We establish a generalized thermodynamic formalism for certain nonhyperbolic maps with countably many preimages. We study existence and uniqueness of conformal measures and statistical properties of the equilibrium states absolutely continuous with respect to the conformal measures. We will see that such measures are not Gibbs but satisfy a version of Gibbs property (weak Gibbs measure). We apply our results to a one-parameter family of one-dimensional maps and a two-dimensional nonconformal map related to number theory. Both of them admit indifferent periodic points.

The paper is devoted to the following problem: \[ w'' (x) + c w'(x)+ F(w(x),x) = 0, \quad x\in{\mathbb R}^1,\quad w(\pm \infty) = w_{\pm}, \] where the non-linear term $F$ depends on the space variable $x$. A classification of non-linearities is given according to the behaviour of the function $F(w,x)$ in a neighbourhood of the points $w_+$ and $w_-$. The classical approach used in the Kolmogorov–Petrovsky–Piskunov paper [10] for an autonomous equation (where $F=F(u)$ does not explicitly depend on $x$), which is based on the geometric analysis on the $(w,w')$-plane, is extended and new methods are developed to analyse the existence and uniqueness of solutions in the non-autonomous case. In particular, we study the case where the function $F(w,x)$ does not have limits as $x \rightarrow \pm \infty$.