Ergodic theory of
meromorphic maps The
theory of multifractal formalism, developed in the
1970s, provides a number of powerful tools for
studying the geometry and ergodic properties of
invariant hyperbolic repellers in the Julia set of
rational functions on the Riemann sphere. The projects
concerns extending the theory to the setup of
transcendental entire and meromorphic maps. Within the
previous work on the project, we have proved, among
others, the Bowen formula for transcendental maps with
a finite number of singular values (Ergodic
Theory Dynam. Systems 32 (2012), no. 4, 11651189).
This
is a joint project with B. Karpińska (Warsaw
University of Technology) and A. Zdunik
(Uniwersity of Warsaw).
In particular, the research was carried out within
the Polish MNiSW / NCN grants N N201 0234 33 and N
N201 607940.


Topological properties
of invariant sets in the dynamics of
transcendental functions The project
concerns the study of iteration of transcendental entire and meromorphic maps
on the complex plane. We are interested in the
relations between the dynamics of the map and the
topology and geometry of invariant subsets, in
particular the Julia set and the escaping set. Within the previous work on
the project, we have proved an open
conjecture concerning the connectivity of the
Julia sets of Newton's method of finding zeroes of
transcendental entire maps (Invent.
Math. 198 (2014), no. 3, 591636).
This is a joint
project with N. Fagella (Universitat de
Barcelona), X. Jarque
(Universitat de Barcelona) and
B. Karpińska (Warsaw University of
Technology),
carried out within the Polish NCN grant
HARMONIA 2012/06/M/ST1/00168.


Dimension of the
graphs of the Weierstrasstype
functions We study the
ergodic properties of the graphs of
nondifferentiable real functions of the
Weierstrass type. They are examples of invariant
repellers of hyperbolic systems with two different
positive Lyapunov exponents, i.e. two different
speeds of expansion in unstable directions. They
can also be regarded as limit sets (attractors) of
some iterated function systems (IFS). Within the previous work
on the project, we have proved
a wellknown open conjecture concerning
the Hausdorff dimension
of such graphs, stated by Mandelbrot in the
1980s (Adv.
Math. 265 (2014), 3259). This
is a joint project with B. Bárány
(Budapest University of Technology and
Economics / Univerity of Warwick) and
J. Romanowska (Uniwersity of
Warsaw).
