# The principal investigator: **Maja Resman**, PhD, Assistant professor,
University of Zagreb, Faculty of Science, Department of mathematics, Zagreb,
Croatia

# The team members:

·
**Maja
Resman, **PhD,
Assistant professor, University of Zagreb, Faculty of Science, Department of
mathematics, Zagreb, Croatia (PI)

https://web.math.pmf.unizg.hr/~mresman/

·
**Goran
Radunović**,
PhD, assistant professor, University of Zagreb, Faculty of Science, Department
of mathematics, Zagreb, Croatia

https://web.math.pmf.unizg.hr/~goranr/

·
**Dino
Peran**,
mag.math., University of Split, Croatia (doctorand)

https://www.pmfst.unist.hr/team/dino-peran/

#
The administering organization: **University of Zagreb, Faculty of Science,
Zagreb, Croatia**

# Total budget of the project: **615.510,00 HRK**

** **

# estimated
duration of the project: **60 months** (15/10/2018-15/10/2023)

** **

__The project summary____: __

The goal of the project is the formation of a new research group around
a new research direction. It is meant as an interplay of areas of the team
members: we plan to relate the qualitative theory of dynamical systems with the
theory of complex dimensions of fractal sets, through fractal analysis of
orbits. By fractal properties of orbits, we mean their box dimension and
generalizations. We are motivated by the ‘inverse question’: can we extract
information on the behavior of a system by fractal analysis of a single
trajectory? The complex dimensions of fractal sets generalize the notion of
their box dimension and reveal the geometrical structure more precisely.
Analysing complex dimensions of orbits seems to be well-adaptedto measuring the
complexity of the system. The theory of complex dimensionsis based on the
analysis of singularities of complex functions (fractal zeta functions
introduced by Lapidus). It is a very interesting field in itself, related to
spectral theory, mathematical physics and number theory (the well-known Riemann
hypothesis).We plan to apply this theory to important questions in discrete
dynamical systems, such as their classifications. In short, key points of the
research are: classifications (formal and analytic) of 1-dim discrete systems
through complex dimension analysis of attached orbits, and understanding their
bifurcations by the set of complex dimensions of attached orbits. Our aim is
also to promote the field of dynamical systems in Croatia by organizing a
workshop and mini-courses. We plan to expand our small group by engaging a new
PhD student and a postdoc. In Croatia, there is only a small group of
researchers in dynamical systems, scattered in different areas, which we intend
to bring closer together. In the course of the project, we will put a
particular effort in scientific growth of the new PhD student. We plan to
provide her/him with the opportunity of international collaborations from the
beginning of her/his career.
More
information can be found at Croatian
Science Foundation page.

__Project activities:__

__ __

**Dubrovnik IX – Topology and dynamical systems conference,
IUC Dubrovnik, June 22-26, 2019**

*Organizers: M. Bestvina, M. Resman, S. Štimac*

Participation of Pavao Mardešić, Maja Resman, Goran
Radunović was supported by project UIP-2017-05-1020. They contributed to
the conference with the following research talks:

*P. Mardešić: *Bounding the length of the principal term of a Poincaré displacement function*

*M.
Resman: *Classifications of Dulac germs*

*G. Radunović: *An overview of the theory of
complex dimensions and fractal zeta functions.*

Additional
information can be found at the conference
web-page.

__ __

__Conferences, workshops, seminars, scientific visits:__

__Conference talks:__

·
March
22-24, 2019 Hawai AMS Sectional meeting,

*G.
Radunović: **Complex
dimensions generated by essential singularities.** (invited talk),*

__web-page__

·
June
22-26, 2019 Dubrovnik IX – Topology and dynamical systems conference, IUC
Dubrovnik

*G. Radunović*: *An
overview of the theory of complex dimensions and fractal zeta functions.*

*M. Resman*: *Classifications
of parabolic Dulac germs*

·
June
17-21, 2019, AQTDE 2019 conference, Castro
Urdiales

*M. Resman*: *Classifications of
parabolic Dulac germs (invited talk)*

__web-page__

*(not financially supported by the project, but talk related to the
research on the project)*

·
July 8-12 2019, Equadiff 2019 conference, Leiden,
Netherlands

*G.
Radunović*: *Towards
bifurcations of complex dimensions*

*M.
Resman**: Formal classification of
parabolic Dulac maps*

**web-page**

** **

__Scientific
visits:__

·
June 03-06 2019**, scientific visit** of Dino Peran
to Université Paris
Diderot (common work with M. Resman, T. Servi, J.P.Rolin on linearization of
hyperbolic transseries, normal forms for power-iterated log transseries
appearing as first return maps for saddle node polycycles)

·
February
15-22 2019, scientific visit of Loic Teyssier, IRMA, Université de Strasbourg,
to University of Zagreb (collaboration with M. Resman on non-ramified analytic
moduli for Dulac saddle maps)

__Workshops:__

·
Dino
Peran, IRP LOW DIMENSIONAL DYNAMICAL SYSTEMS AND APPLICATIONS: ADVANCED COURSE
ON Recent Trends in Nonlinear Science, February 3^{rd} to 7^{th},
2020,** **Centre
de Recerca Matemŕtica , Bellaterra, Barcelona; __http://www.crm.cat/en/Activities/Curs_2019-2020/Pages/IRP_Dynamical_Systems_AC_Participants.aspx__

__Seminar talks:__

__ __

§ 08.10.2020.
Maja Resman: *Zeta functions and complex dimensions of orbits of dynamical
systems*, Seminaire d’Analyse, Université de Haute-Alsace, Mulhouse,http://www.lmia.uha.fr/SeminaireDeMathematiques/Conferences_passees.html

§ *Workgroup on
holomorphic dynamics and resurgence*

§ 09.07.2020.
Dino Peran: *Normal
forms for parabolic transseries of power-iterated log type (in English)* (On-line,
Seminar for dynamical systems, University of Zagreb)

§ 17.06.2020.
D. Panazzolo, Université de Haute Alsace, on-line talk *Resolution of singularities
for differential operators in dimension two* on Scientific
Colloquium of Croatian Mathematical Society, http://degiorgi.math.hr/kolokvij/

# § 28.02.2020. Dino Peran, poster *Formalne normalne forme za transredove
tipa potencija-logaritam* na Simpoziju doktorskih studenata PMF-a 2020.

§ 27.09.2019.
Dino Peran: *Linearizacija hiperboličkih transredova tipa
potencija-logaritam (Linearization of hyperbolic transseries of
power-logarithmic type)* (Seminar for dynamical systems, University of Zagreb)

§ 06.04.2019.
Dino Peran: *Konstrukcija lokalne Fatouove koordinate za parabolicke
difeomorfizme (Construction of the local Fatou coordinate for parabolic
diffeomorphisms)* (Seminar for dynamical systems, University of Zagreb,
from literature)

__ __

__Publications and preprints:__

__ __

**1.
**P.
Mardešić, G. Radunović, M. Resman*, Fractal zeta functions of
orbits of parabolic diffeomorphisms*, submitted (2020), https://arxiv.org/abs/2010.05955

# 2.
Lapidus,
Michel L.; Radunović, Goran; Žubrinić, Darko, *Essential
singularities of fractal zeta functions*, Pure and Applied Functional
Analysis, 5 (2020), 5; 1073-1094 https://arxiv.org/abs/1908.07845v2

__ __

3. P. Mardešić, M. Resman, *Analytic
moduli for parabolic Dulac germs, *accepted for publication in Russian Math.
Surveys (2020)

__https://arxiv.org/abs/1910.06130v1__

__ __

4. P. Mardešić, M. Resman, *Realization
of analytic moduli for parabolic Dulac germs*, a preprint (2020)

__https://arxiv.org/abs/1910.06130v1__

__ __

5. P.
Mardesic, M. Resman, J.P. Rolin, V. Zupanovic, *Tubular neighborhoods of
orbits of power-logarithmic germs*, Journal of dynamics and differential
equations, 1 (2019), 1; 1-49,

https://arxiv.org/pdf/1606.02581v3.pdf

__ __