30,002 research outputs found
Rigidity of critical circle mappings, I
We prove that two critical circle maps with the same rotation number of
bounded type are conjugate for some provided their
successive renormalizations converge together at an exponential rate in the
sense. The number depends only on the rate of convergence. We
also give examples of critical circle maps with the same rotation
number that are not conjugate for any
On Sloane's persistence problem
We investigate the so-called persistence problem of Sloane, exploiting
connections with the dynamics of circle maps and the ergodic theory of
actions. We also formulate a conjecture concerning the
asymptotic distribution of digits in long products of finitely many primes
whose truth would, in particular, solve the persistence problem. The heuristics
that we propose to complement our numerical studies can be thought in terms of
a simple model in statistical mechanics.Comment: 5 figure
Global hyperbolicity of renormalization for C^r unimodal mappings
In this paper we extend M. Lyubich's recent results on the global
hyperbolicity of renormalization of quadratic-like germs to the space of C^r
unimodal maps with quadratic critical point. We show that in this space the
bounded-type limit sets of the renormalization operator have an invariant
hyperbolic structure provided r ge 2+ alpha with alpha close to one. As an
intermediate step between Lyubich's results and ours, we prove that the
renormalization operator is hyperbolic in a Banach space of real analytic maps.
We construct the local stable manifolds and prove that they form a continuous
lamination whose leaves are C^1 codimension one, Banach submanifolds of the
ambient space, and whose holonomy is C^{1+\beta} for some beta >0. We also
prove that the global stable sets are C^1 immersed (codimension one)
submanifolds as well, provided r ge 3+ alpha with alpha close to one. As a
corollary, we deduce that in generic, one-parameter families of C^r unimodal
maps, the set of parameters corresponding to infinitely renormalizable maps of
bounded combinatorial type is a Cantor set with Hausdorff dimension less than
one.Comment: 94 pages, published versio
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