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We consider stochastic dynamical systems on ${\mathbb{R}}$, that is, random processes defined by $X_n^x=\Psi_n(X_{n-1}^x)$, $X_0^x=x$, where $\Psi _n$ are i.i.d. random continuous transformations of some unbounded closed subset of ${\mathbb{R}}$. We assume here that $\Psi_n$ behaves asymptotically like $A_nx$, for some random positive number $A_n$ [the main example is the affine stochastic recursion $\Psi_n(x)=A_nx+B_n$]. Our aim is to describe invariant Radon measures of the process $X_n^x$ in the critical case, when ${\mathbb{E}}\log A_1=0$. We prove that those measures behave at infinity like $\frac{dx}{x}$. We study also the problem of uniqueness of the invariant measure. We improve previous results known for the affine recursions and generalize them to a larger class of stochastic dynamical systems which include, for instance, reflected random walks, stochastic dynamical systems on the unit interval $[0,1]$, additive Markov processes and a variant of the Galton--Watson process., Comment: Published at http://dx.doi.org/10.1214/13-AOP903 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We evaluate the scale at which the multifractal structure of some random Gibbs measures becomes discernible. The value of this scale is obtained through what we call the growth speed in H\"older singularity sets of a Borel measure. This growth speed yields new information on the multifractal behavior of the rescaled copies involved in the structure of statistically self-similar Gibbs measures. Our results are useful to understand the multifractal nature of various heterogeneous jump processes.