}(\tau)=\eta=\varphi_{t(s,\eta)}(s).$

Define a function $\mu:\mathbb{R}\to\mathbb{R}$, by

$\mu(s)=\begin{cases}\phantom{-}t(s,0),&s\geq 0,\\\ -t(0,s),&s<0.\end{cases}$

One can easily check that

$t(s,\tau)=\mu(s)-\mu(\tau),\text{ for }s\geq\tau.$ (15)

Since $t(s,\tau)>0$ whenever $s>\tau$, the above identity yields that $\mu$ is
strictly increasing. We now prove the continuity of $\mu$. Let
$s\in\mathbb{R}$ and choose a strictly increasing sequence
$(s_{n})\subset\mathbb{R}$ with $s_{n}<s$ and $s_{n}\to s$. Set
$t_{n}=\mu(s_{n})$ and $t=\mu(s)$. Since $\mu$ is a strictly increasing
function, we get that $(t_{n})$ is strictly increasing and thus there exists
$t^{\prime}=\lim\limits_{n\to\infty}t_{n}\leq t.$ If $s>0$, then $s_{n}>0$ for
sufficiently large $n\in\mathbb{N}$, hence $t_{n}=t(s_{n},0)$ and, by (14), we
have

$\varphi_{t_{n}}(s_{n})=\varphi_{t(s_{n},0)}(s_{n})=0.$

On the other hand, $\varphi_{t_{n}}(s_{n})\to\varphi_{t^{\prime}}(s)$ and thus
$\varphi_{t^{\prime}}(s)=0$. Since $t=t(s,0)$ is the unique number satisfying
$\varphi_{t}(s)=0$, then $t^{\prime}=t$. If $s\leq 0$, then $s_{n}<0$ for
every $n\in\mathbb{N}$, consequently $t_{n}=-t(0,s_{n})$ for $n\in\mathbb{N}$.
Then, we have

$\varphi_{-t_{n}}(0)=\varphi_{t(0,s_{n})}(0)=s_{n}.$

This yields $\varphi_{-t^{\prime}}(0)=s$. We also have
$\varphi_{-t}(0)=\varphi_{t(0,s)}(0)=s,$ which implies that $t=t^{\prime}$.
All above considerations shows that $\mu$ is left continuous at $s$.
Similarly, one can prove that $\mu$ is right continuous, therefore it is a
continuous function.

Observe now that (14) and (15) imply

$t(s,\tau)=\mu(s)-\mu(\varphi_{t(s,\tau)}(s)),\text{ for }s\geq\tau.$

This yields

$t=\mu(s)-\mu(\varphi_{t}(s)),\text{ for all }t\geq 0\text{ and
}s\in\mathbb{R}.$ (16)

Since $\mu$ is strictly increasing, there exists
$l=\lim\limits_{s\to-\infty}\mu(s)\in\mathbb{R}\cup\\{-\infty\\}.$ If we
assume that $l$ is finite, then letting $t\to\infty$ in the identity

$\mu(s)=t+\mu(\varphi_{t}(s))$

and using Lemma 2.4 and the continuity of $\mu$, we get that
$\mu(s)=\infty+l$, which is false, thus
$\lim\limits_{s\rightarrow-\infty}\mu(s)=-\infty$. If we put

$\ell=\lim\limits_{s\rightarrow\infty}\mu(s)\in\mathbb{R}\cup\\{+\infty\\},$

then the function $\mu:\mathbb{R}\to(-\infty,\ell)$ is invertible, with
continuous inverse, and by (16) we get (13).

Conversely, one can easily check that the identity (13) defines a non-
degenerate semiflow. ∎

In the following, $\mathcal{A}$ denotes the set of all continuous, strictly
increasing functions $\mu:\mathbb{R}\to(-\infty,\ell)$ with

$\lim\limits_{s\rightarrow-\infty}\mu(s)=-\infty\text{ and
}\lim\limits_{s\rightarrow+\infty}\mu(s)=\ell\in\mathbb{R}\cup\\{+\infty\\}.$

Theorem 2.5 shows that $\varphi:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$
is a non-degenerate semiflow if and only if there exits $\mu\in\mathcal{A}$
such that (13) holds. Furthermore, the following alternative holds true:
_either all the maps $s\mapsto\varphi_{t}(s)$, $t>0$, are bounded, if $\mu$ is
bounded by above, i.e., $\ell\in\mathbb{R}$, or they are all unbounded, if
$\ell=+\infty$._

######  Example 2.6.

Setting $\mu(s)=-e^{-s}$, for $s\in\mathbb{R}$, we obtain the first situation
above, with $\ell=0$. On the other hand, if
$\mu(s)=\mathrm{sign}(s)\ln(1+|s|)$ or $\mu(s)=s^{2n+1}$, for some fixed
$n\in\mathbb{N}$, we get $\ell=+\infty$. In particular, for $n=0$, that is
$\mu(s)=s$, we obtain the right translation semiflow.

##  3\. Evolution semigroups

In the following, $X=(X,\|\cdot\|)$ is a Banach space and $\mathcal{B}(X)$
denotes the Banach algebra of all bounded linear operators on $X$.
Furthermore, $C(\mathbb{R},X)$ is the space of all continuous $X$-valued
functions $u:\mathbb{R}\to X$ and by $C_{0}(\mathbb{R},X)$ we mean the Banach
space of all functions in $C(\mathbb{R},X)$ vanishing at $\pm\infty$, endowed
with the sup-norm

$\|u\|_{\infty}=\sup\limits_{s\in\mathbb{R}}\|u(s)\|,$

that is,

$C_{0}(\mathbb{R},X)=\left\\{u\in
C(\mathbb{R},X):\,\lim\limits_{|s|\to\infty}u(s)=0\right\\}.$

It is well-known that $C_{c}(\mathbb{R},X)=\left\\{u\in
C(\mathbb{R},X):\,supp(u)\text{ is compact}\right\\}$ is dense in
$C_{0}(\mathbb{R},X)$.

###  3.1. Generalized evolution semigroups

The main imperfection of the theory of evolution semigroups defined by (3) is
that it does not address to evolution families which are not exponentially
bounded, e.g.,

$U(t,s)(x_{1},x_{2})=(e^{s^{3}-t^{3}}x_{1},\,e^{t^{3}-s^{3}}x_{2}),\text{ for
}(x_{1},x_{2})\in\mathbb{R}^{2}.$

In fact, in this case we get

$T_{t}u(s)=(e^{-3s^{2}t+3st^{2}-t^{3}}u_{1}(s-t),\,e^{3s^{2}t-3st^{2}+t^{3}}u_{2}(s-t)),$

for $t\geq 0,\,u=(u_{1},u_{2})\in
C_{0}(\mathbb{R},\mathbb{R}^{2}),\,s\in\mathbb{R}$. One can easily observe
that $\left\\{T_{t}\right\\}_{t\geq 0}$ does not satisfy the semigroup
inequality

$\|T_{t}\|\leq Ke^{\alpha t},\,t\geq 0,$

for some constants $\alpha>0$ and $K\geq 1$. Therefore, it is not a
$C_{0}$-semigroup on $C_{0}(\mathbb{R},\mathbb{R}^{2})$.

The aim of this section is to generalize the concept of the evolution
semigroup associated to a not necessarily exponentially bounded evolution
family, replacing the right translation semiflow in formula (3) with a certain
real semiflow.

Let $\varphi:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$ be a real semiflow
and $\mathcal{U}=\left\\{U(t,s)\right\\}_{t\geq s}$ be an _evolution family_
on $X$. By a (strongly continuous) _evolution family_ on $X$ we mean a
collection of bounded linear operators $U(t,s)$, for $t\geq s$, acting on $X$,
such that

  * •

$U(t,t)=\mathrm{Id}$, $t\in\mathbb{R}$;

  * •

$U(t,\tau)U(\tau,t_{0})=U(t,t_{0}),$ $t\geq\tau\geq t_{0}$ in $\mathbb{R}$;

  * •

for each $x\in X$, the mapping $(t,s)\mapsto U(t,s)x$ is continuous on

$\Delta=\left\\{(t,s)\in\mathbb{R}^{2}:\,t\geq s\right\\}.$

For any $t\geq 0$ and $u\in C(\mathbb{R},X)$, we define

$T_{t}u(s):=U(s,\varphi_{t}(s))u(\varphi_{t}(s)),\;s\in\mathbb{R}.$ (17)

One can easily observe that $T_{t}u\in C(\mathbb{R},X)$, which yields that the
mapping $T_{t}:C(\mathbb{R},X)\to C(\mathbb{R},X)$ is well-defined for each
$t\geq 0$. Furthermore, we have

$T_{0}=\mathrm{Id}\text{ and }T_{t}T_{\tau}=T_{t+\tau},\text{ for }t,\tau\geq
0.$

######  Remark 3.1.

If $\varphi$ is a degenerate semiflow, then evidently there exists
$s_{0}\in\mathbb{R}$ such that

$T_{t}u\left(s_{0}\right)=u\left(s_{0}\right),\text{ for all $u\in
C(\mathbb{R},X)$ and $t\geq 0$.}$

Conversely, if the above relation holds for some $s_{0}\in\mathbb{R}$, then
$\varphi$ is degenerate.

Indeed, if we assume that the orbit $o(s_{0})$ is non-trivial, then there
exists $t_{0}>0$ with

$s_{1}=\varphi_{t_{0}}\left(s_{0}\right)<s_{0}.$

Thus,

$U\left(s_{0},s_{1}\right)u(s_{1})=T_{t_{0}}u(s_{0})=u(s_{0}),$

for all mappings $u\in C(\mathbb{R},X)$. For arbitrary $x,x_{1},x_{2}\in X$
with $x_{1}\neq x_{2}$, choose $u_{1},u_{2}\in C(\mathbb{R},X)$ such that

$u_{1}(s_{0})=x_{1},\,u_{2}(s_{0})=x_{2},\text{ and
}u_{1}(s_{1})=u_{2}(s_{1})=x.$

This yields

$U\left(s_{0},s_{1}\right)x=x_{1}\text{ and
}U\left(s_{0},s_{1}\right)x=x_{2},$

which contradicts $x_{1}\neq x_{2}$. Therefore, $o(s_{0})$ is trivial and,
consequently, $\varphi$ is degenerate.

######  Remark 3.2.

If $\varphi:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$ is a degenerate
semiflow satisfying any of the following three equivalent conditions:

  1. (i)

$\omega$ is bounded by above,

  2. (ii)

$\omega$ is constant in a neighborhood of $+\infty$,

  3. (iii)

there exist sequences $t_{n}>0$ and $s_{n}\to\infty$ for which
$(\varphi_{t_{n}}(s_{n}))_{n\in\mathbb{N}}$ is bounded by above,

then $\left\\{T_{t}\right\\}_{t\geq 0}$, given by (17), might not be a
$C_{0}$-semigroup on $C_{0}(\mathbb{R},X)$.

Let us first prove that the above conditions are equivalent if $\varphi$ is a
degenerate semiflow, which means that there exists $s_{0}\in\mathbb{R}$ with
$\omega(s_{0})\in\mathbb{R}$.

(i) $\Rightarrow$ (ii) Suppose that there exists $\alpha\in\mathbb{R}$ such
that $\omega(s)\leq\alpha$, for every $s\geq s_{0}$. Then, for $s\geq s_{0}$
we get that $\omega(s)\in\mathbb{R}$ and
$\omega(\omega(s))\leq\omega(\alpha)$. By (12), this yields that
$\omega(s)\leq\omega(\alpha)$, for every $s\geq s_{0}$. On the other hand, for
$s\geq\alpha$ we have $\omega(s)\geq\omega(\alpha)$. Hence,
$\omega(s)=\omega(\alpha),\text{ for every }s\geq\max\\{s_{0},\alpha\\},$
which proves (ii).

(ii) $\Rightarrow$ (iii) Assume that there exist $\alpha\in\mathbb{R}$ and
$\tau\geq s_{0}$ with $\omega(s)=\alpha$ for every $s\geq\tau$. Thus, there
exists $\delta>0$ such that

$\varphi_{t}(s)-\alpha<1,\text{ for all }t>\delta\text{ and }s\geq\tau.$

For each $n\in\mathbb{N}$ choose $t_{n}>\delta$ and $s_{n}\geq\tau$ with
$s_{n}\to\infty$. Then, $\varphi_{t_{n}}(s_{n})<\alpha+1$ for every
$n\in\mathbb{N}$, that is $(\varphi_{t_{n}}(s_{n}))_{n\in\mathbb{N}}$ is
bounded by above.

(iii) $\Rightarrow$ (i) Let $t_{n}>0$ and $s_{n}\in\mathbb{R}$ with
$s_{n}\to\infty$ such that $\varphi_{t_{n}}(s_{n})\leq\alpha$ for every
$n\in\mathbb{N}$ and for some $\alpha\in\mathbb{R}$. Fix $s\geq s_{0}$. Since
$s_{n}\to\infty$, there exists $n_{0}\in\mathbb{N}$ such that $s_{n}>s$ for
every $n\geq n_{0}.$ Then, $\omega(s_{n})\geq\omega(s)$ for $n\geq n_{0}.$ On
the other hand, from (8) we get $\omega(s_{n})\leq\alpha$ and hence
$\omega(s)\leq\alpha$ for every $s\geq s_{0}$.

Now, set $U(t,s)=\mathrm{Id}$, for $t\geq s$, and let $t_{n}\searrow 0$ and
$s_{n}\to+\infty$ such that

$\alpha-1<\varphi_{t_{n}}(s_{n})<\alpha,\text{ for all }n\in\mathbb{N}\text{
and for some }\alpha\in\mathbb{R}.$

Pick $u\in C_{c}(\mathbb{R},X)$ such that $u(t)=\xi\neq 0$ for
$t\in(\alpha-1,\alpha).$ Since $s_{n}\to+\infty$ and $u$ has compact support,
we get that $u(s_{n})=0$ for large $n\in\mathbb{N}$. Then,

$\displaystyle T_{t_{n}}u(s_{n})-u(s_{n})$
$\displaystyle=U(s_{n},\varphi_{t_{n}}(s_{n}))u(\varphi_{t_{n}}(s_{n}))-u(s_{n})=u(\varphi_{t_{n}}(s_{n}))=\xi\nrightarrow
0,$

which proves that $\left\\{T_{t}\right\\}_{t\geq 0}$ is not a
$C_{0}$-semigroup on $C_{0}(\mathbb{R},X)$.

In the following result, we give conditions such that the family
$\left\\{T_{t}\right\\}_{t\geq 0}$ considered in (17) is a $C_{0}$-semigroup
on $C_{0}(\mathbb{R},X)$.

######  Theorem 3.3.

Assume that $\varphi:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$ is a non-
degenerate semiflow with

$\lim\limits_{s\to\infty}\varphi_{t}(s)=+\infty,\text{ for }t\geq 0,$ (18)

or a (degenerate) real semiflow with
$\lim\limits_{s\to\infty}\omega(s)=+\infty$. If the evolution family
$\mathcal{U}=\left\\{U(t,s)\right\\}_{t\geq s}$ satisfies inequality

$\|U(s,\varphi_{t}(s))\|\leq Ke^{\alpha t},\;t\geq 0,\,s\in\mathbb{R},$ (19)

for some reals $\alpha>0$ and $K\geq 1$, then $\left\\{T_{t}\right\\}_{t\geq
0}$ defined in (17) is a $C_{0}$-semigroup on $C_{0}(\mathbb{R},X)$.

###### Proof.

For $u\in C_{0}(\mathbb{R},X)$, by (19), we have

$\|T_{t}u(s)\|=\|U(s,\varphi_{t}(s))u(\varphi_{t}(s))\|\leq Ke^{\alpha
t}\|u(\varphi_{t}(s))\|.$

Let us first remark that if $\lim\limits_{s\to\infty}\omega(s)=+\infty$, then,
by (8), we deduce that (18) holds. Further, (6) and (18) imply
$u(\varphi_{t}(s))\to 0$ as $s\to\pm\infty$, thus $T_{t}u\in
C_{0}(\mathbb{R},X)$ and, consequently, $T_{t}:C_{0}(\mathbb{R},X)\to
C_{0}(\mathbb{R},X)$ is well-defined for each $t\geq 0$.

Since $C_{c}(\mathbb{R},X)$ is dense in $C_{0}(\mathbb{R},X)$, it suffices to
prove that

$\lim_{n\rightarrow\infty}\left(T_{t_{n}}u(s_{n})-u(s_{n})\right)=0,$

for any $t_{n}\to 0^{+}$, $s_{n}\in\mathbb{R}$ and fixed $u\in
C_{c}(\mathbb{R},X)$.

If $s_{n}\rightarrow-\infty$, then $\varphi_{t_{n}}(s_{n})\rightarrow-\infty$,
and if $s_{n}\rightarrow+\infty$, then
$\varphi_{t_{n}}(s_{n})\rightarrow+\infty$. Since $u$ has compact support, in
both situations we get $T_{t_{n}}u(s_{n})-u(s_{n})=0$ for sufficiently large
$n\in\mathbb{N}$.

It remains to assume that the sequence $(s_{n})$ is bounded and pick
$\varepsilon>0$. Then there exists a finite set
$A_{\varepsilon}=\left\\{\eta_{1},\eta_{2},\ldots,\eta_{k}\right\\}\subset\mathbb{R}$
such that for any $n\in\mathbb{N}$ there exists $\xi_{n}\in A_{\varepsilon}$
with

$\|u(s_{n})-\xi_{n}\|<\eta,$ (20)

where $\eta=\frac{\varepsilon}{2(Ke^{\alpha}+1)}.$

Having compact support, $u$ is uniformly continuous, thus there exists
$\delta_{1}(\varepsilon)>0$ such that

$\|u(s^{\prime})-u(s^{\prime\prime})\|<\eta,\text{ if
$|s^{\prime}-s^{\prime\prime}|<\delta_{1}(\varepsilon)$. }$ (21)

On the other hand, as $\Delta\ni(s,\tau)\mapsto U(s,\tau)x\in X$ is continuous
for each $x\in X$, then, setting $x=\eta_{i}\in A_{\varepsilon}$, there exists
$\delta(\varepsilon,\eta_{i})>0$ such that
$|s-\tau|<\delta(\varepsilon,\eta_{i})$ implies

$\|U(s,\tau)\eta_{i}-\eta_{i}\|<\eta,\,i=\overline{1,k}.$ (22)

Put

$\delta(\varepsilon)=\min\left\\{\delta_{1}(\varepsilon),\delta(\varepsilon,\eta_{1}),\ldots,\delta(\varepsilon,\eta_{k})\right\\}.$

Since $(s_{n})$ is bounded, then $\varphi_{t_{n}}(s_{n})-s_{n}\rightarrow 0$,
thus there exists $N(\varepsilon)\in\mathbb{N}$ such that

$|\varphi_{t_{n}}(s_{n})-s_{n}|<\delta(\varepsilon),\text{ for }n\geq
N(\varepsilon),$ (23)

and, by (21), we get

$\|u(\varphi_{t_{n}}(s_{n}))-u(s_{n})\|<\eta,\text{ for }n\geq
N(\varepsilon).$ (24)

On the other hand, (22) and (23) imply

$\|U(s_{n},\varphi_{t_{n}}(s_{n}))\xi_{n}-\xi_{n}\|<\eta,\text{ for }n\geq
N(\varepsilon).$ (25)

Finally, using inequalities (19), (20), (24) and (25), for $n\geq
N(\varepsilon)$ we have

$\displaystyle\|T_{t_{n}}u(s_{n})-u(s_{n})\|$
$\displaystyle\leq\|U(s_{n},\varphi_{t_{n}}(s_{n}))(u(\varphi_{t_{n}}(s_{n}))-u(s_{n}))\|$
$\displaystyle\quad+\|U(s_{n},\varphi_{t_{n}}(s_{n}))(u(s_{n})-\xi_{n})\|$
$\displaystyle\quad+\|U(s_{n},\varphi_{t_{n}}(s_{n}))\xi_{n}-\xi_{n}\|$
$\displaystyle\quad+\|\xi_{n}-u(s_{n})\|$ $\displaystyle\leq 2Ke^{\alpha
t_{n}}\eta+2\eta.$

Therefore, $\|T_{t_{n}}u(s_{n})-u(s_{n})\|<\varepsilon$ for sufficiently large
$n$, which completes the proof. ∎

The $C_{0}$-semigroup $\left\\{T_{t}\right\\}_{t\geq 0}$ given by Theorem 3.3
will be called _the generalized evolution semigroup_ on $C_{0}(\mathbb{R},X)$
associated to the real semiflow $\varphi$ and the evolution family
$\mathcal{U}$.

Notice that if we put $\varphi_{t}(s)=s-t$ (in fact, the right translation
semiflow), we step over the well-known concept of the evolution semigroup on
$C_{0}(\mathbb{R},X)$ associated to an exponentially bounded evolution family
(see (2) and (3)).

######  Example 3.4.

Setting

$\varphi_{t}\left(s\right)=\begin{cases}se^{t},&\text{ if }s<0,\\\ s,&\text{
if }s\geq 0,\end{cases}$

one can easily check that $\varphi$ is a real semiflow with trivial orbits (in
fact, $o(s)=\left\\{s\right\\}$ for all $s\geq 0$) and

$\omega(s)=\begin{cases}-\infty,&\text{ if }s<0,\\\ s,&\text{ if }s\geq
0.\end{cases}$

Furthermore, $\lim\limits_{s\to\infty}\omega(s)=+\infty$. Then, according to
Theorem 3.3, the corresponding family $\left\\{T_{t}\right\\}_{t\geq 0}$
defined by (17) is a $C_{0}$-semigroup on $C_{0}(\mathbb{R},X)$ provided that
the evolution family $\mathcal{U}$ satisfies condition (19). This reduces in
fact to

$\|U(s,\tau)\|\leq
Ke^{\alpha(\ln(-\tau)-\ln(-s))}\;\Leftrightarrow\;\|U(s,\tau)\|\leq
K({\tau}/{s})^{\alpha},$

for some constants $\alpha>0$ and $K\geq 1$, and all negative $s>\tau$. For
instance, the evolution family

$U(s,\tau)=\frac{1+|\tau|}{1+|s|}\,\mathrm{Id},\,s\geq\tau,$

satisfies the above inequality for $\alpha=K=1$.

######  Remark 3.5.

Let $\varphi:\mathbb{R}_{+}\times\mathbb{R}\to\mathbb{R}$ be a continuous
function such that

$\varphi(t,s)\leq s,\text{ for all }t\geq 0\text{ and }s\in\mathbb{R},$ (26)

and set

$\varphi_{t}(s)=\varphi(t,s),\;s\in\mathbb{R},$

the transition map defined by $t\geq 0$. If the family
$\left\\{T_{t}\right\\}_{t\geq 0}$ given by formula (17) is a
$C_{0}$-semigroup on $C_{0}(\mathbb{R},X)$, then $\varphi$ is a real semiflow
provided that $U(t,s)\neq 0$ for all $t>s$.

###### Proof.

Indeed, it suffices to prove the relations (i) and (ii) in Definition 2.1. By
(26) we have that $\varphi_{0}(s)\leq s$ for every $s\in\mathbb{R}$. Assume
that there exists $s_{0}\in\mathbb{R}$ such that $\varphi_{0}(s_{0})<s_{0}$.
For arbitrary $x\in X$, $x\neq 0$, we choose $u\in C_{c}(\mathbb{R},X)$ with

$u(\varphi_{0}(s_{0}))=x\text{ and }u(s_{0})=0.$

Since $T_{0}u(s_{0})=u(s_{0})=0$, we get

$0=U(s_{0},\varphi_{0}(s_{0}))u(\varphi_{0}(s_{0}))=U(s_{0},\varphi_{0}(s_{0}))x,$

and thus $U(s_{0},\varphi_{0}(s_{0}))=0$, which is false. Then,
$\varphi_{0}(s)=s$ for every $s\in\mathbb{R}$. To prove the second relation in
Definition 2.1, one can easily check that

$T_{t}T_{\tau}=T_{t+\tau},\text{ for }t,\tau\geq 0,$

implies

$U(s,\varphi_{\tau}(\varphi_{t}(s)))u(\varphi_{\tau}(\varphi_{t}(s)))=U(s,\varphi_{t+\tau}(s))u(\varphi_{t+\tau}(s)),$

for all $t,\tau\geq 0,\,s\in\mathbb{R},\,u\in C_{0}(\mathbb{R},X),$ and,
proceeding as above, one may prove that
$\varphi_{t}\circ\varphi_{\tau}=\varphi_{t+\tau}$, for all $t,\tau\geq 0$. ∎

Theorem 2.5 shows that if $\varphi$ is a non-degenerate semiflow, then there
exists $\mu\in\mathcal{A}$ such that

$T_{t}u(s)=U(s,\mu^{-1}(\mu(s)-t))u(\mu^{-1}(\mu(s)-t)),$ (27)

for all $t\geq 0$, $u\in C(\mathbb{R},X)$, and $s\in\mathbb{R}$. On the other
hand, (18) is equivalent to

$\ell=\lim\limits_{s\rightarrow+\infty}\mu(s)=+\infty,$ (28)

and we will denote $\mathcal{A}_{\,\infty}$ the set of all $\mu\in\mathcal{A}$
satisfying (28). Furthermore, the inequality (19) becomes

$\|U(s,\tau)\|\leq Ke^{\alpha(\mu(s)-\mu(\tau))},\text{ for }s\geq\tau.$ (29)

Therefore, we deduce a particular class of generalized evolution semigroups
which is emphasized in the next result.

######  Corollary 3.6.

Assume that $\mu\in\mathcal{A}_{\,\infty}$ and the evolution family
$\mathcal{U}$ satisfies (29) for some constants $\alpha>0$ and $K\geq 1$. Then
the family $\left\\{T_{t}\right\\}_{t\geq 0}$ given by (27) is a
$C_{0}$-semigroup on $C_{0}(\mathbb{R},X)$.

If $\rho:\mathbb{R}\to[0,\infty)$ is a continuous function such that the set
$\\{t\in\mathbb{R}:\rho(t)>0\\}$ is dense in $\mathbb{R}$,

$\lim\limits_{t\to\infty}\int_{0}^{t}\rho(\xi)\,d\xi=+\infty\text{ and
}\lim\limits_{t\to-\infty}\int_{t}^{0}\rho(\xi)\,d\xi=+\infty,$ (30)

(see [21, Definition 4 (iii)] or [13, Definition 2.3]) and there exist
constants $\alpha>0$ and $K\geq 1$ with

$\|U(s,\tau)\|\leq Ke^{\alpha\int_{\tau}^{s}\rho(\xi)\,d\xi},\;s\geq\tau,$
(31)

then the function defined by

$\mu(t)=\int_{0}^{t}\rho(\xi)\,d\xi,\,t\in\mathbb{R},$

belongs to $\mathcal{A}_{\,\infty}$, it is continuously differentiable and
(29) holds. Inequality (31) is in fact the hypothesis of $\rho$-bounded growth
assumed in [25]. Notice that any evolution family generated by a differential
equation $\dot{x}=A(t)x$, where $\mathbb{R}\ni t\mapsto A(t)\in\mathcal{B}(X)$
is continuous in uniform operator topology, satisfies (31) for
$\rho(t)=\|A(t)\|$.

###  3.2. Similar semigroups

In the following, let the assumptions of Corollary 3.6 hold, that is
$\mu:\mathbb{R}\to\mathbb{R}$ is a continuous, strictly increasing function
such that

$\lim\limits_{s\rightarrow-\infty}\mu(s)=-\infty\text{ and
}\lim\limits_{s\rightarrow+\infty}\mu(s)=+\infty,$

and the evolution family $\mathcal{U}$ satisfies (29).

Set

$V(t,s)=U(\mu^{-1}(t),\mu^{-1}(s)),\text{ for }t\geq s.$

One can easily check that $\mathcal{V}=\left\\{V(t,s)\right\\}_{t\geq s}$ is
an evolution family on $X$ satisfying

$\|V(t,s)\|\leq Ke^{\alpha(t-s)},\text{ for }t\geq s,$

and thus $\mathcal{V}$ is exponentially bounded.

From the classical theory of evolution semigroups, we are able now to define
the evolution semigroup on $C_{0}(\mathbb{R},X)$ associated to the evolution
family $\mathcal{V}$, that is a $C_{0}$-semigroup on $C_{0}(\mathbb{R},X)$,
given by

$S_{t}v(s)=V(s,s-t)v(s-t),\,t\geq 0,v\in C_{0}(\mathbb{R},X),s\in\mathbb{R},$

and denote $(A,D(A))$ its generator.

We have

$S_{t}v(s)=U(\mu^{-1}(s),\mu^{-1}(s-t))v(s-t)$

and so

$S_{t}v(\mu(s))=U(s,\mu^{-1}(\mu(s)-t))v(\mu(s)-t).$

Letting $v=u\circ\mu^{-1}\in C_{0}(\mathbb{R},X)$ for $u\in
C_{0}(\mathbb{R},X)$, we get

$(S_{t}(u\circ\mu^{-1}))(\mu(s))=U(s,\mu^{-1}(\mu(s)-t))u(\mu^{-1}(\mu(s)-t))=T_{t}u(s).$

Hence,

$(S_{t}(u\circ\mu^{-1}))\circ\mu=T_{t}u,\text{ for every }u\in
C_{0}(\mathbb{R},X).$ (32)

Define now the operator

$\mathcal{F}:C_{0}(\mathbb{R},X)\to
C_{0}(\mathbb{R},X),\;\mathcal{F}u=u\circ\mu^{-1}.$

Obviously, $\mathcal{F}$ is an invertible, bounded linear operator on
$C_{0}(\mathbb{R},X)$, with its inverse

$\mathcal{F}^{-1}v=v\circ\mu.$

From (32) we have

$T_{t}u=(S_{t}(\mathcal{F}u))\circ\mu=\mathcal{F}^{-1}(S_{t}(\mathcal{F}u)),\text{
for every }u\in C_{0}(\mathbb{R},X),$

and thus

$T_{t}=\mathcal{F}^{-1}S_{t}\,\mathcal{F}.$

The above relation shows that _$\left\\{S_{t}\right\\}_{t\geq 0}$ and
$\left\\{T_{t}\right\\}_{t\geq 0}$ are similar semigroups on
$C_{0}(\mathbb{R},X)$_ (see [10, p. 59]). Hence, the generator $G$ of the
generalized evolution semigroup $\left\\{T_{t}\right\\}_{t\geq 0}$ is given by

$Gu=(\mathcal{F}^{-1}A\,\mathcal{F})u=A\,(u\circ\mu^{-1})\circ\mu,$ (33)

with domain

$D(G)=\left\\{u\in C_{0}(\mathbb{R},X):\,u\circ\mu^{-1}\in D(A)\right\\}.$
(34)

Furthermore, $\sigma(G)=\sigma(A)$.

This result is of a significant importance because one can now simply deduce
the spectral mapping theorem for the generalized evolution semigroup, given by
Corollary 3.6, from the classical theory of evolution semigroups.

######  Corollary 3.7.

Under the assumptions of Corollary 3.6, let $G$ be the generator of the
generalized evolution semigroup $\left\\{T_{t}\right\\}_{t\geq 0}$. The
spectrum $\sigma(T_{t})$ is rotationally invariant for $t>0$ and the spectrum
$\sigma(G)$ is invariant under translations along the imaginary axis.
Furthermore, the generalized evolution semigroup satisfies the spectral
mapping theorem

$\sigma\left(T_{t}\right)\setminus\\{0\\}=e^{t\sigma(G)},\text{ for }t\geq 0,$

and, consequently, the following statements are equivalent:

  1. (1)

$\sigma(T_{t})\cap\mathbb{T}=\emptyset$, for some $t>0$, and thus for all
$t>0$;

  2. (2)

$0\in\varrho(G)$, that is $G:D(G)\subseteq C_{0}(\mathbb{R},X)\to
C_{0}(\mathbb{R},X)$ is invertible, where $\varrho(G)$ is the resolvent of
$G$.

In the following result we connect the generator of the generalized evolution
semigroup to the solvability of an integral equation (see [23, Lemma 1] or
[24, Lemma 1.1]).

######  Proposition 3.8.

Let $u,f\in C_{0}(\mathbb{R},X)$. Then $u\in D(G)$ and $Gu=-f$ if and only if

$u(t)=U(t,s)u(s)+\int_{s}^{t}\mu^{\prime}(\xi)U(t,\xi)f(\xi)\,d\xi,\text{ for
}t\geq s.$

###### Proof.

Let $u\in D(G)$ and $f\in C_{0}(\mathbb{R},X)$ such that $Gu=-f$. By (33)–(34)
we get

$v=u\circ\mu^{-1}\in D(A)\text{ and }Av=-f\circ\mu^{-1}.$

Lemma 1 in [23] implies

$v(t)=V(t,s)v(s)+\int_{s}^{t}V(t,\tau)f(\mu^{-1}(\tau))d\tau,\;t\geq s,$

which is equivalent to

$u(\mu^{-1}(t))=U(\mu^{-1}(t),\mu^{-1}(s))u(\mu^{-1}(s))+\int_{s}^{t}U(\mu^{-1}(t),\mu^{-1}(\tau))f(\mu^{-1}(\tau))d\tau.$

Replacing $t$ and $s$ by $\mu(t)$ and $\mu(s)$, respectively, we get

$\displaystyle u(t)$
$\displaystyle=U(t,s)u(s)+\int_{\mu(s)}^{\mu(t)}U(t,\mu^{-1}(\tau))f(\mu^{-1}(\tau))d\tau$
$\displaystyle=U(t,s)u(s)+\int_{s}^{t}\mu^{\prime}(\xi)U(t,\xi)f(\xi)d\xi,$

which proves the desired formula. The converse can be proved by reversing all
the above arguments. ∎

##  4\. Applications of generalized evolution semigroups to general
dichotomies

Using the framework constructed in the previous sections, we are able now to
completely characterize a wide class of dichotomies of an evolution family via
its corresponding generalized evolution semigroup.

We recall that a $C_{0}$-semigroup $\left\\{T_{t}\right\\}_{t\geq 0}$ on a
Banach space $X$ is called _hyperbolic_ if

$\sigma(T_{t})\cap\mathbb{T}=\emptyset\text{ for some/all }t>0,$

where $\mathbb{T}=\\{\lambda\in\mathbb{C}:\,|\lambda|=1\\}$. It is well-known
that the $C_{0}$-semigroup $\left\\{T_{t}\right\\}_{t\geq 0}$ is hyperbolic if
and only if there exists a projection $\mathcal{P}$ on $X$ satisfying

$T_{t}\mathcal{P}=\mathcal{P}T_{t},\;t\geq 0,$

and the following conditions hold:

  * •

the map $T_{t}|_{\mathcal{Q}X}:\mathcal{Q}X\to\mathcal{Q}X$ is invertible for
each $t\geq 0$, where $\mathcal{Q}=\mathrm{Id}-\mathcal{P}$;

  * •

there exists $\nu>0$ and $N\geq 1$ such that

$\|T_{t}\mathcal{P}x\|\leq Ne^{-\nu t}\|x\|\text{ and
}\|\left(T_{t}^{\mathcal{Q}}\right)^{-1}\mathcal{Q}x\|\leq Ne^{-\nu t}\|x\|,$

for $t\geq 0$ and $x\in X$, where $\left(T_{t}^{\mathcal{Q}}\right)^{-1}$ is
the inverse of $T_{t}|_{\mathcal{Q}X}:\mathcal{Q}X\to\mathcal{Q}X$.

In fact, the structural projection $\mathcal{P}$ is the Riesz projection
corresponding to the operator $T_{t_{0}}$ and the spectral set
$\sigma(T_{t_{0}})\cap\mathbb{D}$,

$\mathcal{P}=\frac{1}{2\pi i}\int_{\mathbb{T}}(\lambda-
T_{t_{0}})^{-1}d\lambda,$

for some fixed $t_{0}>0$ (without loss of generality, one may consider
$t_{0}=1$).

In the following, assume that $\mu\in\mathcal{A}_{\,\infty}$ and
$\mathcal{U}=\left\\{U(t,s)\right\\}_{t\geq s}$ is an evolution family on $X$
satisfying (29). Let $\left\\{T_{t}\right\\}_{t\geq 0}$ be the corresponding
generalized evolution semigroup given by Corollary 3.6 and denote $(G,D(G))$
its generator.

######  Proposition 4.1.

If the generalized evolution semigroup $\left\\{T_{t}\right\\}_{t\geq 0}$ is
hyperbolic, having structural projection $\mathcal{P}$, then

$\mathcal{P}(\beta u)=\beta\mathcal{P}u,$

for all $\beta\in C_{b}(\mathbb{R})$ and $u\in C_{0}(\mathbb{R},X)$, where
$C_{b}(\mathbb{R})$ denotes the space of all bounded continuous real-valued
functions.

###### Proof.

We first remark that

$\mathrm{Range}(\mathcal{P})=\left\\{u\in
C_{0}(\mathbb{R},X):\;\lim\limits_{t\to\infty}T_{t}u=0\right\\}.$

Therefore, if $\beta\in C_{b}(\mathbb{R})$ and $u\in C_{0}(\mathbb{R},X)$,
then $\|T_{t}\mathcal{P}u\|_{\infty}\to 0$ as $t\to\infty$ and

$\left\|T_{t}(\beta\mathcal{P}u)(s)\right\|=|\beta(\mu^{-1}(\mu(s)-t))|\,\|T_{t}\mathcal{P}u(s)\|\leq\|\beta\|_{\infty}\,\|T_{t}\mathcal{P}u\|_{\infty}.$

This yields that $\beta\mathcal{P}u\in\mathrm{Range}(\mathcal{P})$ and
$\mathcal{P}(\beta\mathcal{P}u)=\beta\mathcal{P}u.$ On the other hand, we have

$\displaystyle\|\mathcal{P}(\beta\mathcal{Q}u)\|_{\infty}$
$\displaystyle=\left\|\mathcal{P}\left(\beta
T_{t}\left(T_{t}^{\mathcal{Q}}\right)^{-1}\mathcal{Q}u\right)\right\|_{\infty}$
$\displaystyle=\left\|\mathcal{P}\left(T_{t}\beta_{\mu}\left(T_{t}^{\mathcal{Q}}\right)^{-1}\mathcal{Q}u\right)\right\|_{\infty}$
$\displaystyle=\left\|T_{t}\mathcal{P}\left(\beta_{\mu}\left(T_{t}^{\mathcal{Q}}\right)^{-1}\mathcal{Q}u\right)\right\|_{\infty},$

for some function $\beta_{\mu}\in C_{b}(\mathbb{R})$, which depends on $\beta$
and $\mu$, with $\|\beta_{\mu}\|_{\infty}\leq\|\beta\|_{\infty}$. Let us
estimate

$\|\mathcal{P}(\beta\mathcal{Q}u)\|_{\infty}\leq Ne^{-\nu
t}\|\beta\|_{\infty}\,\|\left(T_{t}^{\mathcal{Q}}\right)^{-1}\mathcal{Q}u\|_{\infty}\leq
N^{2}e^{-2\nu t}\|\beta\|_{\infty}\,\|u\|_{\infty}\to 0,$

as $t\to\infty$. Thus, $\mathcal{P}(\beta\mathcal{Q}u)=0$ and, consequently,

$\mathcal{P}(\beta
u)=\mathcal{P}\left(\beta\mathcal{P}u\right)+\mathcal{P}\left(\beta\mathcal{Q}u\right)=\beta\mathcal{P}u.$

∎

An important consequence of the above result is that

$\mathcal{P}u(s)=P(s)u(s),\;u\in C_{0}(\mathbb{R},X),\,s\in\mathbb{R},$ (35)

for some bounded, strongly continuous, projection-valued function
$P:\mathbb{R}\to\mathcal{B}(X)$ (see [10, Proposition 9.13]).

The following definition generalizes the classical notion of (uniform)
exponential dichotomy of an evolution family.

######  Definition 4.2.

The evolution family $\mathcal{U}$ is said to admits a _$\mu$ -exponential
dichotomy_ if there exist projections $P(t):X\to X$, $t\in\mathbb{R}$,
compatible with $\mathcal{U}$ and there exist constants $\nu>0$, $N\geq 1$
such that

$\|U(t,s)P(s)\|\leq Ne^{-\nu(\mu(t)-\mu(s))}\text{ and }\|U_{Q}(s,t)Q(t)\|\leq
Ne^{-\nu(\mu(t)-\mu(s))},$

for all $t\geq s$.

Notice that $\|P(t)\|\leq N$ for every $t\in\mathbb{R}$. Furthermore, as in
[24, Lemma 4.2], one may prove that the mapping $t\mapsto P(t)$ is strongly
continuous and thus $P(\cdot)\in C_{b}(\mathbb{R},\mathcal{B}_{s}(X))$, the
space of all bounded and continuous functions from $\mathbb{R}$ with values in
$\mathcal{B}(X)$ endowed with the topology of strong convergence.

If $\mu$ is continuously differentiable with $\rho(s)=\mu^{\prime}(s)$, the
above inequalities are replaced by

$\|U(t,s)P(s)\|\leq Ne^{-\nu\int_{s}^{t}\rho(\xi)\,d\xi}\text{ and
}\|U_{Q}(s,t)Q(t)\|\leq Ne^{-\nu\int_{s}^{t}\rho(\xi)\,d\xi},$

for $t\geq s$, and thus we extend the concept of a generalized exponential
dichotomy (firstly introduced, to our knowledge, by Martin in [21, Definition
4]) to arbitrary evolution families (see also [13, 14, 25]). In particular, if
$\mu(s)=s$, we step over the usual exponential behavior. On the other hand,
letting

$\mu(s)=\mathrm{sign}(s)\ln(1+|s|),\,s\in\mathbb{R},$ (36)

we get the polynomial behavior on the real line.

The following question is natural and leads to the main goal of this section:
what is the connection between the existence of $\mu$-exponential dichotomy of
an evolution family and the hyperbolicity of the corresponding generalized
evolution semigroup?

Before giving an answer to this question, we consider a simple example.

######  Example 4.3.

Let $\mu$ be the polynomial growth rate defined by (36). Obviously, the
evolution family

$U(t,s)(x_{1},x_{2})=(e^{\mu(s)-\mu(t)}x_{1},\,e^{\mu(t)-\mu(s)}x_{2}),\text{
for }t\geq s\text{ and }(x_{1},x_{2})\in\mathbb{R}^{2},$

admits a $\mu$-exponential dichotomy. On the other hand, the generalized
evolution semigroup corresponding to $\mathcal{U}$, given by

$T_{t}u(s)=(e^{-t}u_{1}(\mu^{-1}(\mu(s)-t)),\,e^{t}u_{2}(\mu^{-1}(\mu(s)-t))),$

for $t\geq 0$, $u=(u_{1},u_{2})\in C_{0}(\mathbb{R},\mathbb{R}^{2})$,
$s\in\mathbb{R}$, is hyperbolic. Furthermore, one may show that the generator
$G$ of $\left\\{T_{t}\right\\}_{t\geq 0}$ is

$G(u_{1},u_{2})(s)=\left(-u_{1}(s)-\frac{1}{\mu^{\prime}(s)}u_{1}^{\prime}(s),u_{2}(s)-\frac{1}{\mu^{\prime}(s)}u_{2}^{\prime}(s)\right),\text{
$s\in\mathbb{R}$},$

with its domain $D(G)=\\{u\in C_{0}(\mathbb{R},\mathbb{R}^{2})\cap
C^{1}(\mathbb{R},\mathbb{R}^{2}):\,\lim\limits_{s\to\pm\infty}u^{\prime}(s)=0\\}$.

The following result characterizes the concept of $\mu$-exponential dichotomy
of an evolution family in terms of spectral properties of the generator of the
corresponding generalized evolution semigroup.

######  Theorem 4.4 (Dichotomy Theorem).

The following statements are equivalent:

  1. (a)

$\mathcal{U}$ admits a $\mu$-exponential dichotomy;

  2. (b)

$\left\\{T_{t}\right\\}_{t\geq 0}$ is hyperbolic;

  3. (c)

$\sigma(G)\cap i\mathbb{R}=\emptyset;$

  4. (d)

$G$ is invertible and, in particular, if $\mu$ is continuously differentiable,
then its inverse is given by the formula

$(G^{-1}f)(t)=-\int_{\mathbb{R}}\mu^{\prime}(\xi)\Gamma(t,\xi)f(\xi)\,d\xi,$
(37)

for all $f\in C_{0}(\mathbb{R},X)\text{ and }t\in\mathbb{R}$, where $\Gamma$
is the Green function associated to $\mathcal{U}$.

###### Proof.

We emphasize that $\mathcal{U}$ admits a $\mu$-exponential dichotomy with
respect to projections $P(t)$ if and only if the evolution family
$\mathcal{V}$ defined in Paragraph 3.2 admits a (uniform) exponential
dichotomy with respect to

$P_{\mu}(t)=P(\mu^{-1}(t)),\;t\in\mathbb{R}.$

Therefore, the equivalences (a) $\Leftrightarrow$ (b) $\Leftrightarrow$ (c)
hold from Paragraph 3.2 and Theorem 1.1. It remains to prove formula (37). For
this pick $f\in C_{0}(\mathbb{R},X)$ and set

$u(t)=-\int_{\mathbb{R}}\mu^{\prime}(\xi)\Gamma(t,\xi)f(\xi)\,d\xi,\;t\in\mathbb{R}.$

By Theorem 1.1 we get

$\displaystyle A^{-1}(f\circ\mu^{-1})(\mu(t))$
$\displaystyle=-\int_{-\infty}^{\mu(t)}V(\mu(t),\tau)P_{\mu}(\tau)f(\mu^{-1}(\tau))\,d\tau$
$\displaystyle\qquad+\int_{\mu(t)}^{+\infty}V_{Q_{\mu}}(\mu(t),\tau)Q_{\mu}(\tau)f(\mu^{-1}(\tau))\,d\tau$
$\displaystyle=-\int_{-\infty}^{\mu(t)}U(t,\mu^{-1}(\tau))P(\mu^{-1}(\tau))f(\mu^{-1}(\tau))\,d\tau$
$\displaystyle\qquad+\int_{\mu(t)}^{+\infty}U_{Q}(t,\mu^{-1}(\tau))Q(\mu^{-1}(\tau))f(\mu^{-1}(\tau))\,d\tau$
$\displaystyle=-\int_{-\infty}^{t}\mu^{\prime}(\xi)U(t,\xi)P(\xi)f(\xi)\,d\xi$
$\displaystyle\qquad+\int_{t}^{+\infty}\mu^{\prime}(\xi)U_{Q}(t,\xi)Q(\xi)f(\xi)\,d\xi$
$\displaystyle=u(t),\,t\in\mathbb{R}.$

This yields

$A^{-1}(f\circ\mu^{-1})\circ\mu=u.$

Thus, $u\circ\mu^{-1}\in D(A)$ and, equivalently, $u\in D(G)$. On the other
hand, by (33) we have $G^{-1}f=A^{-1}(f\circ\mu^{-1})\circ\mu$, which proves
the desired formula. ∎

We emphasize that formula (35) provides the relation between the projection
$\mathcal{P}$, that corresponds to the hyperbolic generalized evolution
semigroup, and the dichotomy projections $P(t)$ given by Definition 4.2.

######  Example 4.5.

For any fixed continuously differentiable function
$\mu\in\mathcal{A}_{\,\infty}$ and for any projection $P\in\mathcal{B}(X)$, we
consider the evolution family

$U(t,s)=e^{\mu(s)-\mu(t)}P+e^{\mu(t)-\mu(s)}Q,$

where $Q=\mathrm{Id}-P$.

Even if one can easily observe that $\mathcal{U}$ admits a $\mu$-exponential
dichotomy, the purpose of this example is to show that the generator $G$
corresponding to the generalized evolution semigroup associated to
$\mathcal{U}$ is invertible, without explicitly constructing it. Pick $f\in
C_{0}(\mathbb{R},X)$, $f\neq 0$, and set

$u(t)=\int_{-\infty}^{t}\mu^{\prime}(\xi)U(t,\xi)Pf(\xi)\,d\xi-\int_{t}^{\infty}\mu^{\prime}(\xi)U(t,\xi)Qf(\xi)\,d\xi,\;t\in\mathbb{R}.$
(38)

One may check that the integrals in (38) are convergent, thus $u$ is well-
defined. Let $\varepsilon>0$. Since $\lim\limits_{t\to\pm\infty}f(t)=0$, there
exists $\delta_{1}>0$ such that

$\|f(t)\|<\frac{\varepsilon}{4},\text{ for every
}t\in(-\infty,-\delta_{1})\cup(\delta_{1},+\infty).$

On the other hand, from the convergence of the integral
$\int_{0}^{\infty}e^{-\xi}d\xi$, there exists $\delta_{2}>0$ such that

$\int_{t^{\prime}}^{t^{\prime\prime}}e^{-\xi}d\xi<\frac{\varepsilon}{4\|f\|_{\infty}},\text{
for every }t^{\prime},t^{\prime\prime}>\delta_{2}.$

Set $\delta=\max\\{\delta_{1},\delta_{2}\\}$ and choose
$t>\mu^{-1}(\mu(\delta)+\delta)$, which implies $t>\delta$. Now, we
successively have

$\displaystyle\|u(t)\|$
$\displaystyle\leq\int_{-\infty}^{t}\mu^{\prime}(\xi)e^{-(\mu(t)-\mu(\xi))}\|f(\xi)\|\,d\xi+\int_{t}^{\infty}\mu^{\prime}(\xi)e^{-(\mu(\xi)-\mu(t))}\|f(\xi)\|\,d\xi$
$\displaystyle\leq\int_{-\infty}^{-\delta}\mu^{\prime}(\xi)e^{-(\mu(t)-\mu(\xi))}\|f(\xi)\|\,d\xi+\int_{-\delta}^{\delta}\mu^{\prime}(\xi)e^{-(\mu(t)-\mu(\xi))}\|f(\xi)\|\,d\xi$
$\displaystyle\qquad+\int_{\delta}^{t}\mu^{\prime}(\xi)e^{-(\mu(t)-\mu(\xi))}\|f(\xi)\|\,d\xi+\frac{\varepsilon}{4}\int_{t}^{\infty}\mu^{\prime}(\xi)e^{-(\mu(\xi)-\mu(t))}\,d\xi$
$\displaystyle\leq\frac{\varepsilon}{4}\int_{-\infty}^{-\delta}\mu^{\prime}(\xi)e^{-(\mu(t)-\mu(\xi))}\,d\xi+\int_{-\delta}^{\delta}\mu^{\prime}(\xi)e^{-(\mu(t)-\mu(\xi))}\,d\xi\,\|f\|_{\infty}$
$\displaystyle\qquad+\frac{\varepsilon}{4}\int_{\delta}^{t}\mu^{\prime}(\xi)e^{-(\mu(t)-\mu(\xi))}\,d\xi+\frac{\varepsilon}{4}$
$\displaystyle\leq\frac{\varepsilon}{2}\int_{-\infty}^{t}\mu^{\prime}(\xi)e^{-(\mu(t)-\mu(\xi))}\,d\xi+\int_{\mu(t)-\mu(\delta)}^{\mu(t)-\mu(-\delta)}e^{-\tau}\,d\tau\,\|f\|_{\infty}+\frac{\varepsilon}{4}<\varepsilon,$

which shows that $\lim\limits_{t\to+\infty}u(t)=0$. Similarly, one may prove
that $\lim\limits_{t\to-\infty}u(t)=0$ and thus $u\in C_{0}(\mathbb{R},X)$.
Furthermore, for $t\geq s$ we get

$u(t)-U(t,s)u(s)=\int_{s}^{t}\mu^{\prime}(\xi)U(t,\xi)f(\xi)\,d\xi.$

From Proposition 3.8, this yields that $u\in D(G)$ and $Gu=-f$. Therefore, $G$
is invertible and the evolution family $\mathcal{U}$ admits a
$\mu$-exponential dichotomy.

## Comments

Throughout this paper, all the results we obtain stand for
$C_{0}(\mathbb{R},X)$. Evidently they can be extended similarly to the general
context of nonuniform behavior, by formally replacing the function space
$C_{0}(\mathbb{R},X)$ with the super-space $C_{*}$ introduced in [2]. However
we think that such an approach would significantly complicate computations,
without adding any essential merit for our main purposes.

On the other hand, we decided to develop our considerations in the case of
evolution families on the whole real line. We believe that the half-line
situation can also be treated similarly, with only a few modification (e.g.,
[24]), but this is not our purpose.

## Open problems

Authors address the following open questions: What happens if the mappings
$s\mapsto\varphi_{t}(s)$, $t>0$, are bounded, i.e. $\ell\in\mathbb{R}$? Can we
define in this case an evolution semigroup on some appropriate Banach function
space? If the answer is positive, how can we apply it in the study of
asymptotic behavior of the underlying evolution family?

## References

  * [1] L. Barreira, L. H. Popescu, C. Valls, Hyperbolic sequences of linear operators and evolution maps, Milan J. Math. 84 (2016), 203–216.
  * [2] L. Barreira, L. H. Popescu, C. Valls, Nonuniform exponential behavior via evolution semigroups, Mathematika 66 (2020), 15–38.
  * [3] L. Barreira, C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Cont. Dyn. Syst. 22 (2008), 509–528.
  * [4] L. Barreira, C. Valls, Polynomial growth rates, Nonlinear Anal. 71 (2009), 5208–5219.
  * [5] L. Barreira, C. Valls, Hyperbolicity via evolution semigroups on $L^{p}$, Qual. Theory Dyn. Syst. 18 (2019), 887–908.
  * [6] A. J. G. Bento, C. Silva, Stable manifolds for nonuniform polynomial dichotomies, J. Funct. Anal. 257 (2009), 122–148.
  * [7] C. Chicone, Y. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys Monogr., Vol. 70, Amer. Math. Soc. (1999).
  * [8] Ju. L. Daleckiǐ, M. G. Kreǐn, Stability of Solutions of Differential Equations in Banach Space, Transl. Math. Monogr., Vol. 43, Amer. Math. Soc., Providence, RI. (1974).
  * [9] N. Dunford, Spectral Theory. I Convergence to Projections, Trans. Amer. Math. Soc. 54 (1943), 185–217.
  * [10] K. J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., Vol. 194, Springer (2000).
  * [11] D. E. Evans, Time dependent perturbations and scattering of strongly continuous groups on Banach spaces, Math. Ann. 221 (1976), 275–290.
  * [12] J. S. Howland, Stationary scattering theory for time-dependent hamiltonians, Math. Ann. 207 (1974), 315–335.
  * [13] L. Jiang, Generalized exponential dichotomy and global linearization, J. Math. Anal. Appl. 315 (2006), 474–490.
  * [14] L. Jiang, Strongly topological linearization with generalized exponential dichotomy, Nonlinear Anal. 67 (2007), 1102–1110.
  * [15] Y. Latushkin, S. Montgomery-Smith, Lyapunov theorems for Banach spaces, Bull. Amer. Math. Soc 31 (1994), 44–49.
  * [16] Y. Latushkin, S. Montgomery-Smith, Evolutionary semigroups and Lyapunov theorems in Banach spaces, J. Funct. Anal. 127 (1995), 173–197.
  * [17] Y. Latushkin, T. Randolph, Dichotomy of differential equations on Banach spaces and an algebra of translation operators, Integral Equations Operator Theory 23 (1995), 472–500.
  * [18] D. L. Lovelady, On the generation of linear evolution operators, Duke Math. J. 42 (1975), 57–69.
  * [19] N. Lupa, L. H. Popescu, A complete characterization of exponential stability for discrete dynamics, J. Difference Equ. Appl. 23 (2017), 2072–2092.
  * [20] N. Lupa, L. H. Popescu, Admissible Banach function spaces for linear dynamics with nonuniform behavior on the half-line, Semigroup Forum 98 (2019), 184–208.
  * [21] R. H. Martin Jr., Conditional stability and separation of solutions to differential equations, J. Differential Equations 13 (1973), 81–105.
  * [22] N. Van Minh, Semigroups and stability of nonautonomous differential equations in Banach spaces, Trans. Amer. Math. Soc. 345 (1994), 223–242.
  * [23] N. Van Minh, On the proof of characterizations of the exponential dichotomy, Proc. Amer. Math. Soc. 127 (1999), 779–782.
  * [24] N. Van Minh, F. Räbiger, R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory 32 (1998), 332–353.
  * [25] J. S. Muldowney, Dichotomies and asymptotic behaviour for linear differential systems, Trans. Amer. Math. Soc. 283 (1984), 465–484.
  * [26] R. Nagel, G. Nickel, Wellposedness for nonautonomous abstract Cauchy problems, Progr. Nonlinear Differ. Equ. Appl. 50 (2002), 279–293.
  * [27] R. Nagel, A. Rhandi, Semigroup applications everywhere, Phil. Trans. R. Soc. A 378: 20190610, 2020.
  * [28] R. Naulin, M. Pinto, Dichotomies and asymptotic solutions of nonlinear differential systems, Nonlinear Anal. 23 (1994), 871–882.
  * [29] R. Naulin, M. Pinto, Roughness of $(h,k)$-dichotomies, J. Differential Equations 118 (1995), 20–35.
  * [30] H. Neidhardt, On abstract linear evolution equations, I, Math. Nachr. 103 (1981), 283–298.
  * [31] J. M. A. M. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Oper. Theory Adv. Appl., Vol. 88, Birkhäuser (1996).
  * [32] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., Vol. 44, Springer (1983).
  * [33] F. Räbiger, A. Rhandi, R. Schnaubelt, Perturbation and an abstract characterization of evolution semigroups, J. Math. Anal. Appl. 198 (1996), 516–533.
  * [34] F. Räbiger, R. Schnaubelt, The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions, Semigroup Forum 52 (1996), 225–239.
  * [35] R. T. Rau, Hyperbolic evolution semigroups on vector valued function spaces, Semigroup Forum 48 (1994), 107–118.
  * [36] R. T. Rau, Hyperbolic evolution groups and dichotomic evolution families, J. Dynam. Differential Equations 6 (1994), 335–350.