Line, spiral, dense

Exponential of exponential of almost every line in the complex plane is dense in the plane. On the other hand, for lines through any point, for a set of angles of Hausdorff dimension one, exponential of exponential of a line with angle from that set is not dense in the plane.Comment: 14 p, 4 figure

Quasistatic dynamical systems

We introduce the notion of a quasistatic dynamical system, which generalizes that of an ordinary dynamical system. Quasistatic dynamical systems are inspired by the namesake processes in thermodynamics, which are idealized processes where the observed system transforms (infinitesimally) slowly due to external influence, tracing out a continuous path of thermodynamic equilibria over an (infinitely) long time span. Time-evolution of states under a quasistatic dynamical system is entirely deterministic, but choosing the initial state randomly renders the process a stochastic one. In the prototypical setting where the time-evolution is specified by strongly chaotic maps on the circle, we obtain a description of the statistical behaviour as a stochastic diffusion process, under surprisingly mild conditions on the initial distribution, by solving a well-posed martingale problem. We also consider various admissible ways of centering the process, with the curious conclusion that the "obvious" centering suggested by the initial distribution sometimes fails to yield the expected diffusion.Comment: 40 page

Non-existence of absolutely continuous invariant probabilities for exponential maps

We show that for entire maps of the form $z \mapsto \lambda \exp(z)$ such that the orbit of zero is bounded and such that Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This answers a long-standing open problem.Comment: 4 pages. Similar to the version published in Fundamenta in February 200

Rallying Round Our Liberty

https://mds.marshall.edu/music_rec/1001/thumbnail.jp

Free energy and equilibrium states for families of interval maps

Funding: MT was partially supported by FCT grant SFRH/BPD/26521/2006 and NSF grants DMS0606343 and DMS 0908093. ND was supported by ERC Bridges project, the Academy of Finland CoE in Analysis and Dynamics Research and an IBM Goldstine fellowship.We study continuity, and lack thereof, of thermodynamical properties for one-dimensional dynamical systems. Under quite general hypotheses, the free energy is shown to be almost upper-semicontinuous: some normalised component of a limit measure will have free energy at least that of the limit of the free energies. From this, we deduce results concerning existence and continuity of equilibrium states (statistical stability). Counterexamples to statistical stability in the absence of strong hypotheses are provided.PostprintPeer reviewe