The Gauss map (or continued fraction map) is an important dissipative one-dimensional discrete-time dynamical system that exhibits chaotic behavior, and it generates a symbolic dynamics consisting of infinitely many different symbols.

Here we introduce a generalization of the Gauss map, which is given by x_{t+1}=1/x_{t}^{α}-[1/x_{t}^{α}] where α≥0 is a parameter and x_{t}∈[0,1] (t=0,1,2,3,…). The symbol [⋯] denotes the integer part.

This map reduces to the ordinary Gauss map for α=1. The system exhibits a sudden “jump into chaos” at the critical parameter value α=α_{c}≡0.241485141808811⋯ which we analyze in detail in this paper.

Several analytical and numerical results are established for this new map as a function of the parameter α. In particular, we show that, at the critical point, the invariant density approaches a q-Gaussian with q=2 (i.e., the Cauchy distribution), which becomes infinitely narrow as α→α_{c}^{+}.

Moreover, in the chaotic region for large values of the parameter α we analytically derive approximate formulas for the invariant density, by solving the corresponding Perron-Frobenius equation. For α→∞ the uniform density is approached.

We provide arguments that some features of this transition scenario are universal and are relevant for other, more general systems as well.

C. Beck, U. Tirnakli, C. Tsallis, Generalization of the Gauss map: A jump into chaos with universal features, Physical Review E 110(6) (2024) 064213.