Conformal Fractals

In this chapter we shall consider a compact metric space X with an open, distance-expanding map T on it, embedded isometrically into a smooth Riemannian manifold M. We shall assume that T extends to a neighbourhood U of X to a mapping f of class C1+ε for some 0 < ε ≤ 1 or smoother, including real-analytic. C1+ε and more general Cr+ε for r = 1, 2, … means that the r-th derivative is Hölder continuous with the exponent ε for ε < 1 and Lipschitz continuous for ε = 1. We shall also assume that there exists a constant λ > 1 such that for every x ϵ U and for every non-zero vector v tangent to M at x, ∥Df(v)∥ > λ∥v∥ holds, where ∥·∥ is the norm induced by the Riemannian metric. The pair (X, f) will be called an expanding repeller and f an expanding map. If f is of some class A, e.g. Cα or analytic, we shall say that the expanding repeller is of that class, or that this is an A-expanding repeller. In particular, if f is conformal we call (X, f) a conformal expanding repeller, abbreviated to CER. Finally, if we skip the assumption that T = f∣x is open on X, we shall call (X, f) an expanding set. Sometimes, to distinguish the domain of f, we shall write (X, f, U).

In Sections 6.2 and 6.3 we provide some introduction to conformal expanding repellers, studying the transfer operator, postponing the main study to Chapters 9 and 10, where we shall use tools of geometric measure theory.