INVARIANT MEASURES THAT ARE ABSOLUTELY CONTINUOUS INdxFOR ONE-PARAMETER FAMILIES OF ONE-DIMENSIONAL MAPS

Abstract

We express stress tensor correlators using the Schwinger-Keldysh formalism. The absence of off-diagonal counterterms in this formalism ensures that the +− and −+ correlators are free of primitive divergences. We use dimensional regularization in position space to explicitly check this at one loop order for a massless scalar on a flat space background. We use the same procedure to show that the ++ correlator contains the divergences first computed by 't Hooft and Veltman for the scalar contribution to the graviton self-energy.

Bifurcation frequency for unimodal maps

“…This section is dedicated to proving Theorem B. As in Yakobson [8], Collet and Eckmann [1], and Guckenheimer [2] we will define an inductive process. In general it is necessary to eliminate parameters to assure metric control throughout the induction.…”

“…For a qualitative understanding of the dynamical complexity of interval maps, it is important to know how frequent the axiom A case is among the members of an one-parameter family. Many other similar questions have been asked and some of them answered; we mention Yakobson [8], Collet and Eckmann [1], and Guckenheimer [2]. Also, Newhouse and Palis [5] and Palis and Takens [6] have answered similar questions in the context of diffeomorphisms of a compact surface.…”

“…We have two cases depending on the topological structure of the Julia set Σ 0 of/o (the complement of the basin of the periodic sinks of/ 0 ): if Σ 0 does not contain intervals, among other facts, it follows from our theorems that f μ is axiom A with frequency one at μ = 0. If Σ 0 contains intervals, it follows from Yakobson [8] that f μ has an absolutely continuous ergodic invariant probability measure with frequency one at μ = 0.…”