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result(s) for
"Dobbs, Neil"
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The company you keep
Jim Grant is a public interest lawyer and single father raising his daughter in the tranquil suburbs of Albany, New York. Grant's world is turned upside down when a brash young reporter exposes his true identity as a former 1970s antiwar radical fugitive wanted for murder.
DVD
Knobbly but nice
Our main result states that, under an exponential map whose Julia set is the whole complex plane, on each piecewise smooth Jordan curve there is a point whose orbit is dense. This has consequences for the boundaries of nice sets, used in induction methods to study ergodic and geometric properties of the dynamics.
Journal Article
Lattice-free sets, multi-branch split disjunctions, and mixed-integer programming
In this paper we study the relationship between valid inequalities for mixed-integer sets, lattice-free sets associated with these inequalities and the multi-branch split cuts introduced by Li and Richard (Discret Optim 5:724–734,
2008
). By analyzing
-dimensional lattice-free sets, we prove that for every integer
there exists a positive integer
such that every facet-defining inequality of the convex hull of a mixed-integer polyhedral set with
integer variables is a
-branch split cut. We use this result to give a finite cutting-plane algorithm to solve mixed-integer programs. We also show that the minimum value
, for which all facets of polyhedral mixed-integer sets with
integer variables can be generated as
-branch split cuts, grows exponentially with
. In particular, when
, we observe that not all facet-defining inequalities are 6-branch split cuts.
Journal Article
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 behavior 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.
Journal Article
Hausdorff Dimension of Julia Sets in the Logistic Family
A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the Mandelbrot set
M
, the Hausdorff dimension is generally discontinuous. Answering a question of J-C. Yoccoz in the conformal setting, we observe that the Hausdorff dimension of quadratic Julia sets depends continuously on
c
and find explicit bounds at the tip of
M
for most real parameters in the the sense of 1-dimensional Lebesgue measure.
Journal Article
Measures with positive Lyapunov exponent and conformal measures in rational dynamics
Ergodic properties of rational maps are studied, generalising the work of F. Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for an ergodic invariant probability measure with positive Lyapunov exponent to be absolutely continuous with respect to a general conformal measure. If they hold, we can construct an induced expanding Markov map with integrable return time which generates the invariant measure.
Journal Article
Free Energy and Equilibrium States for Families of Interval Maps
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 (including statistical stability). Metric entropy, not semicontinuous as a general multimodal map varies, is shown to be upper
semicontinuous under an appropriate hypothesis on critical orbits. Equilibrium states vary continuously, under mild hypotheses, as one
varies the parameter and the map. We give a general method for constructing induced maps which automatically give strong exponential
tail estimates. This also allows us to recover, and further generalise, recent results concerning statistical properties (decay of
correlations, etc.). Counterexamples to statistical stability are given which also show sharpness of the main results.
eBook
Nice sets and invariant densities in complex dynamics
In complex dynamics, we construct a so-called nice set (one for which the first return map is Markov) around any point which is in the Julia set but not in the post-singular set, adapting a construction of Rivera–Letelier. This simplifies the study of absolutely continuous invariant measures. We prove a converse to a recent theorem of Kotus and Świątek, so for a certain class of meromorphic maps the absolutely continuous invariant measure is finite if and only if an integrability condition is satisfied.
Journal Article
Outer billiard around a curvilinear triangle with a fixed diameter
We consider an outer billiard around a Reulaux triangle. We prove the existence of infinitely many periodic points accumulating at infinity. To do so we con- struct a return map from a strip into itself and we study its properties. We also show some numerical simulations which, in particular, display heteroclinic intersections and Smale's horseshoes.
Journal Article