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One-dimensional Bernoulli mapping with hole is suggested to describe theregularities of the appearance of a chaotic set under the saddle-node scenarioof the birth of the Smale--Williams hyperbolic attractor. In such a mapping, anon-trivial chaotic set (with non-zero Hausdorff dimension) arises in thegeneral case as a result of a cascade of period-adding bifurcationscharacterized by geometric scaling both in the phase space and in the parameterspace. Numerical analysis of the behavior of models demonstrating thesaddle-node scenario of birth of a hyperbolic chaotic Smale--Williams attractorshows that these regularities are preserved in the case of multidimensionalsystems. Limits of applicability of the approximate 1D model are discussed.

Bernoulli mapping with hole and a saddle-node scenario of the birth of hyperbolic Smale--Williams attractor