93 research outputs found
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 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
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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
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