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We consider the limit set of generalised iterated function systems. Under the assumption of a natural potential, the so-called cylinder function, we prove the existence of the invariant probability measure satisfying the equilibrium state. We motivate this approach by showing that for typical self-affine sets there exists an ergodic invariant measure having… (More)

- ANTTI KÄENMÄKI
- 2004

We study how the Hausdorff measure is distributed in nonsymmetric narrow cones in Rn. As an application, we find an upper bound close to n−k for the Hausdorff dimension of sets with large k-porosity. With k-porous sets we mean sets which have holes in k different directions on every small scale.

- Antti Käenmäki
- 2003

We consider infinite conformal function systems on Rd. We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some l-dimensional C1-submanifold with positive Hausdorff t-dimensional measure, where 0 < l < d and t is the Hausdorff dimension of the limit set. We then show that the closure of the limit set belongs… (More)

- Antti Käenmäki
- 2017

We present an application of the recently developed ergodic theoretic machinery on scenery flows to a classical geometric measure theoretic problem in Euclidean spaces. We also review the enhancements to the theory required in our work. Our main result is a sharp version of the conical density theorem, which we reduce to a question on rectifiability.

- TAPIO RAJALA, Pekka Koskela, +9 authors Hannele Säntti
- 2009

- Antti Käenmäki, Tuomas Sahlsten, Pablo Shmerkin
- J. London Math. Society
- 2015

We expand the ergodic theory developed by Furstenberg and Hochman on dynamical systems that are obtained from magnifications of measures. We prove that any fractal distribution in the sense of Hochman is generated by a uniformly scaling measure, which provides a converse to a regularity theorem on the structure of distributions generated by the scenery… (More)

- ANTTI KÄENMÄKI
- 2016

We prove that generically, for a self-affine set in R, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question.

for μ-almost all points x ∈ R? Here the right-hand sides of (1.1) and (1.2) are denoted by dimloc(μ, x) and dimloc(μ, x), and they are the upper and lower local dimensions of the measure μ at x ∈ R, respectively. We prove that if C is a cone with opening angle at least π, then (1.1) and (1.2) hold for all measures μ and for μ-almost all x ∈ R. Moreover,… (More)

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