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.

We generalize to the case of finitely generated groups of homeomorphisms the notion of a local measure entropy introduced by Brin and Katok [7] for a single map. We apply the theory of dimensional type characteristics of a dynamical system elaborated by Pesin [25] to obtain a relationship between the topological entropy of a pseudogroup and a group of homeomorphisms of a metric space, defined by Ghys, Langevin and Walczak in [12], and its local measure entropies. We prove an analogue of the Variational...

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)].

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.

We deal with locally connected exceptional minimal sets of surface homeomorphisms. If the
surface is different from the torus, such a minimal set is either finite or a finite
disjoint union of simple closed curves. On the torus, such a set can admit also a
structure similar to that of the Sierpiński curve.

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