7,541 research outputs found

    On the Plaque Expansivity Conjecture

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    It is one of the main properties of uniformly hyperbolic dynamics that points of two distinct trajectories cannot be uniformly close one to another. This characteristics of hyperbolic dynamics is called expansivity. Hirsch, Pugh and Shub, 1977, formulated the so-called Plaque Expansivity Conjecture, assuming that two invariant sequences of leaves of central manifolds, corresponding to a partially hyperbolic diffeomorphism, cannot be locally close. There are many important statements in the theory of partial hyperbolicity that can be proved provided Plaque Expansivity Conjecture holds true. Here we are proving this conjecture in its general form.Comment: The proof written here was wrong. I hope to replace this with a correct on

    On the omega-limit sets of tent maps

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    For a continuous map f on a compact metric space (X,d), a subset D of X is internally chain transitive if for every x and y in D and every delta > 0 there is a sequence of points {x=x_0,x_1, ...,x_n=y} such that d(f(x_i),x_{i+1}) < delta for i=0,1, ...,n-1. It is known that every omega-limit set is internally chain transitive; in earlier work it was shown that for X a shift of finite type, a closed subset D of X is internally chain transitive if and only if D is an omega-limit set for some point in X, and that the same is also true for the tent map with slope equal to 2. In this paper, we prove that for tent maps whose critical point c=1/2 is periodic, every closed, internally chain transitive set is necessarily an omega-limit set. Furthermore, we show that there are at least countably many tent maps with non-recurrent critical point for which there is a closed, internally chain transitive set which is not an omega-limit set. Together, these results lead us to conjecture that for those tent maps with shadowing (or pseudo-orbit tracing), the omega-limit sets are precisely those sets having internal chain transitivity.Comment: 17 page

    Seurat games on Stockmeyer graphs

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    We define a family of vertex colouring games played over a pair of graphs or digraphs (G, H) by players ∀ and ∃. These games arise from work on a longstanding open problem in algebraic logic. It is conjectured that there is a natural number n such that ∀ always has a winning strategy in the game with n colours whenever G 6∼= H. This is related to the reconstruction conjecture for graphs and the degree-associated reconstruction conjecture for digraphs. We show that the reconstruction conjecture implies our game conjecture with n = 3 for graphs, and the same is true for the degree-associated reconstruction conjecture and our conjecture for digraphs. We show (for any k < ω) that the 2-colour game can distinguish certain non-isomorphic pairs of graphs that cannot be distinguished by the k-dimensional Weisfeiler-Leman algorithm. We also show that the 2-colour game can distinguish the non-isomorphic pairs of graphs in the families defined by Stockmeyer as counterexamples to the original digraph reconstruction conjecture

    What is the true charge transfer gap in parent insulating cuprates?

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    A large body of experimental data point towards a charge transfer instability of parent insulating cuprates to be their unique property. We argue that the true charge transfer gap in these compounds is as small as 0.4-0.5\,eV rather than 1.5-2.0\,eV as usually derived from the optical gap measurements. In fact we deal with a competition of the conventional (3d9^9) ground state and a charge transfer (CT) state with formation of electron-hole dimers which evolves under doping to an unconventional bosonic system. Our conjecture does provide an unified standpoint on the main experimental findings for parent cuprates including linear and nonlinear optical, Raman, photoemission, photoabsorption, and transport properties anyhow related with the CT excitations. In addition we suggest a scenario for the evolution of the CuO2_2 planes in the CT unstable cuprates under a nonisovalent doping.Comment: 13 pages, 5 figures, submitted to PR
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