761 research outputs found
A temporal Central Limit Theorem for real-valued cocycles over rotations
We consider deterministic random walks on the real line driven by irrational
rotations, or equivalently, skew product extensions of a rotation by
where the skewing cocycle is a piecewise constant mean zero function with a
jump by one at a point . When is badly approximable and
is badly approximable with respect to , we prove a Temporal Central
Limit theorem (in the terminology recently introduced by D.Dolgopyat and
O.Sarig), namely we show that for any fixed initial point, the occupancy random
variables, suitably rescaled, converge to a Gaussian random variable. This
result generalizes and extends a theorem by J. Beck for the special case when
is quadratic irrational, is rational and the initial point is
the origin, recently reproved and then generalized to cover any initial point
using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig
(Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use
renormalization, but in order to treat irrational values of , instead of
geometric arguments, we use the renormalization associated to the continued
fraction algorithm and dynamical Ostrowski expansions. This yields a suitable
symbolic coding framework which allows us to reduce the main result to a CLT
for non homogeneous Markov chains.Comment: a few typos corrected, 28 pages, 4 figure
High distance Heegaard splittings via fat train tracks
We define "fat" train tracks and use them to give a combinatorial criterion
for the Hempel distance of Heegaard splittings for closed orientable
3-manifolds. We apply this criterion to 3-manifolds obtained from surgery on
knots in the three sphere.Comment: 25 pages no figures. to appear in Proceedings of "Knots Groups and
3-manifolds" Marseilles France 200
Cycle lengths in sparse graphs
Let C(G) denote the set of lengths of cycles in a graph G. In the first part
of this paper, we study the minimum possible value of |C(G)| over all graphs G
of average degree d and girth g. Erdos conjectured that |C(G)|
=\Omega(d^{\lfloor (g-1)/2\rfloor}) for all such graphs, and we prove this
conjecture. In particular, the longest cycle in a graph of average degree d and
girth g has length \Omega(d^{\lfloor (g-1)/2\rfloor}). The study of this
problem was initiated by Ore in 1967 and our result improves all previously
known lower bounds on the length of the longest cycle. Moreover, our bound
cannot be improved in general, since known constructions of d-regular Moore
Graphs of girth g have roughly that many vertices. We also show that
\Omega(d^{\lfloor (g-1)/2\rfloor}) is a lower bound for the number of odd cycle
lengths in a graph of chromatic number d and girth g. Further results are
obtained for the number of cycle lengths in H-free graphs of average degree d.
In the second part of the paper, motivated by the conjecture of Erdos and
Gyarfas that every graph of minimum degree at least three contains a cycle of
length a power of two, we prove a general theorem which gives an upper bound on
the average degree of an n-vertex graph with no cycle of even length in a
prescribed infinite sequence of integers. For many sequences, including the
powers of two, our theorem gives the upper bound e^{O(\log^* n)} on the average
degree of graph of order n with no cycle of length in the sequence, where
\log^* n is the number of times the binary logarithm must be applied to n to
get a number which is at mos
Detecting right-veering diffeomorphisms
A result of Honda, Kazez, and Mati\'{c} states that a contact structure is
tight if and only if all its supporting open books are right-veering. We show a
combinatorial way of detecting the left-veering arcs in open books, implying
the existence of an algorithm that detects the right-veering property for
compact surfaces with boundary.Comment: Comments are welcom
Category forcings, , and generic absoluteness for the theory of strong forcing axioms
We introduce a category whose objects are stationary set preserving complete
boolean algebras and whose arrows are complete homomorphisms with a stationary
set preserving quotient. We show that the cut of this category at a rank
initial segment of the universe of height a super compact which is a limit of
super compact cardinals is a stationary set preserving partial order which
forces and collapses its size to become the second uncountable
cardinal. Next we argue that any of the known methods to produce a model of
collapsing a superhuge cardinal to become the second uncountable
cardinal produces a model in which the cutoff of the category of stationary set
preserving forcings at any rank initial segment of the universe of large enough
height is forcing equivalent to a presaturated tower of normal filters. We let
denote this statement and we prove that the theory of
with parameters in is generically invariant
for stationary set preserving forcings that preserve . Finally we
argue that the work of Larson and Asper\'o shows that this is a next to optimal
generalization to the Chang model of Woodin's generic
absoluteness results for the Chang model . It remains open
whether and are equivalent axioms modulo large cardinals
and whether suffices to prove the same generic absoluteness results
for the Chang model .Comment: - to appear on the Journal of the American Mathemtical Societ
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