For large , we consider the ordinary continued fraction of =/ with 1≤≤≤, or, equivalently, Euclid’s gcd algorithm for two integers 1≤≤≤, putting the uniform distribution on the set of and s. We study the distribution of the total cost of execution of the algorithm for an additive cost function on the set ℤ
of possible digits, asymptotically for →∞. If is nonlattice and satisfies mild growth conditions, the local limit theorem was proved previously by the second named author. Introducing...
We consider families of unimodal maps whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure of depends differentiably on , as a distribution of order . The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of for a Benedicks-Carleson map , in terms of a single smooth function and the inverse branches...
We study spectral properties of transfer operators for diffeomorphisms on a Riemannian manifold . Suppose that is an isolated hyperbolic subset for , with a compact isolating neighborhood . We first introduce Banach spaces of distributions supported on , which are anisotropic versions of the usual space of functions and of the generalized Sobolev spaces , respectively. We then show that the transfer operators associated to and a smooth weight extend boundedly to these spaces, and...
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