Page 386 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 386
TOPOLOGICAL METHODS IN DYNAMICAL SYSTEMS (MS-65)
[2] Jean-Marc Gambaudo and Maxime Lagrange, Topological lower bounds on the distance
between area preserving diffeomorphisms. Boletim da Socieda de Brasileira de Matemat-
ica, 31(1):9–27, 2000.
On the boundary of the basins of attraction for the secant method applied
to polynomials
Xavier Jarque, xavier.jarque@ub.edu
Universitat de Barcelona, Spain
Coauthors: Antonio Garijo, Laura Gardini
We investigate the discrete dynamical system S defined on R2 given by the secant method
applied to a real polynomial p. Every simple root α of p has associated its basin of attraction
A(α) formed by the set of points converging under S towards α and A∗(α) its immediate basin
of attraction.
We focus on the structure and dynamical behaviour of the boundary of the immediate basin
of attraction of a root of p. We call external roots of p the smallest and largest value and internal
all the rest. If α is an internal root of p then ∂A∗(α) is given by the stable manifold of a 4-cycle.
Moreover we show that, under some hypothesis, those internal basins are simply connected.
Wecken property and boundary preserving coincidences
Michael Kelly, kelly@loyno.edu
Loyola University New Orleans, United States
A result of B. Jiang divides all surfaces into two groups depending on the realizability of the
Nielsen number. A surface is said to be Wecken if each homotopy class of self-maps has a rep-
resentative which realizes the Nielsen number. It turns out that all non-Wecken surfaces admit a
sequence of maps for which the difference tends to infinity. The setting of boundary preserving
self-maps has a relative Nielsen number. The analogous problem was then studied in a series of
papers by B. Brown, B Sanderson and M. Kelly, resulting in a slightly different classification.
Here, there are two non-Wecken surfaces for which the difference remains bounded.
This talk considers this same problem, but now in the setting of coincidences for a pair of
maps using the corresponding Nielsen numbers for coincidence. The results obtained are joint
work with Leticia Silva and Joao Vieira (UNESP-Rio Claro, Brasil). We focus on the boundary
preserving case and as a result produce a class of pairs of bounded surfaces which satisfy the
Wecken property.
384
[2] Jean-Marc Gambaudo and Maxime Lagrange, Topological lower bounds on the distance
between area preserving diffeomorphisms. Boletim da Socieda de Brasileira de Matemat-
ica, 31(1):9–27, 2000.
On the boundary of the basins of attraction for the secant method applied
to polynomials
Xavier Jarque, xavier.jarque@ub.edu
Universitat de Barcelona, Spain
Coauthors: Antonio Garijo, Laura Gardini
We investigate the discrete dynamical system S defined on R2 given by the secant method
applied to a real polynomial p. Every simple root α of p has associated its basin of attraction
A(α) formed by the set of points converging under S towards α and A∗(α) its immediate basin
of attraction.
We focus on the structure and dynamical behaviour of the boundary of the immediate basin
of attraction of a root of p. We call external roots of p the smallest and largest value and internal
all the rest. If α is an internal root of p then ∂A∗(α) is given by the stable manifold of a 4-cycle.
Moreover we show that, under some hypothesis, those internal basins are simply connected.
Wecken property and boundary preserving coincidences
Michael Kelly, kelly@loyno.edu
Loyola University New Orleans, United States
A result of B. Jiang divides all surfaces into two groups depending on the realizability of the
Nielsen number. A surface is said to be Wecken if each homotopy class of self-maps has a rep-
resentative which realizes the Nielsen number. It turns out that all non-Wecken surfaces admit a
sequence of maps for which the difference tends to infinity. The setting of boundary preserving
self-maps has a relative Nielsen number. The analogous problem was then studied in a series of
papers by B. Brown, B Sanderson and M. Kelly, resulting in a slightly different classification.
Here, there are two non-Wecken surfaces for which the difference remains bounded.
This talk considers this same problem, but now in the setting of coincidences for a pair of
maps using the corresponding Nielsen numbers for coincidence. The results obtained are joint
work with Leticia Silva and Joao Vieira (UNESP-Rio Claro, Brasil). We focus on the boundary
preserving case and as a result produce a class of pairs of bounded surfaces which satisfy the
Wecken property.
384
