165 (American Mathematical Society, Providence, RI, pp. In: Concerning the Hilbert 16th problem American Mathematical Society Translation Series 2, vol.
Kotova, A., Stanzo, V.: On few-parameter genericfamilies of vector fields on the two-dimensional sphere. Turaev D.: Polynomial approximations of symplectic dynamics and richness of chaos in nonhyperbolic area-preserving maps.
Newhouse, S., Ruelle, D., Takens, F.: Occurrence of strange axiom A attractors near quasiperiodic flows on T m, m > 3. Ruelle D., Takens F.: On the nature of turbulence. In particular, every such universal map has an infinite set of coexisting hyperbolic attractors and repellers. As an application, we show that any C r-generic two-dimensional map that belongs to the Newhouse domain (i.e., it has a so-called wild hyperbolic set, so it is not uniformly-hyperbolic, nor uniformly partially-hyperbolic) and that neither contracts, nor expands areas, is C r-universal in the sense that its iterations, after an appropriate coordinate transformation, C r-approximate every orientation-preserving two-dimensional diffeomorphism arbitrarily well. In other words, any generic n-dimensional dynamical phenomenon can be obtained by iterations of C r-close to identity maps, with the same dimension of the phase space. We show that for any r ≥ 1 the renormalized iterations of C r-close to identity maps of an n-dimensional unit ball B n ( n ≥ 2) form a residual set among all orientation-preserving C r-diffeomorphisms B n→ R n. Given an n-dimensional C r-diffeomorphism g, its renormalized iteration is an iteration of g, restricted to a certain n-dimensional ball and taken in some C r-coordinates in which the ball acquires radius 1.
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