We show that given $${n \geqslant 3}$$ , $${q \geqslant 1}$$ , and a finite set $${\{y_1, \ldots, y_q \}}$$ in $${\mathbb{R}^n}$$ there exists a quasiregular mapping $${\mathbb{R}^n\to \mathbb{R}^n}$$ omitting exactly points $${y_1, \ldots, y_q}$$ .

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The commonly used spatial entropy hr(U) of the multi-dimensional shift space U is the limit of growth rate of admissible local patterns on finite rectangular sublattices which expands to whole space Zd, d ≥ 2. This work studies spatial entropy hΩ(U) of shift space U on general expanding system Ω = {Ω(n)}∞ n=1 where Ω(n) is increasing finite sublattices and expands to Zd. Ω is called genuinely d-dimensional if Ω(n) contains no lower-dimensional part whose size is comparable to that of its d-dimensional part. We show that hr(U) is the supremum of hΩ(U) for all genuinely two-dimensional Ω. Furthermore, when Ω is genuinely d-dimensional and satisfies certain conditions, then hΩ(U) = hr(U). On the contrary, when Ω(n) contains a lower-dimensional part, then hr(U) < hΩ(U) for some U. Therefore, hr(U) is appropriate to be the d-dimensional spatial entropy.

Going beyond the linearized study has been a longstanding problem in the theory of Landau damping. In this paper we establish exponential Landau damping in analytic regularity. The damping phenomenon is reinterpreted in terms of transfer of regularity between kinetic and spatial variables, rather than exchanges of energy; phase mixing is the driving mechanism. The analysis involves new families of analytic norms, measuring regularity by comparison with solutions of the free transport equation; new functional inequalities; a control of non-linear echoes; sharp “deflection” estimates; and a Newton approximation scheme. Our results hold for any potential no more singular than Coulomb or Newton interaction; the limit cases are included with specific technical effort. As a side result, the stability of homogeneous equilibria of the non-linear Vlasov equation is established under sharp assumptions. We point out the strong analogy with the KAM theory, and discuss physical implications. Finally, we extend these results to some Gevrey (non-analytic) distribution functions.

We study infinite iterated functions systems (IIFSs) consisting of bi-Lipschitz mappings instead of conformal contractions, focusing on IFSs that do not satisfy the open set condition. By assuming the logarithmic distortion property and some cardinality growth condition, we obtain a formula for the Hausdorff, box, and packing dimensions of the limit set in terms of certain topological pressure. By assuming, in addition, the weak separation condition, we show that these dimensions are equal to the growth dimension of the limit set.