Let A be a topologically transitive invariant subset of an expanding piecewise monotonic map on [0,1] with the Darboux property. We investigate existence and uniqueness of conformal measures on A and relate Hausdorff and conformal measures on A to each other.
We consider the set of expanding maps of the circle which have a unique absolutely continuous invariant probability measure whose density is unbounded, and show that this set is dense in the space of expanding maps with the topology. This is in contrast with results for or maps, where the invariant densities can be shown to be continuous.
We consider piecewise monotonic and expanding transformations τ of a real interval (not necessarily bounded) into itself with countable number of points of discontinuity of τ’ and with some conditions on the variation which need not be a bounded function (although it is bounded on any compact interval). We prove that such transformations have absolutely continuous invariant measures. This result generalizes all previous “bounded variation” existence theorems.
For a class of quadratic polynomial endomorphisms close to the standard torus map , we show that the Julia set J(f) is homeomorphic to the torus. We identify J(f) as the closure ℛ of the set of repelling periodic points and as the Shilov boundary of the set K(f) of points with bounded forward orbit. Moreover, it turns out that (J(f),f) is a mixing repeller and supports a measure of maximal entropy for f which is uniquely determined as the harmonic measure for K(f).