Spiraling of approximations and spherical averages of Siegel transforms

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.We consider the question of how approximations satisfying Dirichlet’s theorem spiral around vectors in Rd. We give pointwise almost everywhere results (using only the Birkhoff ergodic theorem on the space of lattices). In addition, we show that for every unimodular lattice, on average, the directions of approximates spiral in a uniformly distributed fashion on the d − 1 dimensional unit sphere. For this second result, we adapt a very recent proof of Marklof and Strombergsson [19] to show a spherical average result for Siegel transforms on SLd+1(R)/ SLd+1(Z). Our techniques are elementary. Results like this date back to the work of Eskin-Margulis-Mozes [9] and KleinbockMargulis [14] and have wide-ranging applications. We also explicitly construct examples in which the directions are not uniformly distributedJ.S.A. partially supported by NSF grant DMS 1069153, and NSF grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). A.G. partially supported by the Royal Society. J.T. acknowledges the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 291147

Decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors

We consider two dimensional maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium (iterates of Lebesgue measure converge exponentially fast to physical measure). We prove that these maps have exponential decay of correlations over a large class of observables. We use this result to deduce exponential decay of correlations for the Poincare maps of a large class of singular hyperbolic flows. From this we deduce logarithm laws for these flows.Comment: 39 pages; 03 figures; proof of Theorem 1 corrected; many typos corrected; improvements on the statements and comments suggested by a referee. Keywords: singular flows, singular-hyperbolic attractor, exponential decay of correlations, exact dimensionality, logarithm la

Cancer recurrence times from a branching process model

As cancer advances, cells often spread from the primary tumor to other parts of the body and form metastases. This is the main cause of cancer related mortality. Here we investigate a conceptually simple model of metastasis formation where metastatic lesions are initiated at a rate which depends on the size of the primary tumor. The evolution of each metastasis is described as an independent branching process. We assume that the primary tumor is resected at a given size and study the earliest time at which any metastasis reaches a minimal detectable size. The parameters of our model are estimated independently for breast, colorectal, headneck, lung and prostate cancers. We use these estimates to compare predictions from our model with values reported in clinical literature. For some cancer types, we find a remarkably wide range of resection sizes such that metastases are very likely to be present, but none of them are detectable. Our model predicts that only very early resections can prevent recurrence, and that small delays in the time of surgery can significantly increase the recurrence probability.Comment: 26 pages, 9 figures, 4 table