Neil Dobbs' preprints/publications

and his short Curriculum Vitae, without the list of publications, from 2009 while at KTH.

My PhD thesis: Critical points, cusps and induced expansion in dimension one
Supervisor: Jacek Graczyk.
Defended on 11th December 2006, Orsay.

The first five of the following are on my Arxiv page.

• Nice sets and invariant densities in complex dynamics. Accepted for publication, Math. Proc. Camb. Phil. Soc.
  
• Measures with positive Lyapunov exponent and conformal measures in rational dynamics. Accepted for publication, Trans. AMS.
• On cusps and flat tops.
• Renormalisation-induced phase transitions for unimodal maps. Commun. Math. Phys. 286, 377-387 (2009)
• Non-existence of absolutely continuous invariant probabilities for exponential maps (Joint with Bartlomiej Skorulski). Fundamenta Mathematicae, 198(3):283-287, 2008
• Visible Measures of Maximal Entropy in Dimension One
Abstract: A real one-dimensional analogue of Zdunik's dichotomy is proven, giving conditions for a multimodal map to have a measure of maximal entropy of dimension one.
Published in the Bulletin of the London Mathematical Society. Bull. London Math. Soc. 2007 39: 366-376.

• Hyperbolic Dimension for Interval Maps
Abstract: The hyperbolic and Hausdorff dimensions are shown to coincide for $C^2$ maps without recurrent critical points. The maps may have parabolic periodic points. The Julia set for certain such maps may have hyperbolic dimension equal to 1 but Lebesgue measure equal to 0.
Published in Nonlinearity, Volume 19, Issue 12.

• Expanding Cocyles for Interval Maps
Abstract: We give a cocycle expansivity result for $C^2$ multimodal interval maps with non-flat critical points. It extends the Mané hyperbolicity theorem to also describe orbits which pass near critical points.
Cocycles dilatants pour des transformations de l'intervalle.
Résumé: On étend le théorème d'hyperbolicité de Mané pour traiter des orbites qui passent par des voisinages critiques pour des applications multimodales de l'intervalle. On démontre que, pour des cocycles bien adaptés, ces applications sont dilatantes.
Published in C. R. Math. Acad. Sci. Paris 345 (2007), no. 1, 39--44.