Abstract. 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 wellposed 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.
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. 1 Definition 3. We denote by M(f ) the collection of ergodic, f -invariant probability measures.Definition 4. The Lyapunov exponent of a measure µ ∈ M(f ) is defined as χ µ := log |Df |dµ. The entropy of µ we denote by h µ .Of course −∞ ≤ χ µ < ∞. If µ is not supported on a periodic attractor, then χ µ ≥ 0 by [11].Definition 5. We denote by M + (f ) the collection of µ ∈ M(f ) for which χ µ > 0. Definition 6. We call a (φ, t)-conformal measure m exceptional if Supp(m) = J , and call it non-exceptional if it admits no restriction which, when normalised, is an exceptional conformal measure.Remark: If t ≥ 0 then m is non-exceptional by the eventually onto property of rational maps. Moreover, unless we are in a special case where there is a finite Proof. We know already p ≥ P (t). By Theorem 8, p = h µ − tχ µ , thus p ≤ P (t).Corollary 10. Let f be a rational map of the Riemann sphere, t ≥ 0, let m be a (P (t), t)-conformal measure and let µ ∈ M + (f ). Suppose h µ − tχ µ = P (t). Then µ is equivalent to m. In particular, µ is the unique equilibrium state with positive Lyapunov exponent for the potential −t log |Df | and m is ergodic.Remark: Przytycki ([12]) conjectured that, for t equal to the first zero of the pressure function, there was a unique equilibrium state with positive exponent. For that particular value of t, the answer can be shown using the work of Ledrappier ([5]). Rephrasing the previous corollary, we answer a stronger question:Corollary 11. For each t ≥ 0, there is at most one equilibrium state with positive Lyapunov exponent for the potential −t log |Df |.Remark: Provided P (t) = 0, any equilibrium state for the potential −t log |Df | has positive Lyapunov exponent.
Non-invertible Pesin theory is developed for a class of piecewise smooth interval maps which may have unbounded derivative, but satisfy a property analogous to C 1+ǫ . The critical points are not required to verify a non-flatness condition, so the results are applicable to C 1+ǫ maps with flat critical points. If the critical points are too flat, then no absolutely continuous invariant probability measure can exist. This generalises a result of Benedicks and Misiurewicz.
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 andŚwiatek, so for a certain class of meromorphic maps the absolutely continuous invariant measure is finite if and only if an integrability condition is satisfied.
The Joint Replenishment Problem (JRP) is a fundamental optimization problem in supplychain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers' waiting costs.We study the approximability of JRP-D, the version of JRP with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of 1.207, a stronger, computerassisted lower bound of 1.245, as well as an upper bound and approximation ratio of 1.574. The best previous upper bound and approximation ratio was 1.667; no lower bound was previously published. For the special case when all demand periods are of equal length we give an upper bound of 1.5, a lower bound of 1.2, and show APX-hardness.
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