74 research outputs found
On the Lebesgue measure of the Julia set of a quadratic polynomial
The goal of this note is to prove the following theorem: Let
be a quadratic polynomial which has no irrational indifferent periodic points,
and is not infinitely renormalizable. Then the Lebesgue measure of the Julia
set is equal to zero.
As part of the proof we discuss a property of the critical point to be {\it
persistently recurrent}, and relate our results to corresponding ones for real
one dimensional maps. In particular, we show that in the persistently recurrent
case the restriction is topologically minimal and has zero
topological entropy. The Douady-Hubbard-Yoccoz rigidity theorem follows this
result
On cycles and coverings associated to a knot
We consider the space of all representations of the commutator subgroup of a
knot group into a finite abelian group {\Sigma}, together with a shift map
{\sigma}_x. This is a finite dynamical system, introduced by D.Silver and S.
Williams. We describe the lengths of its cycles in terms of the roots of the
Alexander polynomial of the knot. This generalizes our previous result for
{\Sigma}= Z/p, p is prime, and gives a complete classification of depth 2
solvable coverings of the knot complement
Note on the geometry of generalized parabolic towers
The goal of this technical note is to show that the geometry of generalized
parabolic towers cannot be essentially bounded. It fills a gap in author's
paper "Combinatorics, geomerty and attractors of quasi-quadratic maps", Annals
of Math., 1992
Dynamics of quadratic polynomials, III: Parapuzzle and SBR measures
This is a continuation of notes on dynamics of quadratic polynomials. In this
part we transfer the our prior geometric result to the parameter plane. To any
parameter value c in the Mandelbrot set (which lies outside of the main
cardioid and little Mandelbrot sets attached to it) we associate a ``principal
nest of parapuzzle pieces'' and show that the moduli of the annuli grow at
least linearly. The main motivation for this work was to prove the following:
Theorem B (joint with Martens and Nowicki). Lebesgue almost every real
quadratic polynomial which is non-hyperbolic and at most finitely
renormalizable has a finite absolutely continuous invariant measure
Combinatorics, geometry and attractors of quasi-quadratic maps
The Milnor problem on one-dimensional attractors is solved for S-unimodal
maps with a non-degenerate critical point c. It provides us with a complete
understanding of the possible limit behavior for Lebesgue almost every point.
This theorem follows from a geometric study of the critical set of
a "non-renormalizable" map. It is proven that the scaling factors
characterizing the geometry of this set go down to 0 at least exponentially.
This resolves the problem of the non-linearity control in small scales. The
proofs strongly involve ideas from renormalization theory and holomorphic
dynamics
Ergodic theory for smooth one-dimensional dynamical systems
In this paper we study measurable dynamics for the widest reasonable class of
smooth one dimensional maps. Three principle decompositions are described in
this class : decomposition of the global measure-theoretical attractor into
primitive ones, ergodic decomposition and Hopf decomposition. For maps with
negative Schwarzian derivative this was done in the series of papers [BL1-BL5],
but the approach to the general smooth case must be different
Teichm\"uller space of Fibonacci maps
According to Sullivan, a space of unimodal maps with the same
combinatorics (modulo smooth conjugacy) should be treated as an
infinitely-dimensional Teichm\"{u}ller space. This is a basic idea in
Sullivan's approach to the Renormalization Conjecture. One of its principle
ingredients is to supply with the Teichm\"{u}ller metric. To have
such a metric one has to know, first of all, that all maps of are
quasi-symmetrically conjugate. This was proved [Ji] and [JS] for some classes
of non-renormalizable maps (when the critical point is not too recurrent). Here
we consider a space of non-renormalizable unimodal maps with in a sense fastest
possible recurrence of the critical point (called Fibonacci). Our goal is to
supply this space with the Teichm\"{u}ller metric
Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle
This work studies combinatorics and geometry of the Yoccoz puzzle for
quadratic polynomials. It is proven that the moduli of the ``principal nest''
of annuli grow at linear rate. As a corollary we obtain complex a priori bounds
and local connectivity of the Julia set for many infinitely renormalizable
quadratics
Dynamics of quadratic polynomials II: rigidity
This is a continuation of the series of notes on the dynamics of quadratic
polynomials. We show the following
Rigidity Theorem: Any combinatorial class contains at most one quadratic
polynomial satisfying the secondary limbs condition with a-priori bounds.
As a corollary, such maps are combinatorially and topologically rigid, and as
a consequence, the Mandelbrot set is locally connected at the correspoinding
parameter values
The Quasi-Additivity Law in Conformal Geometry
On a Riemann surface of finite type containing a family of disjoint
disks (``islands''), we consider several natural conformal invariants
measuring the distance from the islands to \di S and separation between
different islands. In a near degenerate situation we establish a relation
between them called the Quasi-Additivity Law. We then generalize it to a
Quasi-Invariance Law providing us with a transformation rule of the moduli in
question under covering maps. This rule (and in particular, its special case
called the Covering Lemma) has important applications in holomorphic dynamics
which will be addressed in the forthcoming notes.Comment: LaTeX, 36 pages, 7 figure
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