__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:__

__ __

·
D. Peran: *Analytic normalization of strongly hyperbolic
(complex) Dulac germs (on-line talk), *ICDEA 2022, 27th INTERNATIONAL
CONFERENCE ON DIFFERENCE EQUATIONS AND APPLICATIONS, 18-22 July 2022, Paris,
France, https://icdea2022.sciencesconf.org/data/pages/icdea2022_leaflet_v5_1_.pdf

·
D. Peran: *Analytic
linearization of hyperbolic (complex) Dulac germs,* Sedmi hrvatski
matematički kongres, 15-18 June 2022, Split, Croatia, https://www.pmfst.unist.hr/cromc2020/hr/naslovna/

· D: Peran: *Formal
linearization of logarithmic transseries and analytic linearization of Dulac
germs (invited online talk)*, __http://www.fields.utoronto.ca/activities/21-22/tame-dynamical,
__Mini-Workshop on Transseries and Dynamical Systems, May 30-June 1, 2022,
The Fields Institute, Toronto, Canada, __http://www.fields.utoronto.ca/activities/21-22/tame-dynamical__

· D. Peran: *Normal forms for logarithmic transseries and
Dulac germs*, Bifurcations of dynamical systems workshop February 9-12,
2022, Zagreb, http://frabdyn.math.hr/

· G. Radunović: *Fractal
zeta functions of orbits of parabolic diffeomorphisms (invited talk),* AMS-EMS-SMF 2022 - 18-22 July, 2022 - Grenoble – UGA,
https://ams-ems-smf2022.inviteo.fr/

· G. Radunović: *Fractal
zeta functions generated by orbits of parabolic diffeomorphisms, *7th
Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals, Ithaca, NY, June 4-8, 2022,
https://math.cornell.edu/7th-cornell-conference-analysis-probability-and-mathematical-physics-fractals

· G. Radunović: *Fractal
zeta functions of orbits of parabolic diffeomorphisms, *Bifurcations of dynamical systems workshop February 9-12,
2022, Zagreb, http://frabdyn.math.hr/

· M. Resman: *Zeta functions and complex dimensions
of orbits of dynamical systems** ,*
Dynamics Days Europe, August 22-26, 2022, Aberdeen University, Aberdeen,
Scotland, https://www.abdn.ac.uk/events/conferences/dynamics-days-2022.php

·
M. Resman: *Rigidity of saddle-loops*,
Mini-workshop on Transseries and Dynamical Systems, May 30 - June 1,
2022, Fields Institute, Toronto, Canada,

http://www.fields.utoronto.ca/activities/21-22/tame-dynamical

__ __

·
M. Resman: *Analytic
invariants of Dulac germs*, Bifurcations of dynamical systems , Workshop,
9th - 12th February, 2022, Zagreb, Croatia, http://frabdyn.math.hr/

__ __

·
*D. Peran*: *Normal forms for hyperbolic logarithmic transseries,* NoLineal conference 20/21, June 30-July 02, 2021, Madrid,
Spain (on-line)

* *

·
*D. Peran*: *Linearization of hyperbolic logarithmic transseries and Dulac germs*,__
__26th International
Conference on Difference Equations and Applications - ICDEA 2021, July 26-30, 2021, Sarajevo, BIH
(on-line)

__ __

·
*M. Resman*:
*Complex dimensions and lengths of epsilon-neighborhoods of orbits*
(on-line, invited talk), Journées 2021 du GDR EFI, May 31- June 2 2021,
University of Strasbourg, https://efi2021.sciencesconf.org/

* *

·
*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:__

· April-June, 2022, scientific visit of M. Resman
to Fields Institute, Toronto in scope of thematic semester
http://www.fields.utoronto.ca/activities/21-22/tame (organization of
mini-workshop and collaboration with Rolin, Servi, Panazzolo)

· September 14-23, 2022, scientific visit of M.
Resman, Hasselt University, collaboration with R. Huzak ; August 28-September
7, 2022, scientific visit of M. Resman, Université de Bourgogne, collaboration
with P. Mardešić (work on Chebyshev scales for saddle-node bifurcation)

·
27.3.-4.4.
Hasselt: Scientific collaboration of G. Radunovic with R. Huzak, P. De

Maesschalck and A. Janssens on defining the fractal codimension for

nilpotent contact points in two-dimensional slow-fast systems.

·
4.-12.4. Dijon:
Scientific collaboration of M. Resman with P. Mardesic and M. Klimes

on finishing the joint paper with M. Resman "Reading analytic invariants

of parabolic diffeomorphisms from their orbits".

·
30.4.-7.5.
Karlsruhe: Scientific collaboration of G. Radunovic with S. Winter on

connecting the theory of support measures (and fractal curvature

measures) to the theory of fractal zeta functions. Preliminary results

obtained. Radunovic also gave a seminar talk "An overview of the theory

of complex dimensions" in scope of the AG Stochastische Geometrie

seminar of the KIT Institute of stochastics.

·
2.7.-10.7.
Karlsruhe: Scientific collaboration of G. Radunovic with S. Winter on

connecting the theory of support measures (and fractal curvature

measures) to the theory of fractal zeta functions. Draft of the paper

written. Radunovic also gave a seminar talk "Applications of fractal

zeta functions" in scope of the AG Stochastische Geometrie seminar of

the KIT Institute of stochastics.

·
18.8.-25.8.
Hasselt: Scientific collaboration of G. Radunovic with R. Huzak, P. De

Maesschalck and A. Janssens resulting in submission of the joint paper

"Fractal codimension of nilpotent contact points in two-dimensional

slow-fast systems".

·
September
20-30, 2021, **scientific visit of G. Radunović, M. Resman**, to
Université de Bourgogne, Dijon, to work with Pavao Mardešić on reading
analytic invariants through fractal zeta functions of orbits, research paper in
preparation

·
October
25-31, 2020, **scientific visit of P.Mardešić** to Zagreb, talk at
Colloquium of Croatian Math Society and scientific work with G. Radunović,
M. Resman

·
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:__

·
M. Resman, Mini-workshop on Transseries and Dynamical
Systems, in scope of Thematic Program on Tame Geometry, Transseries and
Applications to Analysis and Geometry, May 30 - June 01, 2022 (M. Resman
organizer with J.P.Rolin)

·
Dino Peran, G. Radunović, M.
Resman, Bifurcations of dynamical systems Workshop, 9th - 12th February,
2022, Zagreb, Croatia (part of organizing commitee)

·
Dino Peran, Advancing
Bridges in Complex Dynamics - INSTITUT DE MATHÉMATIQUES DE MARSEILLE
(univ-amu.fr), September 20-24 2021, CIRM, Luminy,
France, https://www.i2m.univ-amu.fr/events/advancing-bridges-in-complex-dynamics/

·
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:__

· M. Resman: *Classifications of Dulac germs*,
Geometry and Model Theory seminar,

Fields Institute, Toronto, Canada, April 2022,
https://www.youtube.com/watch?v=BXTeDWbHCOw

·
M.
Resman: *Fiksne tocke difeomorfizama i njihove orbite *, Scientific
colloquium of the Split Mathematical Society, Split, Croatia, November 2021,

·
G.
Radunović: *Fractal zeta functions
of orbits of parabolic diffeomorphisms*, UC Riverside, CA,
FRG seminar, 28.4.22. (online seminar)

·
G.
Radunović: *An overview of the theory of complex dimensions, *KIT -
Fakultät für Mathematik, Institut für Stochastik, AG Stochastische Geometrie, 6.5.2022

·
G. Radunović*:
Applications of fractal zeta functions*, Fakultät für Mathematik, Institut
für Stochastik, AG Stochastische Geometrie, 7.7.2022

·
D. Peran: *Normal
forms for logarithmic transseries* (online seminar), Geometry and Model
Theory Seminar, The Fields Institute, Toronto, Canada, 24.05.22., http://www.fields.utoronto.ca/activities/21-22/geometry-and-model-theory-seminar

·
May 2021, M.
Resman: *Transasymptotic expansions in classification problems for Dulac
germs*, seminar at Hasselt University, Belgium (on-line), May 2021

·
April 2021,
M. Resman: *Analytic moduli for parabolic Dulac germs, construction and
realization*, GeomTop Weizmann seminar, Weizmann Institute of Science
(on-line),

·
23/02/2021, D. Peran:
*Normal forms for transseries and Dulac germs*, Seminar for
Dynamical systems, PMF-MO, Zagreb (also the public defense of the dissertation
topic)

·
02/02/2021, M. Resman: *Zeta functions and complex
dimensions of orbits of dynamical systems,* Seminar for dynamical systems,
PMF-MO, Zagreb

·
26/01/2021, G. Radunović: *Fractional integro-derivative
of fractal zeta functions and Log-Minkowski measurability*, Seminar for
dynamical Systems, PMF-MO, Zagreb

·
20/10/2020, P. Mardešić: *The tennis racket
effect and Abelian integrals*, The Colloquium of the Croatian Mathematical
Society, Zagreb

·
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:__

__ __

1. P. Mardešic, G. Radunovic, M.
Resman, *Fractal zeta functions of orbits of parabolic diffeomorphisms*,
Analysis and Mathematical Physics, 12, DOI 10.1007/s13324-022-00724-3, pp. 1-
70 (2022); https://arxiv.org/abs/2010.05955

2. D. Peran, M. Resman,
J.P.Rolin, T. Servi, *Linearization of complex hyperbolic Dulac germs*,
Journal of mathematical analysis and applications, 508, pp. 1--27 (2022), https://arxiv.org/abs/2109.00284v1

__ __

3. Mardešić, Pavao; Resman, Maja
Realization of analytic moduli for parabolic Dulac germs, Ergodic Theory and
Dynamical Systems, First View , pp. 1 - 55, 2020 __https://arxiv.org/abs/1910.06130v1__

__ __

4. Mardešić, P., Resman, M.,
Analytic moduli for parabolic Dulac germs, Russian Math. Surveys, Volume 76 ,
Number 3, June 2021, p. 389-460

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

__ __

# 5. 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

__ __

6. 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

__ __

__ __

__Preprints:__

1. Peran, D., Resman, M., Rolin, J.P.,
Servi, T. *Normal forms of hyperbolic logarithmic transseries*, submitted
(2021), https://arxiv.org/abs/2105.10660v2

2. Klimes, M., Mardesic, P., Radunovic, G.,
Resman, M., Analytic invariants of a parabolic diffeomorphism from its orbit,
submitted (2021), arxiv

3. Panazzolo, D., Resman, M.,
Teyssier, L., Rigidity of saddle loops, submitted (2022), arxiv

4. Fractal codimension of nilpotent contact points in
two-dimensional

slow-fast systems, Peter De Maesschalck, Renato Huzak, Ansfried Janssens, Goran
Radunović, https://arxiv.org/abs/2208.10173,
submitted (2022)

5. Quasiperiodic sets at infinity and meromorphic
extensions of their

fractal zeta functions, Goran Radunović, https://arxiv.org/abs/2208.11245,
submitted (2022)

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