The paper is devoted to investigation of the statistical properties of skew-product dynamical systems \(F:[0,1]^2\circlearrowleft\) of the form \(F(x,y)=(T(x), G(x,y))\) with the following properties:

(1)

\(F\) is uniformly contracting on each fiber of the natural vertical foliation of the square \([0,1]^2\);

(2)

\(G\) has bounded variation in one direction;

(3)

\(T:[0,1]\circlearrowleft\) is piecewise monotonic with finite \(C^1\) increasing branches and \(\inf_{x\in[0,1]}| T'(x)|>1\), and \(1/T'\) has finite universal \(p\)-bounded variation;

(4)

\(T\) is weakly mixing under unique a.c.i.m.

For such systems the authors prove the existence of a unique physical measure with exponential decay of correlations with respect to Lipschitz observables. Then this result is applied to suitable Poincaré maps of singular hyperbolic (Lorenz-like) flows to deduce exponential decay of correlations and logarithm law for the hitting times to small targets.
See also the erratum to this paper in [ibid. 282, No. 1–2, 615–621 (2016; Zbl 06538600)].