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1. Renormalization for Lorenz maps of monotone combinatorial types

Gaidashev, Denis

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Applied Mathematics and Statistics.

Lorenz maps are maps of the unit interval with one critical point of order rho > 1 and a discontinuity at that point. They appear as return maps of sections of the geometric Lorenz flow. We construct real a priori bounds for renormalizable Lorenz maps with certain monotone combinatorics and a sufficiently flat critical point, and use these bounds to show existence of periodic points of renormalization, as well as existence of Cantor attractors for dynamics of infinitely renormalizable Lorenz maps.

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.

It has been observed that the famous Feigenbaum–Coullet–Tresser period-doubling universality has a counterpart for area-preserving maps of ℝ^{2}. A renormalization approach has been used in a computer-assisted proof of existence of an area-preserving map with orbits of all binary periods in Eckmann et al [Existence of a fixed point of the doubling transformation for area-preserving maps of the plane. Phys. Rev. A26(1) (1982), 720–722; A computer-assisted proof of universality for area-preserving maps. Mem. Amer. Math. Soc.47 (1984), 1–121]. As is the case with all non-trivial universality problems in non-dissipative systems in dimensions more than one, no analytic proof of this period-doubling universality exists to date. We argue that the period-doubling renormalization fixed point for area-preserving maps is almost one dimensional, in the sense that it is close to the following Hénon-like (after a coordinate change) map:

where ϕ solves We then give a ‘proof’ of existence of solutions of small analytic perturbations of this one-dimensional problem, and describe some of the properties of this solution. The ‘proof’ consists of an analytic argument for factorized inverse branches of ϕ together with verification of several inequalities and inclusions of subsets of ℂ numerically. Finally, we suggest an analytic approach to the full period-doubling problem for area-preserving maps based on its proximity to the one-dimensional case. In this respect, the paper is an exploration of possible analytic machinery for a non-trivial renormalization problem in a conservative two-dimensional system.

We study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random V-variable and homogeneous Markov constructions.

4. Square Summability of Variations and Convergence of the Transfer OperatorJohansson, Anders

et al.

Öberg, Anders

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Applied Mathematics.

In this paper we study the one-sided shift operator on a state space defined by a finite alphabet. Using a scheme developed by Walters [P. Walters. Trans. Amer Math. Soc. 353(l) (2001), 327-347], we prove that the sequence of iterates of the transfer operator converges under square summability of variations of the g-function, a condition which gave uniqueness of a g-measure in our earlier work [A. Johansson and A. Oberg. Math. Res. Lett. 10(5-6) (2003), 587-601]. We also prove uniqueness of the so-called G-measures, introduced by Brown and Dooley [G. Brown and A. H. Dooley. Ergod. Th. & Dynam. Sys. 11 (1991), 279-307], under square summability of variations.

We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of g-measures. Indag. Math. (N.S.) 22 (2011), 315-329] to a weaker condition, Berbee's condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a g-measure with a continuous g-function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.

6. Random affine code tree fractals and Falconer-Sloan condition

Järvenpää, Esa

et al.

Univ Oulu, Dept Math Sci, POB 3000, Oulu 90014, Finland..

Järvenpää, Maarit

Univ Oulu, Dept Math Sci, POB 3000, Oulu 90014, Finland..

Li, Bing

Univ Oulu, Dept Math Sci, POB 3000, Oulu 90014, Finland.;South China Univ Technol, Dept Math, Guangzhou 510641, Guangdong, Peoples R China..

Stenflo, Örjan

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.

We calculate the almost sure dimension for a general class of random affine code tree fractals in R-d. The result is based on a probabilistic version of the Falconer-Sloan condition C(s) introduced in Falconer and Sloan [Continuity of subadditive pressure for self-affine sets. Real Anal. Exchange 34 (2009), 413-427]. We verify that, in general, systems having a small number of maps do not satisfy condition C(s). However, there exists a natural number n such that for typical systems the family of all iterates up to level n satisfies condition C(s).