Normal forms for transseries and Dulac germs

doctoral thesis

• Title (croatian): Normalne forme za transredove i Dulacove klice
• Author: Dino Peran
• Mentor: Maja Resman (mentor)
• Mentor: Tamara Servi (komentor); VIAF: 61361939
• Committee member: Vesna Županović (predsjednik povjerenstva)
• Committee member: Goran Radunović (član povjerenstva)
• Committee member: Pavao Mardešić (član povjerenstva); VIAF: 7534604
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-12-13, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics
• Universal decimal classification (UDC): 51 - Mathematics

Abstract

Transseries are formal (possibly infinite) sums of monomials that are formal products of iterated exponentials, powers and iterated logarithms, with real coefficients (see e.g. [6], [21]). We consider here a subclass of logarithmic transseries which contain only powers and iterated logarithms. Transseries appear in many problems in mathematics ( [3], [11]) and physics ( [1]) as asymptotic expansions of certain important maps. In dynamics, for example, transseries are related to the famous... More Dulac’s problem ( [7]) of non-accumulation of limit cycles on a hyperbolic or semi-hyperbolic polycycle of a planar analytic vector field. The problem was solved independently by Ilyashenko ( [10], [11], [12]) and Écalle ( [3]), but the proofs are so far not well-understood, at least in the semi-hyperbolic case. The study of the accumulation of limit cycles on a polycycle is naturally related to the study of fixed points of the first return map of a polycycle (see e.g. [32]). The first return map of a hyperbolic polycycle is an analytic map on interval (0;d), d > 0, which has a transserial asymptotic expansion at zero. In particular, its asymptotic expansion at zero is a logarithmic series involving only polynomials in logarithms attached to each power, which is called a Dulac series (see e.g. [12], [32]). The proof of the Dulac problem strongly relies on the existence of a logarithmic asymptotic expansion of the first return map. Although Dulac gave the proof ( [7]) of the mentioned problem, there was an imprecision in his proof. In particular, at some point in the proof, the statement that every first return map of a hyperbolic polycycle is uniquely determined by its asymptotic expansion is used. This is not correct in general for non-analytic maps on the real line, due to the possibility of adding exponentially small terms, as opposed to the case of analytic maps and their Taylor expansions. Ilyashenko corrected this imprecision in [11] by proving that every such map can be analytically extended to a sufficiently large complex domain called a standard quadratic domain and by applying the Phragmen-Lindelöf Theorem (a maximum modulus principle on an unbounded complex domain, see e.g. [11], [12]). The existence of such analytic extension makes possible to conclude that the first return map is uniquely determined by its logarithmic asymptotic expansion. In this dissertation, we consider the so-called Dulac germs (called almost regular germs in [12]), i.e., analytic germs on (0;d), d > 0, that have a Dulac series as their asymptotic expansion at zero, and that can be analytically extended to a standard quadratic domain. On the one hand, we consider normal forms of logarithmic transseries (the formal part), and, on the other hand, analytic normalizations of hyperbolic and strongly hyperbolic Dulac germs (the analytic part). We also generalize to their complex counterparts, called hyperbolic and strongly hyperbolic complex Dulac germs. We prove as well that, for hyperbolic and strongly hyperbolic Dulac germs, the formal normalizations are asymptotic expansions of their analytic normalizations. In proving that the formal transserial normalization is an asymptotic expansion of the analytic normalization, in general, there is a problem of a choice of the summation rule at the limit ordinal steps. In particular, given some map f on open interval (0;d), d > 0, we want to assign to the map f its asymptotic expansion at zero in power-iterated logarithm scale. Up to the first limit ordinal it can be done following the usual Poincaré algorithm, contrary to the limit ordinal steps where we have multiple choices of intermediate sums. Therefore, we have to determine a summation rule at limit ordinal steps, which vary from problem to problem (see e.g. integral summation rule in [20], [22]). Luckily, for hyperbolic and strongly hyperbolic Dulac germs the formal normalizations are Dulac series, so standard Poincaré algorithm suffices. On the other hand, in case of parabolic Dulac germs, it is proved in [22] that, in general, the formal normalization is of order type strictly bigger than w. Normal forms and normalizations of standard power series are already known (see e.g. [4], [12], [16]). Furthermore, in previous papers ( [21], [22]), normal forms for logarithmic transseries were obtained only for power-logarithm transseries, i.e., logarithmic transseries that contain only powers and the first iterate of logarithm. The techniques used in [21] are based on a transfinite algorithm of successive changes of variables. Here, we generalize these results to a larger class of logarithmic transseries containing also iterated logarithms. Additionally, as a normalization process we use fixed point theorems on various complete metric spaces of logarithmic transseries. In this way, normalizations are given as limits (in appropriate topologies) of Picard sequences. This is important for the future work because we think that this approach is better for revealing the summation rule at limit ordinal steps. In proving the existence of the analytic normalization of a hyperbolic Dulac germ, we generalize the classical Koenigs Theorem (see e.g. [4], [14], [24]) for linearization of analytic hyperbolic diffeomorphisms at zero. Recently, there have been some improvements of this result for various classes of maps not necessarily analytic at 0. One such generalization is a result of Dewsnap and Fischer [5] for C1 real maps on an open interval around zero with power-logarithmic asymptotic bounds. In this dissertation, we prove a linearization theorem for analytic maps with power-logarithmic asymptotic bounds on invariant complex domains, that can be seen as a generalization of both Koenigs Theorem and the result of Dewsnap and Fischer from [5, Theorem 2.2]. In particular, we apply the mentioned linearization theorem to obtain the analytic linearization of hyperbolic (complex) Dulac germs. Finally, we also generalize the Bottcher Theorem ¨ (see e.g. [4], [24]) for germs of strongly hyperbolic analytic diffeomorphisms at zero to strongly hyperbolic complex Dulac germs on standard quadratic domains. Less

Abstract (croatian)

Transredovi su formalne (beskonačne) sume formalnih umnožaka iteriranih eksponencijalnih funkcija, općih potencija i iteriranih logaritama (koje nazivamo monomi) s realnim koeficijentima (vidjeti npr. [6], [21]). U ovoj disertaciji bavimo se podklasom takozvanih logaritamskih transredova čiji monomi sadrže samo opće potencije i iterirane logaritme. Transredovi se pojavljuju u mnogim problemima u matematici ( [3], [11]) i fizici ( [1]) kao asimptotski razvoji nekih značajnih... More preslikavanja. U dinamičkim sustavima transredovi su primjerice povezani s poznatim Dulacovim problemom ( [7]) o neakumulaciji graničnih ciklusa na hiperbolički ili semi-hiperbolički policiklus ravninskog analitičkog vektorskog polja. Iako su navedeni problem nezavisno riješili Ilyashenko ( [10], [11], [12]) i Écalle ( [3]), rješenja semi-hiperboličkog slučaja i dalje nisu u potpunosti shvaćena. Akumulacija graničnih ciklusa na policiklus se prirodno povezuje s teorijom fiksnih točaka preslikavanja povrata (ili Poincaréovog preslikavanja ) danog policiklusa (vidjeti npr. [32]). Preslikavanje povrata hiperboličkog policiklusa je analitičko preslikavanje na intervalu (0;d), d > 0, s transredom kao asimptotskim razvojem u nuli. Preciznije, njegov asimptotski razvoj u nuli je logaritamski red u kojem, uz svaku opću potenciju, stoji polinom u logaritmima. Takav red nazivamo Dulacov red (vidjeti npr. [12], [32]). Rješenje Dulacovog problema uvelike se oslanja na postojanje logaritamskog asimptotskog razvoja preslikavanja povrata u nuli. Iako je Dulac dao rješenje navedenog problema, u njegovom rješenju ( [7]) je postojala nepreciznost. Naime, bez dokaza je korištena tvrdnja da je svako preslikavanje povrata hiperboličkog policiklusa jedinstveno određeno svojim asimptotskim razvojem. Navedena tvrdnja općenito nije istinita za realna preslikavanja koja nisu analitička u nuli zbog mogućnosti dodavanja eksponencijalno malih članova u razvoju, za razliku od analitičkih preslikavanja u nuli i njihovih Taylorovih razvoja. Ilyashenko je u [11] otklonio navedenu nepreciznost dokazavši da se svako preslikavanje povrata može analitički proširiti na dovoljno veliku kompleksnu domenu koju nazivamo standardna kvadratna domena. Phragmen-Lindelöfov teorem (varijanta principa maksimuma na neomeđenoj kompleksnoj domeni, vidjeti npr. [11], [12]) tada daje injektivnost asimptotskog razvoja za preslikavanja povrata. Na taj način možemo zaključiti da je preslikavanje povrata jedinstveno određeno svojim logaritamskim asimptotskim razvojem. U ovoj disertaciji promatramo takozvane Dulacove klice (skoro regularne klice u [12]), tj. analitičke klice na (0;d), d > 0, kojima je asimptotski razvoj u nuli Dulacov red te koje se mogu analitički proširiti na neku standardnu kvadratnu domenu. S jedne strane, promatramo normalne forme logaritamskih transredova (formalni dio). S druge strane, promatramo analitičke normalizacije (jako) hiperboličkih Dulacovih klica (analitički dio) te njihovih generalizacija koje nazivamo (jako) hiperboličkim kompleksnim Dulacovim klicama. Također dokazujemo da je formalna normalizacija asimptotski razvoj analitičke normalizacije (jako) hiperboličkih Dulacovih klica. U dokazu da je formalna normalizacija asimptotski razvoj analitičke normalizacije općenito se javlja problem izbora sumacijskog pravila u koracima indeksiranim graničnim ordinalima. Naime, pretpostavimo da želimo odrediti asimptotski razvoj u nuli u skali općih potencija i iteriranih logaritama za dano preslikavanje f na otvorenom intervalu (0;d), d > 0. Koristeći standardni Poincaréov algoritam to se može napraviti do prvog graničnog ordinala. U koracima određenim graničnim ordinalima postoje višestruki izbori takozvanih međusuma. Upravo zbog toga je potrebno odrediti pravila sumacije, koja ovise o problemu kojeg promatramo (vidjeti integralno pravilo sumacije u [20], [22]). Kod hiperboličkih i jako hiperboličkih Dulacovih klica, formalne normalizacije su, srećom, Dulacovi redovi, pa nam je dovoljan Poincareov algoritam za asimptotski razvoj. S druge strane, u [22] je dokazano da je formalna normalizacija paraboličkih Dulacovih klica transred indeksiran ordinalom strogo većim od w. Normalne forme i normalizacije standardnih redova potencija su otprije poznate (vidjeti npr. [4], [12], [16]). Nadalje, u prijašnjim radovima ( [21], [22]) normalne forme su određene samo za transredove tipa potencija-logaritam, tj. za logaritamske transredove koji sadržavaju samo opće potencije i prvu iteraciju logaritma. Tehnike korištene u [21] temelje se na transfinitoj kompoziciji elementarnih zamjena varijabli. U ovoj disertaciji generaliziramo navedene rezultate na širu klasu svih logaritamskih transredova koji mogu sadržavati i iterirane logaritme. Nadalje, u postupku normalizacije koristimo teoreme fiksne točke na potpunim metričkim prostorima logaritamskih transredova. Time su normalizacije dane kao limesi (u odgovarajućim topologijama) Picardovih iteracija. Smatramo da je ovaj pristup problemu normalizacije bolji pri određivanju sumacijskog pravila na mjestima graničnih ordinala, što ga čini bitnim za naš budući rad. U dokazu postojanja analitičke normalizacije hiperboličkih Dulacovih klica, generaliziramo klasični Koenigsov teorem (vidjeti npr. [4], [14], [24]) koji daje linearizaciju analitičkih hiperboličkih difeomorfizama u nuli. Nedavno je ovaj rezultat poboljšan za razne klase preslikavanja koja nisu nužno analitička u nuli. Jedno takvo poboljšanje je Dewsnap-Fischerov rezultat [5] za realna preslikavanja klase C1 na otvorenom intervalu oko nule, s asimptotskim ocjenama tipa potencija-logaritam. U ovoj disertaciji dokazujemo linearizacijski teorem za analitička preslikavanja s asimptotskim ocjenama tipa potencija-logaritam na invariantnim kompleksnim domenama, koji možemo smatrati generalizacijom i Koenigsovog teorema i Dewsnap-Fischerovog rezultata iz [5, Theorem 2.2]. Nadalje, navedeni linearizacijski teorem primjenjujemo prilikom analitičke linearizacije hiperboličkih (kompleksnih) Dulacovih klica. Naposljetku, generaliziramo Böttcherov teorem (vidjeti npr. [4], [24]) za jako hiperboličke difeomorfizme u nuli, za klasu jako hiperboličkih kompleksnih Dulacovih klica na standardnim kvadratnim domenama. Less

• Keywords: logarithmic transseries; order of transseries; normal forms; normalization; linearization; formal and analytic classification; (complex) Dulac germs; Dulac series; standard quadratic domains; local fixed point theory; fixed point theorems; iteration theory; Koenigs sequence
• Keywords (croatian): logaritamski transredovi; red logaritamskih transredova; normalne forme; normalizacija; linearizacija; formalna i analitička klasifikacija; (kompleksne) Dulacove klice; Dulacovi redovi; standardne kvadratne domene; lokalna teorija fiksne točke; teoremi fiksne točke; teorija iteracija; Koenigsov niz
• Language: english
• URN:NBN: urn:nbn:hr:217:394321
• Promotion: 2021
• Study programme: Title: Mathematics; Study programme type: university; Study level: postgraduate; Academic / professional title: doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika (doktor/doktorica znanosti, područje prirodnih znanosti, polje matematika)
• Type of resource: Text
• Extent: xviii, 248 str.
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-12-22 12:44:01