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Partial variational principle for finitely generated groups of polynomial growth and some foliated spaces

100%
Biś A.
Colloquium Mathematicum
|
2008
|
tom 110
|
nr 2
431-449
EN
We generalize the notion of topological pressure to the case of a finitely generated group of continuous maps and introduce group measure entropy. Also, we provide an elementary proof that any finitely generated group of polynomial growth admits a group invariant measure and show that for a group of polynomial growth its measure entropy is less than or equal to its topological entropy. The dynamical properties of groups of polynomial growth are reflected in the dynamics of some foliated spaces.
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Entropy of distal groups, pseudogroups, foliations and laminations

63%
Biś A., Walczak P.
Annales Polonici Mathematici
|
2011
|
tom 100
|
nr 1
45-54
EN
A distality property for pseudogroups and foliations is defined. Distal foliated bundles satisfying some growth conditions are shown to have zero geometric entropy in the sense of É. Ghys, R. Langevin and P. Walczak [Acta Math. 160 (1988)].
3
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Geometry of Markov systems and codimension one foliations

63%
Biś A., Urbański M.
Annales Polonici Mathematici
|
2008
|
tom 94
|
nr 2
187-196
EN
We show that the theory of graph directed Markov systems can be used to study exceptional minimal sets of some foliated manifolds. A C¹ smooth embedding of a contracting or parabolic Markov system into the holonomy pseudogroup of a codimension one foliation allows us to describe in detail the h-dimensional Hausdorff and packing measures of the intersection of a complete transversal with exceptional minimal sets.
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