Fractal analysis of discrete dynamical systems (Acronym: FRACLASS)

Fraktalna analiza diskretnih dinamičkih sustava

Croatian Science Foundation (HRZZ) Installation project, call UIP-05-2017 no. UIP-2017-05-1020

Project description

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

# The team members:

· Maja Resman, PhD, Associate 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, PhD, University of Split, Croatia

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:

· 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 Matemàtica , 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)