Abstract
We prove that a $C^3$ critical circle map without periodic points has zero Lyapunov exponentwith respect to its unique invariant Borel probability measure. Moreover, no critical point of such a mapsatisfies the Collet-Eckmann condition. This result is proved directly from the well-known real a-priori bounds,without using Pesin's theory. We also show how our methods yield an analogous result for infinitely renormalizableunimodal maps of any combinatorial type. Finally we discuss an application of these facts to the study of neutralmeasures of certain rational maps of the Riemann sphere.
Highlights
This paper studies critical circle maps from the differentiable ergodic theory viewpoint
In+1 ⊂ f qn+1 (In) this article we show that the Lyapunov exponent of a C3 critical circle map is always zero
The general approach leading to zero Lyapunov exponents is by arguing by contradiction and using Pesin’s theory: non-zero Lyapunov exponent implies the existence of periodic orbits, and that would be a contradiction for critical circle maps with irrational rotation number
Summary
This paper studies critical circle maps (as well as infinitely renormalizable unimodal maps) from the differentiable ergodic theory viewpoint. The geometric rigidity and renormalization aspects of the theory remain open for maps with more than one critical point, see Question 7.3 Such brief account bypasses important numerical studies by several physicists, as well as computer-assisted and conceptual work by Feigenbaum, Kadanoff, Lanford, Rand, Epstein and others; see [10] and references therein.
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