Abstract In this paper we study a family of limsup sets that are defined using iterated function systems. Our main result is an analogue Khintchine’s theorem for these sets. We then apply to the topic intrinsic Diophantine Approximation on self-similar particular, define new height element $${\mathbb {Q}}^d$$ <mml:m...

Iterated Function Systems (IFS) are interesting parametric models for generating fractal sets and functions. The general idea is to compress, deform and translate a given set or function with a collection of operators and to iterate the procedure. Under weak assumptions, IFS possess a unique fixed point which is in general fractal. IFS were introduced in a deterministic context, then were gener...

Nonlinear systems, defined by a system of ODE’s, usually contain nonlinear terms. Under some circumstances these terms can be locally approximated by linear factors and by discretization can be transformed in the sequence of (hyperbolic) Iterated Function Systems (IFS). Then, this IFS generate a unique attractor that uses as a kind of characteristic set that reflects the dynamics in the vicinit...

Let {Si}i∈Λ be a finite contracting affine iterated function system (IFS) on Rd. (Σ,σ) denote the two-sided full shift over alphabet Λ, and let π:Σ→Rd coding map associated with IFS. We prove that projection of an ergodic σ-invariant measure Σ under π is always exact dimensional, its Hausdorff dimension satisfies Ledrappier–Young-type formula. Furthermore, result extends to average IFSs. This c...

Given a sequence of i.i.d. random functions Ψn:R→R, n∈N, we consider the iterated function system and Markov chain, which is recursively defined by X0x:=x Xnx:=Ψn−1(Xn−1x) for x∈R n∈N. Under two basic assumptions that Ψn are a.s. continuous at any point in R asymptotically linear “endpoints” ±∞, study tail behavior stationary laws such chains means renewal theory. Our approach provides an exten...