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Register a new userOn Coupling Lemma and Stochastic Properties with Unbounded Observables for 1-d Expanding Maps
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In this paper, we establish a coupling lemma for standard families in the setting of piecewise expanding interval maps with countably many branches. Our method merely requires that the expanding map satisfies Chernovs one-step expansion at $q$-scale and eventually covers a magnet interval. Therefore, our approach is particularly powerful for maps whose inverse Jacobian has low regularity and those who does not satisfy the big image property. The main ingredients of our coupling method are two crucial lemmas: the growth lemma in terms of the characteristic $cZ$ function and the covering ratio lemma over the magnet interval. We first prove the existence of an absolutely continuous invariant measure. What is more important, we further show that the growth lemma enables the liftablity of the Lebesgue measure to the associated Hofbauer tower, and the resulting invariant measure on the tower admits a decomposition of Pesin-Sinai type. Furthermore, we obtain the exponential decay of correlations and the almost sure invariance principle (which is a functional version of the central limit theorem). For the first time, we are able to make a direct relation between the mixing rates and the $cZ$ function, see (ref{equ:totalvariation1}). The novelty of our results relies on establishing the regularity of invariant density, as well as verifying the stochastic properties for a large class of unbounded observables. Finally, we verify our assumptions for several well known examples that were previously studied in the literature, and unify results to these examples in our framework.
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We consider families of dynamics that can be described in terms of Perron-Frobenius operators with exponential mixing properties. For piecewise C^2 expanding interval maps we rigorously prove continuity properties of the drift J(l) and of the diffusion coefficient D(l) under parameter variation. Our main result is that D(l) has a modulus of continuity of order O(|dl||log|dl|)^2), i.e. D(l) is Lipschitz continuous up to quadratic logarithmic corrections. For a special class of piecewise linear maps we provide more precise estimates at specific parameter values. Our analytical findings are verified numerically for the latter class of maps by using exact formulas for the transport coefficients. We numerically observe strong local variations of all continuity properties.
For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so called quantum limits, and one would like to understand which invariant measures can occur that way, thereby classifying the semiclassical behaviour of eigenfunctions. We introduce a class of maps on the torus for whose quantisations we can understand the set of quantum limits in great detail. In particular we can construct examples of ergodic maps which have singular ergodic measures as quantum limits, and examples of non-ergodic maps where arbitrary convex combinations of absolutely continuous ergodic measures can occur as quantum limits. The maps we quantise are obtained by cutting and stacking.
The goal of this paper is to construct invariant dynamical objects for a (not necessarily invertible) smooth self map of a compact manifold. We prove a result that takes advantage of differences in rates of expansion in the terms of a sheaf cohomolog
ical long exact sequence to create unique lifts of finite dimensional invariant subspaces of one term of the sequence to invariant subspaces of the preceding term. This allows us to take invariant cohomological classes and under the right circumstances construct unique currents of a given type, including unique measures of a given type, that represent those classes and are invariant under pullback. A dynamically interesting self map may have a plethora of invariant measures, so the uniquess of the constructed currents is important. It means that if local growth is not too big compared to the growth rate of the cohomological class then the expanding cohomological class gives sufficient marching orders to the system to prohibit the formation of any other such invariant current of the same type (say from some local dynamical subsystem). Because we use subsheaves of the sheaf of currents we give conditions under which a subsheaf will have the same cohomology as the sheaf containing it. Using a smoothing argument this allows us to show that the sheaf cohomology of the currents under consideration can be canonically identified with the deRham cohomology groups. Our main theorem can be applied in both the smooth and holomorphic setting.
We show that for any $lambda in mathbb{C}$ with $|lambda|<1$ there exists an analytic expanding circle map such that the eigenvalues of the associated transfer operator (acting on holomorphic functions) are precisely the nonnegative powers of $lambda$ and $bar{lambda}$. As a consequence we obtain a counterexample to a variant of a conjecture of Mayer on the reality of spectra of transfer operators.
We study the topological properties of attractors of Iterated Function Systems (I.F.S.) on the real line, consisting of affine maps of homogeneous contraction ratio. These maps define what we call a second generation I.F.S.: they are uncountably many and the set of their fixed points is a Cantor set. We prove that when this latter either is the attractor of a finite, non-singular, hyperbolic, I.F.S. (of first generation), or it possesses a particular dissection property, the attractor of the second generation I.F.S. consists of finitely many closed intervals.
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