Abstract. In this paper we study the asymptotic behaviour of the escape rate of a Gibbs measure supported on a conformal repeller through a small hole. There are additional applications to the convergence of the Hausdorff dimension of the survivor set. IntroductionGiven any transformation T : X → X preserving an ergodic probability measure µ and any Borel set A ⊂ X , the escape rate quantifies the asymptotic behaviour of the measure of the set of points x ∈ X for which none of the first n terms in the orbit intersects U . Bunimovich and Yurchenko [7] considered the fundamental case of the doubling map and Haar measure, and where U is a dyadic interval. Subsequently, Keller and Liverani [16] proved a more general perturbation result which was then used to show, amongst other things, that a similar formula holds in the case that T is an expanding interval map and µ the absolutely continuous invariant probability measure. Other papers regarding this topic include [1,6,9,19] and references therein.In this paper, we prove analogous results in the more general setting of Gibbs measures supported on conformal repellers. Much of the analysis is undertaken in the setting of subshifts of finite type; this not only allows us to prove similar results for a broad class of maps which can be modelled symbolically but also improve on the work of Lind [18] who considered the convergence of topological entropy for a topologically mixing subshift.Another interesting aspect of our analysis is the connection with the work of Hirata [12] on the exponential law for first return times for Axiom A diffeomorphisms. Some of the ingredients in our approach were suggested by Hirata's paper, although we had to significantly modify the actual details.Let M be a Riemannian manifold and f : M → M a C 1 -map. Let J be a compact subset of M such that f (J ) = J . We say that the pair (J, f ) is a conformal repeller if: (1) f | J is a conformal map; (2) there exist c > 0 and λ > 1 such that d f n x v ≥ cλ n v for all x ∈ J , v ∈ T x M, and n ≥ 1; (3) f is topologically mixing on J ;

Abstract. We study the scaling scenery and limit geometry of invariant measures for the non-conformal toral endomorphism (x, y) → (mx mod 1, ny mod 1) that are Bernoulli measures for the natural Markov partition. We show that the statistics of the scaling can be described by an ergodic CP-chain in the sense of Furstenberg. Invoking the machinery of CP-chains yields a projection theorem for Bernoulli measures, which generalises in part earlier results by Hochman-Shmerkin and Ferguson-Jordan-Shmerkin. We also give an ergodic theoretic criterion for the dimension part of Falconer's distance set conjecture for general sets with positive length using CP-chains and hence verify it for various classes of fractals such as self-affine carpets of Bedford-McMullen, Lalley-Gatzouras and Barański class and all planar self-similar sets.

Abstract. We study the orthogonal projections of a large class of self-affine carpets, which contains the carpets of Bedford and McMullen as special cases. Our main result is that if Λ is such a carpet, and certain natural irrationality conditions hold, then every orthogonal projection of Λ in a non-principal direction has Hausdorff dimension min(γ, 1), where γ is the Hausdorff dimension of Λ. This generalizes a recent result of Peres and Shmerkin on sums of Cantor sets.

Abstract. We investigate the dimension of intersections of the Sierpiński gasket with lines. Our first main result describes a countable, dense set of angles that are exceptional for Marstrand's theorem. We then provide a multifractal analysis for the set of points in the projection for which the associated slice has a prescribed dimension.

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Abstract. In this paper we compute the dimension of a class of dynamically defined nonconformal sets. Let X ⊆ T 2 denote a Bedford-McMullen set and T : X → X the natural expanding toral endomorphism which leaves X invariant. For an open set U ⊂ X we letWe investigate the box and Hausdorff dimensions of X U for both a fixed Markov hole and also when U is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through U , while the Hausdorff dimension depends on the escape rate of the measure of maximal dimension.

One-block quantifier elimination is comprised of computing a semi-algebraic description of the projection of a semi-algebraic set or of deciding the truth of a semi-algebraic formula with a single quantifier.

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