35,148 research outputs found
Non-uniform hyperbolicity and existence of absolutely continuous invariant measures
We prove that for certain partially hyperbolic skew-products, non-uniform
hyperbolicity along the leaves implies existence of a finite number of ergodic
absolutely continuous invariant probability measures which describe the
asymptotics of almost every point. The main technical tool is an extension for
sequences of maps of a result of de Melo and van Strien relating hyperbolicity
to recurrence properties of orbits. As a consequence of our main result, we
also obtain a partial extension of Keller's theorem guaranteeing the existence
of absolutely continuous invariant measures for non-uniformly hyperbolic one
dimensional maps.Comment: 24 page
Construction of the discrete hull for the combinatorics of a regular pentagonal tiling of the plane
The article 'A "regular" pentagonal tiling of the plane' by P. L. Bowers and
K. Stephenson defines a conformal pentagonal tiling. This is a tiling of the
plane with remarkable combinatorial and geometric properties. However, it
doesn't have finite local complexity in any usual sense, and therefore we
cannot study it with the usual tiling theory. The appeal of the tiling is that
all the tiles are conformally regular pentagons. But conformal maps are not
allowable under finite local complexity. On the other hand, the tiling can be
described completely by its combinatorial data, which rather automatically has
finite local complexity. In this paper we give a construction of the discrete
hull just from the combinatorial data. The main result of this paper is that
the discrete hull is a Cantor space
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