Page 385 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 385
TOPOLOGICAL METHODS IN DYNAMICAL SYSTEMS (MS-65)
the Vietoris topology). In this talk I will present a result which reveals that pseudo-arc is a
generic object also in a measure theoretical setting; namely, I will show that the inverse limit
of the generic Lebesgue measure preserving interval map is the pseudo-arc. Building on this
result I will construct a parametrised family of planar homeomorphisms with attractors being
the pseudo-arc and several interesting topological and measure-theoretical properties.
Attractors of dissipative homeomorphisms of the infinite surface
homeomorphic to a punctured sphere
Grzegorz Graff, grzegorz.graff@pg.edu.pl
Gdan´sk University of Technology, Poland
Coauthors: Rafael Ortega, Alfonso Ruiz-Herrera
A class of dissipative orientation preserving homeomorphisms of the infinite annulus, pairs of
pants, and generally infinite surface homeomorphic to a punctured sphere is considered. We
prove that in some isotopy classes the local behavior of such homeomorphisms at a fixed point,
namely the existence of so-called inverse saddle, impacts the topology of the attractor - it cannot
be arcwise connected.
This work was supported by the National Science Centre, Poland within the grant Sheng 1
UMO-2018/30/Q/ST1/00228 (G. Graff).
Magnetic Helicity and the Calabi Invariant
Gunnar Hornig, ghornig@dundee.ac.uk
University of Dundee, United Kingdom
Coauthor: Callum Birkett
Magnetic helicity (also called Asymptotic Hopf invariant, V.I Arnold 1974) is an important
tool in the study of both astrophysical and laboratory plasmas. Helicity is an integral over the
helicity density h = A ∧ B, dA = B, that measures the average asymptotic linkage of magnetic
flux B in a given domain. It is a topological invariant of the magnetic field and provides a
lower bound for the energy. However, the use of the helicity integral has been hampered by the
fact that it only measures the average linkage over the whole domain and does not provide any
more detailed information about linkage within parts of the domain. Attempts to extract more
information about the linkage of flux, for instance by considering the helicity density, or line
integrals over the helicity density (field line helicity), encounter the problem that h is not gauge
invariant. In this talk we introduce the Calabi invariant, an integral quantity closely associated
with helicity (Calabi 1970, Gambaudo et al. 2000) and show that this leads to interesting new
ways to interpret the helicity integral and allows to calculate a gauge invariant asymptotic field
line helicity.
References
[1] Eugenio Calabi, On the group of automorphisms of a symplectic manifold, from: “Prob-
lems in Analysis" (Lectures at the Sympos. in Honor of Salomon Bochner, Princeton
Univ., Princeton, NJ, 1969), 1970.
383
the Vietoris topology). In this talk I will present a result which reveals that pseudo-arc is a
generic object also in a measure theoretical setting; namely, I will show that the inverse limit
of the generic Lebesgue measure preserving interval map is the pseudo-arc. Building on this
result I will construct a parametrised family of planar homeomorphisms with attractors being
the pseudo-arc and several interesting topological and measure-theoretical properties.
Attractors of dissipative homeomorphisms of the infinite surface
homeomorphic to a punctured sphere
Grzegorz Graff, grzegorz.graff@pg.edu.pl
Gdan´sk University of Technology, Poland
Coauthors: Rafael Ortega, Alfonso Ruiz-Herrera
A class of dissipative orientation preserving homeomorphisms of the infinite annulus, pairs of
pants, and generally infinite surface homeomorphic to a punctured sphere is considered. We
prove that in some isotopy classes the local behavior of such homeomorphisms at a fixed point,
namely the existence of so-called inverse saddle, impacts the topology of the attractor - it cannot
be arcwise connected.
This work was supported by the National Science Centre, Poland within the grant Sheng 1
UMO-2018/30/Q/ST1/00228 (G. Graff).
Magnetic Helicity and the Calabi Invariant
Gunnar Hornig, ghornig@dundee.ac.uk
University of Dundee, United Kingdom
Coauthor: Callum Birkett
Magnetic helicity (also called Asymptotic Hopf invariant, V.I Arnold 1974) is an important
tool in the study of both astrophysical and laboratory plasmas. Helicity is an integral over the
helicity density h = A ∧ B, dA = B, that measures the average asymptotic linkage of magnetic
flux B in a given domain. It is a topological invariant of the magnetic field and provides a
lower bound for the energy. However, the use of the helicity integral has been hampered by the
fact that it only measures the average linkage over the whole domain and does not provide any
more detailed information about linkage within parts of the domain. Attempts to extract more
information about the linkage of flux, for instance by considering the helicity density, or line
integrals over the helicity density (field line helicity), encounter the problem that h is not gauge
invariant. In this talk we introduce the Calabi invariant, an integral quantity closely associated
with helicity (Calabi 1970, Gambaudo et al. 2000) and show that this leads to interesting new
ways to interpret the helicity integral and allows to calculate a gauge invariant asymptotic field
line helicity.
References
[1] Eugenio Calabi, On the group of automorphisms of a symplectic manifold, from: “Prob-
lems in Analysis" (Lectures at the Sympos. in Honor of Salomon Bochner, Princeton
Univ., Princeton, NJ, 1969), 1970.
383
