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**The periodic points of renormalization.**
*(English)*
Zbl 0936.37017

In several places in the theory of dynamical systems one finds universal scalings; quantitative scaling behavior that is independent of details of the system. The most familiar case appears in cascades of homoclinic doubling bifurcations in unimodal interval maps, where both the sequence of period doubling bifurcation points on the parameter line and the structure of the invariant Cantor set that exists at the accumulation point of the cascade, has universal scaling properties. An explanation of this phenomenon was given independently by Coullet-Tresser and Feigenbaum. They introduced a renormalization operator (which rescales a second iterate of the interval map restricted to a subinterval around the critical point) and could explain the universality from the existence of a hyperbolic fixed point for this operator, possessing one unstable direction.

To prove the existence of the hyperbolic fixed point, turned out to be difficult. That it exists was shown in the class of analytic unimodal maps by O. E. Lanford III in the early eighties, using numerical approximations. Since the early nineties an analytical theory, covering universality in unimodal dynamics in a much broader context than period doubling cascades, was developed by D. Sullivan and further by C. McMullen and M. Lyubich. This theory uses complex analytic methods.

In the well written article under review, the author proves the existence (but not hyperbolicity) of fixed and periodic points of renormalization operators for unimodal interval maps \(f\) of the form \(f = \phi \circ q_t\) with \(\phi\) a \(C^2\) diffeomorphism and \(q_t(x) = -2 t |x|^\alpha + 2t - 1\) for any critical order \(\alpha >1\). His approach relies on estimates of distortion bounds, using only real analysis, and a topological fixed point theorem. Instead of working in the class of unimodal maps, the author works with ‘decomposed unimodal maps’ consisting of maps arranged in a binary tree which are to be composed to provide a unimodal map. It is on this space of decomposed unimodal maps that renormalization is studied.

To prove the existence of the hyperbolic fixed point, turned out to be difficult. That it exists was shown in the class of analytic unimodal maps by O. E. Lanford III in the early eighties, using numerical approximations. Since the early nineties an analytical theory, covering universality in unimodal dynamics in a much broader context than period doubling cascades, was developed by D. Sullivan and further by C. McMullen and M. Lyubich. This theory uses complex analytic methods.

In the well written article under review, the author proves the existence (but not hyperbolicity) of fixed and periodic points of renormalization operators for unimodal interval maps \(f\) of the form \(f = \phi \circ q_t\) with \(\phi\) a \(C^2\) diffeomorphism and \(q_t(x) = -2 t |x|^\alpha + 2t - 1\) for any critical order \(\alpha >1\). His approach relies on estimates of distortion bounds, using only real analysis, and a topological fixed point theorem. Instead of working in the class of unimodal maps, the author works with ‘decomposed unimodal maps’ consisting of maps arranged in a binary tree which are to be composed to provide a unimodal map. It is on this space of decomposed unimodal maps that renormalization is studied.

Reviewer: Ale Jan Homburg (Utrecht)

### MSC:

37E20 | Universality and renormalization of dynamical systems |

37F25 | Renormalization of holomorphic dynamical systems |

37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |

37E05 | Dynamical systems involving maps of the interval |