7,541 research outputs found
On the Plaque Expansivity Conjecture
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
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
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?
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 (3d) 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 CuO planes in the CT unstable
cuprates under a nonisovalent doping.Comment: 13 pages, 5 figures, submitted to PR
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