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900 |
Conformal dynamics problem list
|
math.DS
|
This is a list of unsolved problems given at the Conformal Dynamics
Conference which was held at SUNY Stony Brook in November 1989. Problems were
contributed by the editor and the other authors.
|
math
|
901 |
Remarks on iterated cubic maps
|
math.DS
|
This note will discuss the dynamics of iterated cubic maps from the real or
complex line to itself, and will describe the geography of the parameter space
for such maps. It is a rough survey with few precise statements or proofs, and
depends strongly on work by Douady, Hubbard, Branner and Rees.
|
math
|
902 |
One-dimensional maps and Poincaré metric
|
math.DS
|
Invertible compositions of one-dimensional maps are studied which are assumed
to include maps with non-positive Schwarzian derivative and others whose sum of
distortions is bounded. If the assumptions of the Koebe principle hold, we show
that the joint distortion of the composition is bounded. On the other hand, if
all maps with possibly non-negative Schwarzian derivative are almost
linear-fractional and their nonlinearities tend to cancel leaving only a small
total, then they can all be replaced with affine maps with the same domains and
images and the resulting composition is a very good approximation of the
original one. These technical tools are then applied to prove a theorem about
critical circle maps.
|
math
|
903 |
Dynamics of certain smooth one-dimensional mappings I: The $C^{1+α}$-Denjoy-Koebe distortion lemma
|
math.DS
|
We prove a technical lemma, the $C^{1+\alpha }$-Denjoy-Koebe distortion
lemma, estimating the distortion of a long composition of a $C^{1+\alpha }$
one-dimensional mapping $f:M\mapsto M$ with finitely many, non-recurrent, power
law critical points. The proof of this lemma combines the ideas of the
distortion lemmas of Denjoy and Koebe.
|
math
|
904 |
Dynamics of certain smooth one-dimensional mappings II: geometrically finite one-dimensional mappings
|
math.DS
|
We study geometrically finite one-dimensional mappings. These are a subspace
of $C^{1+\alpha}$ one-dimensional mappings with finitely many, critically
finite critical points. We study some geometric properties of a mapping in this
subspace. We prove that this subspace is closed under quasisymmetrical
conjugacy. We also prove that if two mappings in this subspace are
topologically conjugate, they are then quasisymmetrically conjugate. We show
some examples of geometrically finite one-dimensional mappings.
|
math
|
905 |
A partial description of the parameter space of rational maps of degree two: Part 2
|
math.DS
|
This continues the investigation of a combinatorial model for the variation
of dynamics in the family of rational maps of degree two, by concentrating on
those varieties in which one critical point is periodic. We prove some general
results about nonrational critically finite degree two branched coverings, and
finally identify the boundary of the rational maps in the combinatorial model,
thus completing the proofs of results announced in Part 1.
|
math
|
906 |
Expanding direction of the period doubling operator
|
math.DS
|
We prove that the period doubling operator has an expanding direction at the
fixed point. We use the induced operator, a ``Perron-Frobenius type operator'',
to study the linearization of the period doubling operator at its fixed point.
We then use a sequence of linear operators with finite ranks to study this
induced operator. The proof is constructive. One can calculate the expanding
direction and the rate of expansion of the period doubling operator at the
fixed point.
|
math
|
907 |
The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets
|
math.DS
|
It is shown that the boundary of the Mandelbrot set $M$ has Hausdorff
dimension two and that for a generic $c \in \bM$, the Julia set of $z \mapsto
z^2+c$ also has Hausdorff dimension two. The proof is based on the study of the
bifurcation of parabolic periodic points.
|
math
|
908 |
Critical circle maps near bifurcation
|
math.DS
|
We estimate harmonic scalings in the parameter space of a one-parameter
family of critical circle maps. These estimates lead to the conclusion that the
Hausdorff dimension of the complement of the frequency-locking set is less than
$1$ but not less than $1/3$. Moreover, the rotation number is a H\"{o}lder
continuous function of the parameter.
|
math
|
909 |
The Teichmüller space of an Anosov diffeomorphism of $T^2$
|
math.DS
|
In this paper we consider the space of smooth conjugacy classes of an Anosov
diffeomorphism of the two-torus. The only 2-manifold that supports an Anosov
diffeomorphism is the 2-torus, and Franks and Manning showed that every such
diffeomorphism is topologically conjugate to a linear example, and furthermore,
the eigenvalues at periodic points are a complete smooth invariant. The
question arises: what sets of eigenvalues occur as the Anosov diffeomorphism
ranges over a topological conjugacy class? This question can be reformulated:
what pairs of cohomology classes (one determined by the expanding eigenvalues,
and one by the contracting eigenvalues) occur as the diffeomorphism ranges over
a topological conjugacy class? The purpose of this paper is to answer this
question: all pairs of H\"{o}lder reduced cohomology classes occur.
|
math
|
910 |
On the Lebesgue measure of the Julia set of a quadratic polynomial
|
math.DS
|
The goal of this note is to prove the following theorem: Let $p_a(z) = z^2+a$
be a quadratic polynomial which has no irrational indifferent periodic points,
and is not infinitely renormalizable. Then the Lebesgue measure of the Julia
set $J(p_a)$ 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 $p_a|\omega(0)$ is topologically minimal and has zero
topological entropy. The Douady-Hubbard-Yoccoz rigidity theorem follows this
result.
|
math
|
911 |
Ergodic theory for smooth one-dimensional dynamical systems
|
math.DS
|
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.
|
math
|
912 |
Dynamics of certain smooth one-dimensional mappings III: Scaling function geometry
|
math.DS
|
We study scaling function geometry. We show the existence of the scaling
function of a geometrically finite one-dimensional mapping. This scaling
function is discontinuous. We prove that the scaling function and the
asymmetries at the critical points of a geometrically finite one-dimensional
mapping form a complete set of $C^{1}$-invariants within a topological
conjugacy class.
|
math
|
913 |
Dynamics of certain smooth one-dimensional mappings IV: Asymptotic geometry of Cantor sets
|
math.DS
|
We study hyperbolic mappings depending on a parameter $\varepsilon $. Each of
them has an invariant Cantor set. As $\varepsilon $ tends to zero, the mapping
approaches the boundary of hyperbolicity. We analyze the asymptotics of the gap
geometry and the scaling function geometry of the invariant Cantor set as
$\varepsilon $ goes to zero. For example, in the quadratic case, we show that
all the gaps close uniformly with speed $\sqrt {\varepsilon}$. There is a
limiting scaling function of the limiting mapping and this scaling function has
dense jump discontinuities because the limiting mapping is not expanding.
Removing these discontinuities by continuous extension, we show that we obtain
the scaling function of the limiting mapping with respect to the Ulam-von
Neumann type metric.
|
math
|
914 |
Periods implying almost all periods, trees with snowflakes, and zero entropy maps
|
math.DS
|
Let $X$ be a compact tree, $f$ be a continuous map from $X$ to itself,
$End(X)$ be the number of endpoints and $Edg(X)$ be the number of edges of $X$.
We show that if $n>1$ has no prime divisors less than $End(X)+1$ and $f$ has a
cycle of period $n$, then $f$ has cycles of all periods greater than
$2End(X)(n-1)$ and topological entropy $h(f)>0$; so if $p$ is the least prime
number greater than $End(X)$ and $f$ has cycles of all periods from 1 to
$2End(X)(p-1)$, then $f$ has cycles of all periods (this verifies a conjecture
of Misiurewicz for tree maps). Together with the spectral decomposition theorem
for graph maps it implies that $h(f)>0$ iff there exists $n$ such that $f$ has
a cycle of period $mn$ for any $m$. We also define {\it snowflakes} for tree
maps and show that $h(f)=0$ iff every cycle of $f$ is a snowflake or iff the
period of every cycle of $f$ is of form $2^lm$ where $m\le Edg(X)$ is an odd
integer with prime divisors less than $End(X)+1$.
|
math
|
915 |
The "spectral" decomposition for one-dimensional maps
|
math.DS
|
We construct the "spectral" decomposition of the sets $\bar{Per\,f}$,
$\omega(f)=\cup\omega(x)$ and $\Omega(f)$ for a continuous map $f$ of the
interval to itself. Several corollaries are obtained; the main ones describe
the generic properties of $f$-invariant measures, the structure of the set
$\Omega(f)\setminus \bar{Per\,f}$ and the generic limit behavior of an orbit
for maps without wandering intervals. The "spectral" decomposition for
piecewise-monotone maps is deduced from the Decomposition Theorem. Finally we
explain how to extend the results of the present paper for a continuous map of
a one-dimensional branched manifold into itself.
|
math
|
916 |
The Fibonacci unimodal map
|
math.DS
|
This paper will study topological, geometrical and measure-theoretical
properties of the real Fibonacci map. Our goal was to figure out if this type
of recurrence really gives any pathological examples and to compare it with the
infinitely renormalizable patterns of recurrence studied by Sullivan. It turns
out that the situation can be understood completely and is of quite regular
nature. In particular, any Fibonacci map (with negative Schwarzian and
non-degenerate critical point) has an absolutely continuous invariant measure
(so, we deal with a ``regular'' type of chaotic dynamics). It turns out also
that geometrical properties of the closure of the critical orbit are quite
different from those of the Feigenbaum map: its Hausdorff dimension is equal to
zero and its geometry is not rigid but depends on one parameter.
|
math
|
917 |
Quasisymmetric conjugacies between unimodal maps
|
math.DS
|
It is shown that some topological equivalency classes of S-unimodal maps are
equal to quasisymmetric conjugacy classes. This includes some infinitely
renormalizable polynomials of unbounded type.
|
math
|
918 |
Dynamics of certain non-conformal degree two maps on the plane
|
math.DS
|
In this paper we consider maps on the plane which are similar to quadratic
maps in that they are degree 2 branched covers of the plane. In fact, consider
for $\alpha$ fixed, maps $f_c$ which have the following form (in polar
coordinates):
$$f_c(r\,e^{i\theta})\;=\;r^{2\alpha}\,e^{2i\theta}\,+\,c$$
When $\alpha=1$, these maps are quadratic ($z \maps z^2 + c$), and their
dynamics and bifurcation theory are to some degree understood. When $\alpha$ is
different from one, the dynamics is no longer conformal. In particular, the
dynamics is not completely determined by the orbit of the critical point.
Nevertheless, for many values of the parameter c, the dynamics has strong
similarities to that of the quadratic family. For other parameter values the
dynamics is dominated by 2 dimensional behavior: saddles and the like.
The objects of study are Julia sets, filled-in Julia sets and the
connectedness locus. These are defined in analogy to the conformal case. The
main drive in this study is to see to what extent the results in the conformal
case generalize to that of maps which are topologically like quadratic maps
(and when $\alpha$ is close to one, close to being quadratic).
|
math
|
919 |
On the quasisymmetrical classification of infinitely renormalizable maps: I. Maps with Feigenbaum's topology.
|
math.DS
|
A semigroup (dynamical system) generated by $C^{1+\alpha}$-contracting
mappings is considered. We call a such semigroup regular if the maximum $K$ of
the conformal dilatations of generators, the maximum $l$ of the norms of the
derivatives of generators and the smoothness $\alpha$ of the generators satisfy
a compatibility condition $K< 1/l^{\alpha}$. We prove the {\em geometric
distortion lemma} for a regular semigroup generated by
$C^{1+\alpha}$-contracting mappings.
|
math
|
920 |
On the quasisymmetrical classification of infinitely renormalizable maps: II. remarks on maps with a bounded type topology.
|
math.DS
|
We use the upper and lower potential functions and Bowen's formula estimating
the Hausdorff dimension of the limit set of a regular semigroup generated by
finitely many $C^{1+\alpha}$-contracting mappings. This result is an
application of the geometric distortion lemma in the first paper at this
series.
|
math
|
921 |
On the realization of fixed point portraits (an addendum to Goldberg, Milnor: Fixed point portraits)
|
math.DS
|
We establish that every formal critical portrait (as defined by Goldberg and
Milnor), can be realized by a postcritically finite polynomial.
|
math
|
922 |
Periodic orbits for Hamiltonian systems in cotangent bundles
|
math.DS
|
We prove the existence of at least $cl(M)$ periodic orbits for certain time
dependant Hamiltonian systems on the cotangent bundle of an arbitrary compact
manifold $M$. These Hamiltonians are not necessarily convex but they satisfy a
certain boundary condition given by a Riemannian metric on $M$. We discretize
the variational problem by decomposing the time 1 map into a product of
``symplectic twist maps''. A second theorem deals with homotopically non
trivial orbits in manifolds of negative curvature.
|
math
|
923 |
On removable sets for Sobolev spaces in the plane
|
math.DS
|
Let $K$ be a compact subset of $\bar{\bold C} ={\bold R}^2$ and let $K^c$
denote its complement. We say $K\in HR$, $K$ is holomorphically removable, if
whenever $F:\bar{\bold C} \to\bar{\bold C}$ is a homeomorphism and $F$ is
holomorphic off $K$, then $F$ is a M\"obius transformation. By composing with a
M\"obius transform, we may assume $F(\infty )=\infty$. The contribution of this
paper is to show that a large class of sets are $HR$. Our motivation for these
results is that these sets occur naturally (e.g. as certain Julia sets) in
dynamical systems, and the property of being $HR$ plays an important role in
the Douady-Hubbard description of their structure.
|
math
|
924 |
The existence of sigma-finite invariant measures, applications to real one-dimensional dynamics
|
math.DS
|
A general construction for $\sigma-$finite absolutely continuous invariant
measure will be presented. It will be shown that the local bounded distortion
of the Radon-Nykodym derivatives of $f^n_*(\lambda)$ will imply the existence
of a $\sigma-$finite invariant measure for the map $f$ which is absolutely
continuous with respect to $\lambda$, a measure on the phase space describing
the sets of measure zero. Furthermore we will discuss sufficient conditions for
the existence of $\sigma-$finite invariant absolutely continuous measures for
real 1-dimensional dynamical systems.
|
math
|
925 |
Scalings in circle maps III
|
math.DS
|
Circle maps with a flat spot are studied which are differentiable, even on
the boundary of the flat spot. Estimates on the Lebesgue measure and the
Hausdorff dimension of the non-wandering set are obtained. Also, a sharp
transition is found from degenerate geometry similar to what was found earlier
for non-differentiable maps with a flat spot to bounded geometry as in critical
maps without a flat spot.
|
math
|
926 |
Hyperbolic components in spaces of polynomial maps
|
math.DS
|
We consider polynomial maps $f:\C\to\C$ of degree $d\ge 2$, or more generally
polynomial maps from a finite union of copies of $\C$ to itself which have
degree two or more on each copy. In any space $\p^{S}$ of suitably normalized
maps of this type, the post-critically bounded maps form a compact subset
$\cl^{S}$ called the connectedness locus, and the hyperbolic maps in $\cl^{S}$
form an open set $\hl^{S}$ called the hyperbolic connectedness locus. The
various connected components $H_\alpha\subset \hl^{S}$ are called hyperbolic
components. It is shown that each hyperbolic component is a topological cell,
containing a unique post-critically finite map which is called its center
point. These hyperbolic components can be separated into finitely many distinct
``types'', each of which is characterized by a suitable reduced mapping schema
$\bar S(f)$. This is a rather crude invariant, which depends only on the
topology of $f$ restricted to the complement of the Julia set. Any two
components with the same reduced mapping schema are canonically biholomorphic
to each other. There are similar statements for real polynomial maps, or for
maps with marked critical points.
|
math
|
927 |
The Teichmüller space of the standard action of $SL(2,Z)$ on $T^2$ is trivial
|
math.DS
|
The group $SL(n,{\bf Z})$ acts linearly on $\R^n$, preserving the integer
lattice $\Z^{n} \subset \R^{n}$. The induced (left) action on the n-torus
$\T^{n} = \R^{n}/\Z^{n}$ will be referred to as the ``standard action''.
It has recently been shown that the standard action of $SL(n,\Z)$ on $\T^n$,
for $n \geq 3$, is both topologically and smoothly rigid. That is, nearby
actions in the space of representations of $SL(n,\Z)$ into ${\rm
Diff}^{+}(\T^{n})$ are smoothly conjugate to the standard action. In fact, this
rigidity persists for the standard action of a subgroup of finite index. On the
other hand, while the $\Z$ action on $\T^{n}$ defined by a single hyperbolic
element of $SL(n,\Z)$ is topologically rigid, an infinite dimensional space of
smooth conjugacy classes occur in a neighborhood of the linear action.
The standard action of $SL(2, \Z)$ on $\T^2$ forms an intermediate case, with
different rigidity properties from either extreme. One can construct continuous
deformations of the standard action to obtain an (arbritrarily near) action to
which it is not topologically conjugate. The purpose of the present paper is to
show that if a nearby action, or more generally, an action with some mild
Anosov properties, is conjugate to the standard action of $SL(2, \Z)$ on $\T^2$
by a homeomorphism $h$, then $h$ is smooth. In fact, it will be shown that this
rigidity holds for any non-cyclic subgroup of $SL(2, \Z)$.
|
math
|
928 |
Dynamics of certain non-conformal semigroups
|
math.DS
|
A semigroup generated by two dimensional $C^{1+\alpha}$ contracting maps is
considered. We call a such semigroup regular if the maximum $K$ of the
conformal dilatations of generators, the maximum $l$ of the norms of the
derivatives of generators and the smoothness $\alpha$ of the generators satisfy
a compatibility condition $K< 1/l^{\alpha}$. We prove that the shape of the
image of the core of a ball under any element of a regular semigroup is good
(bounded geometric distortion like the Koebe $1/4$-lemma \cite{a}). And we use
it to show a lower and a upper bounds of the Hausdorff dimension of the limit
set of a regular semigroup. We also consider a semigroup generated by higher
dimensional maps.
|
math
|
929 |
Cantor sets in the line: scaling function and the smoothness of the shift map
|
math.DS
|
Consider $d$ disjoint closed subintervals of the unit interval and consider
an orientation preserving expanding map which maps each of these subintervals
to the whole unit interval. The set of points where all iterates of this
expanding map are defined is a Cantor set. Associated to the construction of
this Cantor set is the scaling function which records the infinitely deep
geometry of this Cantor set. This scaling function is an invariant of $C^1$
conjugation. We solve the inverse problem posed by Dennis Sullivan: given a
scaling function, determine the maximal possible smoothness of any expanding
map which produces it.
|
math
|
930 |
Polynomial diffeomorphisms of C^2, IV: The measure of maximal entropy and laminar currents
|
math.DS
|
This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$.
One can associate to such an automorphism two currents $\mu^\pm$ and the
equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some
geometric and dynamical properties of these objects. First, we characterize
$\mu$ as the unique measure of maximal entropy. Then we show that the measure
$\mu$ has a local product structure and that the currents $\mu^\pm$ have a
laminar structure. This allows us to deduce information about periodic points
and heteroclinic intersections. For example, we prove that the support of $\mu$
coincides with the closure of the set of saddle points. The methods used
combine the pluripotential theory with the theory of non-uniformly hyperbolic
dynamical systems.
|
math
|
931 |
Singular measures in circle dynamics
|
math.DS
|
Critical circle homeomorphisms have an invariant measure totally singular
with respect to the Lebesgue measure. We prove that singularities of the
invariant measure are of Holder type. The Hausdorff dimension of the invariant
measure is less than 1 but greater than 0.
|
math
|
932 |
Hyperbolicity is dense in the real quadratic family
|
math.DS
|
It is shown that for non-hyperbolic real quadratic polynomials topological
and quasisymmetric conjugacy classes are the same. By quasiconformal rigidity,
each class has only one representative in the quadratic family, which proves
that hyperbolic maps are dense.
|
math
|
933 |
Local connectivity of Julia sets: expository lectures
|
math.DS
|
The following notes provide an introduction to recent work of Branner,
Hubbard and Yoccoz on the geometry of polynomial Julia sets. They are an
expanded version of lectures given in Stony Brook in Spring 1992. I am indebted
to help from the audience.
Section 1 describes unpublished work by J.-C. Yoccoz on local connectivity of
quadratic Julia sets. It presents only the "easy" part of his work, in the
sense that it considers only non-renormalizable polynomials, and makes no
effort to describe the much more difficult arguments which are needed to deal
with local connectivity in parameter space. It is based on second hand sources,
namely Hubbard together with lectures by Branner and Douady. Hence the
presentation is surely quite different from that of Yoccoz.
Section 2 describes the analogous arguments used by Branner and Hubbard to
study higher degree polynomials for which all but one of the critical orbits
escape to infinity. In this case, the associated Julia set J is never locally
connected. The basic problem is rather to decide when J is totally
disconnected. This Branner-Hubbard work came before Yoccoz, and its technical
details are not as difficult. However, in these notes their work is presented
simply as another application of the same geometric ideas.
Chapter 3 complements the Yoccoz results by describing a family of examples,
due to Douady and Hubbard (unpublished), showing that an infinitely
renormalizable quadratic polynomial may have non-locally-connected Julia set.
An Appendix describes needed tools from complex analysis, including the
Gr\"otzsch inequality.
|
math
|
934 |
Hubbard forests
|
math.DS
|
The theory of Hubbard trees provides an effective classification of
non-linear post-critically finite polynomial maps from \C to itself. This note
will extend this classification to the case of maps from a finite union of
copies of \C to itself. Maps which are post-critically finite and nowhere
linear will be characterized by a ``forest'', which is made up out of one tree
in each copy of \C.
|
math
|
935 |
Weak disks of Denjoy minimal sets
|
math.DS
|
This paper investigates the existence of Denjoy minimal sets and, more
generally, strictly ergodic sets in the dynamics of iterated homeomorphisms. It
is shown that for the full two-shift, the collection of such invariant sets
with the weak topology contains topological balls of all finite dimensions. One
implication is an analogous result that holds for diffeomorphisms with
transverse homoclinic points. It is also shown that the union of Denjoy minimal
sets is dense in the two-shift and that the set of unique probability measures
supported on these sets is weakly dense in the set of all shift-invariant,
Borel probability measures.
|
math
|
936 |
Remarks on quadratic rational maps
|
math.DS
|
This will is an expository description of quadratic rational maps. Sections 2
through 6 are concerned with the geometry and topology of such maps. Sections
7--10 survey of some topics from the dynamics of quadratic rational maps. There
are few proofs. Section 9 attempts to explore and picture moduli space by means
of complex one-dimensional slices. Section 10 describes the theory of real
quadratic rational maps. For convenience in exposition, some technical details
have been relegated to appendices: Appendix A outlines some classical algebra.
Appendix B describes the topology of the space of rational maps of degree
\[d\]. Appendix C outlines several convenient normal forms for quadratic
rational maps, and computes relations between various invariants.\break
Appendix D describes some geometry associated with the curves
\[\Per_n(\mu)\subset\M\]. Appendix E describes totally disconnected Julia sets
containing no critical points. Finally, Appendix F, written in collaboration
with Tan Lei, describes an example of a connected quadratic Julia set for which
no two components of the complement have a common boundary point.
|
math
|
937 |
A shooting approach to the Lorenz equations
|
math.DS
|
We announce and outline a proof of the existence of a homoclinic orbit of the
Lorenz equations. In addition, we develop a shooting technique and two key
conditions, which lead to the existence of a one-to-one correspondence between
a set of solutions and the set of all infinite sequences of 1's and 3's.
|
math
|
938 |
Ergodicity in Hamiltonian systems
|
math.DS
|
We discuss the Sinai method of proving ergodicity of a discontinuous
Hamiltonian system with (non-uniform) hyperbolic behavior.
|
math
|
939 |
Distortion results and invariant cantor sets of unimodal maps
|
math.DS
|
A distortion theory is developed for $S-$unimodal maps. It will be used to
get some geometric understanding of invariant Cantor sets. In particular
attracting Cantor sets turn out to have Lebesgue measure zero. Furthermore the
ergodic behavior of $S-$unimodal maps is classified according to a distortion
property, called the Markov-property.
|
math
|
940 |
Combinatorics, geometry and attractors of quasi-quadratic maps
|
math.DS
|
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 $\omega(c)$ 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.
|
math
|
941 |
Distribution of periodic points of polynomial diffeomorphisms of C^2
|
math.DS
|
This paper deals with the dynamics of a simple family of holomorphic
diffeomorphisms of $\C^2$: the polynomial automorphisms. This family of maps
has been studied by a number of authors. We refer to [BLS] for a general
introduction to this class of dynamical systems. An interesting object from the
point of view of potential theory is the equilibrium measure $\mu$ of the set
$K$ of points with bounded orbits. In [BLS] $\mu$ is also characterized
dynamically as the unique measure of maximal entropy. Thus $\mu$ is also an
equilibrium measure from the point of view of the thermodynamical formalism. In
the present paper we give another dynamical interpretation of $\mu$ as the
limit distribution of the periodic points of $f$.
|
math
|
942 |
Accessability of typical points for invariant measures of positive Lyapunov exponents for iterations of holomorphic maps
|
math.DS
|
We prove that if A is the basin of immediate attraction to a periodic
attracting or parabolic point for a rational map f on the Riemann sphere, if
$A$ is completely invariant (i.e. $f^{-1}(A)=A$), and if $\mu$ is an arbitrary
$f$-invariant measure with positive Lyapunov exponents on the boundary of $A$,
then $\mu$-almost every point $q$ in the boundary of $A$ is accessible along a
curve from $A$. In fact we prove the accessability of every "good" $q$ i.e.
such $q$ for which "small neighbourhoods arrive at large scale" under iteration
of $f$. This generalizes Douady-Eremenko-Levin-Petersen theorem on the
accessability of periodic sources.
|
math
|
943 |
Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps, geometric coding trees technique
|
math.DS
|
We prove that if A is the basin of immediate attraction to a periodic
attracting or parabolic point for a rational map f on the Riemann sphere, then
periodic points in the boundary of A are dense in this boundary. To prove this
in the non simply- connected or parabolic situations we prove a more abstract,
geometric coding trees version.
|
math
|
944 |
On postcritically finite polynomials, part 1: critical portraits
|
math.DS
|
We extend the work of Bielefeld, Fisher and Hubbard on Critical Portraits to
the case of arbitrary postcritically finite polynomials. This determines an
effective classification of postcritically finite polynomials as dynamical
systems. This paper is the first in a series of two based on the author's
thesis, which deals with the classification of postcritically finite
polynomials. In this first part we conclude the study of critical portraits
initiated by Fisher and continued by Bielefeld, Fisher and Hubbard.
|
math
|
945 |
On postcritically finite polynomials, part 2: Hubbard trees
|
math.DS
|
We provide an effective classification of postcritically finite polynomials
as dynamical systems by means of Hubbard Trees. This can be viewed as an
application of the results developed in part 1 (Stony Brook IMS 1993/5).
|
math
|
946 |
Induced expansion for quadratic polynomials
|
math.DS
|
We prove that non-hyperbolic non-renormalizable quadratic polynomials are
expansion inducing. For renormalizable polynomials a counterpart of this
statement is that in the case of unbounded combinatorics renormalized mappings
become almost quadratic. Technically, this follows from the decay of the box
geometry. Specific estimates of the rate of this decay are shown which are
sharp in a class of S-unimodal mappings combinatorially related to rotations of
bounded type. We use real methods based on cross-ratios and Schwarzian
derivative complemented by complex-analytic estimates in terms of conformal
moduli.
|
math
|
947 |
Geometry of quadratic polynomials: moduli, rigidity and local connectivity
|
math.DS
|
A while ago MLC (the conjecture that the Mandelbrot set is locally connected)
was proven for quasi-hyperbolic points by Douady and Hubbard, and for
boundaries of hyperbolic components by Yoccoz. More recently Yoccoz proved MLC
for all at most finitely renormalizable parameter values. One of our goals is
to prove MLC for some infinitely renormalizable parameter values. Loosely
speaking, we need all renormalizations to have bounded combinatorial rotation
number (assumption C1) and sufficiently high combinatorial type (assumption
C2).
For real quadratic polynomials of bounded combinatorial type the complex a
priori bounds were obtained by Sullivan. Our result complements the Sullivan's
result in the unbounded case. Moreover, it gives a background for Sullivan's
renormalization theory for some bounded type polynomials outside the real line
where the problem of a priori bounds was not handled before for any single
polynomial.
An important consequence of a priori bounds is absence of invariant
measurable line fields on the Julia set (McMullen) which is equivalent to
quasi-conformal (qc) rigidity. To prove stronger topological rigidity we
construct a qc conjugacy between any two topologically conjugate polynomials
(Theorem III). We do this by means of a pull-back argument, based on the linear
growth of moduli and a priori bounds. Actually the argument gives the stronger
combinatorial rigidity which implies MLC.
|
math
|
948 |
A monotonicity conjecture for real cubic maps
|
math.DS
|
This is an outline of work in progress. We study the conjecture that the
topological entropy of a real cubic map depends ``monotonely'' on its
parameters, in the sense that each locus of constant entropy in parameter space
is a connected set. This material will be presented in more detail in a later
paper.
|
math
|
949 |
Teichmüller space of Fibonacci maps
|
math.DS
|
According to Sullivan, a space ${\cal E}$ 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 ${\cal E}$ with the Teichm\"{u}ller metric. To have
such a metric one has to know, first of all, that all maps of ${\cal E}$ 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.
|
math
|
950 |
Henon mappings in the complex domain II: projective and inductive limits of polynomials
|
math.DS
|
Let H: C^2 -> C^2 be the Henon mapping given by (x,y) --> (p(x) - ay,x). The
key invariant subsets are K_+/-, the sets of points with bounded forward
images, J_+/- = the boundary of K_+/-, J = the union of J_+ and J_-, and K =
the union of K_+ and K_-. In this paper we identify the topological structure
of these sets when p is hyperbolic and |a| is sufficiently small, ie, when H is
a small perturbation of the polynomial p. The description involves projective
and inductive limits of objects defined in terms of p alone.
|
math
|
951 |
Absorbing Cantor sets in dynamical systems: Fibonacci maps
|
math.DS
|
In this paper we shall show that there exists a polynomial unimodal map f:
[0,1] -> [0,1] which is
1) non-renormalizable(therefore for each x from a residual set, $\omega(x)$
is equal to an interval),
2) for which $\omega(c)$ is a Cantor set, and
3) for which $\omega(x)=\omega(c)$ for Lebesgue almost all x.
So the topological and the metric attractor of such a map do not coincide.
This gives the answer to a question posed by Milnor.
|
math
|
952 |
Polynomial maps with a Julia set of positive measure
|
math.DS
|
In this paper we shall show that there exists L_0 such that for each even
integer L >= L_0 there exists $c_1 \in \rz$ for which the Julia set of $z -->
z^L + c_1$ has positive Lebesgue measure. This solves an old problem.
Editor's note: In 1997, it was shown by Xavier Buff that there was a serious
flaw in the argument, leaving a gap in the proof. Currently (1999), the
question of polynomials with a positive measure Julia sets remains open.
|
math
|
953 |
Inducing, slopes, and conjugacy classes
|
math.DS
|
We show that the conjugacy class of an eventually expanding continuous
piecewise affine interval map is contained in a smooth codimension 1
submanifold of parameter space. In particular conjugacy classes have empty
interior. This is based on a study of the relation between induced Markov maps
and ergodic theoretical behavior.
|
math
|
954 |
Hausdorff dimension and Kleinian groups
|
math.DS
|
Let G be a non-elementary, finitely generated Kleinian group, Lambda(G) its
limit set and Omega(G) = S \ Lambda(G) (S = the sphere) its set of
discontinuity. Let delta(G) be the critical exponent for the Poincar\'e series
and let Lambda_c be the conical limit set of G. Suppose Omega_0 is a simply
connected component of Omega(G). We prove that
(1) delta(G) = dim(Lambda_c).
(2) A simply connected component Omega is either a disk or dim(Omega)>1$.
(3) Lambda(G) is either totally disconnected, a circle or has dimension > 1,
(4) G is geometrically infinite iff dim(Lambda)=2.
(5) If G_n \to G algebraically then dim(Lambda) <= \liminf dim(Lambda_n).
(6) The Minkowski dimension of Lambda equals the Hausdorff dimension.
(7) If Area(Lambda)=0 then delta(G) = dim(Lambda(G)).
The proof also shows that \dim(Lambda(G)) > 1 iff the conical limit set has
dimension > 1 iff the Poincar\'e exponent of the group is > 1. Furthermore, a
simply connected component of Omega(G) either is a disk or has
non-differentiable boundary in the the sense that the (inner) tangent points of
\partial Omega have zero 1-dimensional measure. Almost every point (with
respect to harmonic measure) is a twist point.
|
math
|
955 |
Dynamical zeta functions for maps of the interval
|
math.DS
|
A dynamical zeta function $\zeta$ and a transfer operator $\scr L$ are
associated with a piecewise monotone map $f$ of the interval $[0,1]$ and a
weight function $g$. The analytic properties of $\zeta$ and the spectral
properties of $\scr L$ are related by a theorem of Baladi and Keller under an
assumption of ``generating partition''. It is shown here how to remove this
assumption and, in particular, extend the theorem of Baladi and Keller to the
case when $f$ has negative Schwarzian derivative.
|
math
|
956 |
Iterations of rational functions: which hyperbolic components contain polynomials?
|
math.DS
|
Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann
sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or
all but one critical points (or values) are in the immediate basin of
attraction to an attracting fixed point then there exists a polynomial in the
component $H(f)$ of $H^d$ containing $f$. If all critical points are in the
immediate basin of attraction to an attracting fixed point or parabolic fixed
point then $f$ restricted to Julia set is conjugate to the shift on the
one-sided shift space of $d$ symbols.
We give exotic examples of maps of an arbitrary degree $d$ with a non-simply
connected, completely invariant basin of attraction and arbitrary number $k \ge
2$ of critical points in the basin. For such a map $f\in H^d$ with $k<d$ there
is no polynomial in $H(f)$.
Finally we describe a computer experiment joining an exotic example to a
Newton's method (for a polynomial) rational function with a 1-parameter family
of rational maps.
|
math
|
957 |
A toral diffeomorphism with a non-polygonal rotation set
|
math.DS
|
We construct a diffeomorphism of the two-dimensional torus which is isotopic
to the identity and whose rotation set is not a polygon.
|
math
|
958 |
The set of maps F_{a,b}: x -> x+a+{b/{2 pi}} sin(2 pi x) with any given rotation interval is contractible
|
math.DS
|
Consider the two-parameter family of real analytic maps $F_{a,b}:x \mapsto x+
a+{b\over 2\pi} \sin(2\pi x)$ which are lifts of degree one endomorphisms of
the circle. The purpose of this paper is to provide a proof that for any closed
interval $I$, the set of maps $F_{a,b}$ whose rotation interval is $I$, form a
contractible set.
|
math
|
959 |
Ratio geometry, rigidity and the scenery process for hyperbolic Cantor sets
|
math.DS
|
Given a $C^{1+\gamma}$ hyperbolic Cantor set $C$, we study the sequence
$C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show
that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process.
The values of this process, called limit sets, are indexed by a H\"older
continuous set-valued function defined on D. Sullivan's dual Cantor set. We
show the limit sets are themselves $C^{k+\gamma}, C^\infty$ or $C^\omega$
hyperbolic Cantor sets, with the highest degree of smoothness which occurs in
the $C^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to the
following rigidity theorem: if two $C^{k+\gamma}, C^\infty$ or $C^\omega$
hyperbolic Cantor sets are $C^1$-conjugate, then the conjugacy (with a
different extension) is in fact already $C^{k+\gamma}, C^\infty$ or $C^\omega$.
Within one $C^{1+\gamma}$ conjugacy class, each smoothness class is a Banach
manifold, which is acted on by the semigroup given by rescaling subintervals.
Conjugacy classes nest down, and contained in the intersection of them all is a
compact set which is the attractor for the semigroup: the collection of limit
sets. Convergence is exponentially fast, in the $C^1$ norm.
|
math
|
960 |
Coexistence of critical orbit types in sub-hyperbolic polynomial maps
|
math.DS
|
We establish necessary and sufficient conditions for the realization of
mapping schemata as post-critically finite polynomials, or more generally, as
post-critically finite polynomial maps from a finite union of copies of the
complex numbers {\bf C} to itself which have degree two or more in each copy.
As a consequence of these results we prove a transitivity relation between
hyperbolic components in parameter space which was conjectured by Milnor.
|
math
|
961 |
Open sets of diffeomorphisms having two attractors, each with an everywhere dense basin
|
math.DS
|
We announce the discovery of a diffeomorphism of a three-dimensional manifold
with boundary which has two disjoint attractors. Each attractor attracts a set
of positive $3$-dimensional Lebesgue measure whose points of Lebesgue density
are dense in the whole manifold. This situation is stable under small
perturbations.
|
math
|
962 |
Acceleration of bouncing balls in external fields
|
math.DS
|
We introduce two models, the Fermi-Ulam model in an external field and a one
dimensional system of bouncing balls in an external field above a periodically
oscillating plate. For both models we investigate the possibility of unbounded
motion. In a special case the two models are equivalent.
|
math
|
963 |
Dual billiards, twist maps, and impact oscillators
|
math.DS
|
In this paper techniques of twist map theory are applied to the annulus maps
arising from dual billiards on a strictly convex closed curve G in the plane.
It is shown that there do not exist invariant circles near G when there is a
point on G where the radius of curvature vanishes or is discontinuous. In
addition, when the radius of curvature is not $C^1$ there are examples with
orbits that converge to a point of G. If the derivative of the radius of
curvature is bounded, such orbits cannot exist. The final section of the paper
concerns an impact oscillator whose dynamics are the same as a dual billiards
map. The appendix is a remark on the connection of the inverse problems for
invariant circles in billiards and dual billiards.
|
math
|
964 |
Some remarks on periodic billiard orbits in rational polygons
|
math.DS
|
A polygon is called rational if the angle between each pair of sides is a
rational multiple of $\pi.$ The main theorem we will prove is
Theorem 1: For rational polygons, periodic points of the billiard flow are
dense in the phase space of the billiard flow.
This is a strengthening of Masur's theorem, who has shown that any rational
polygon has ``many'' periodic billiard trajectories; more precisely, the set of
directions of the periodic trajectories are dense in the set of velocity
directions $\S^1.$ We will also prove some refinements of Theorem 1: the ``well
distribution'' of periodic orbits in the polygon and the residuality of the
points $q \in Q$ with a dense set of periodic directions.
|
math
|
965 |
Forcing of periodic orbits for interval maps and renormalization of piecewise affine maps
|
math.DS
|
We prove that for continuous maps on the interval, the existence of an
n-cycle, implies the existence of n-1 points which interwind the original ones
and are permuted by the map. We then use this combinatorial result to show that
piecewise affine maps (with no zero slope) cannot be infinitely renormalizable.
|
math
|
966 |
Measures with infinite Lyapunov exponents for the periodic Lorentz gas
|
math.DS
|
In \cite{Ch91a} it was shown that the billiard ball map for the periodic
Lorentz gas has infinite topological entropy. In this article we study the set
of points with infinite Lyapunov exponents. Using the cell structure developed
in \cite{BSC90,Ku} we construct an ergodic invariant probability measure with
infinite topological entropy supported on this set. Since the topological
entropy is infinite this is a measure of maximal entropy. From the construction
it is clear that there many such measures can coexist on a single component of
topological transitivity. We also construct an ergodic invariant probability
measure with finite entropy which is supported on this set showing that
infinite exponents do not necessarily lead to infinite entropy.
|
math
|
967 |
Internal addresses in the Mandelbrot set and Galois groups of polynomials
|
math.DS
|
We describe an interesting interplay between symbolic dynamics, the structure
of the Mandelbrot set, permutations of periodic points achieved by analytic
continuation, and Galois groups of certain polynomials.
Internal addresses are a convenient and efficient way of describing the
combinatorial structure of the Mandelbrot set, and of giving geometric meaning
to the ubiquitous kneading sequences in human-readable form (Sections 3 and 4).
A simple extension, \emph{angled internal addresses}, distinguishes
combinatorial classes of the Mandelbrot set and in particular distinguishes
hyperbolic components in a concise and dynamically meaningful way.
This combinatorial description of the Mandelbrot set makes it possible to
derive existence theorems for certain kneading sequences and internal addresses
in the Mandelbrot set (Section~6) and to give an explicit description of the
associated parameters. These in turn help to establish some algebraic results
about permutations of periodic points and to determine Galois groups of certain
polynomials (Section~7).
Through internal addresses, various areas of mathematics are thus related in
this manuscript, including symbolic dynamics and permutations, combinatorics of
the Mandelbrot set, and Galois groups.
|
math
|
968 |
Rational Maps Whose Fatou Components Are Jordan Domains
|
math.DS
|
We prove: If $f(z)$ is a critically finite rational map which has exactly two
critical points and which is not conjugate to a polynomial, then the boundary
of every Fatou component of $f$ is a Jordan curve. If $f(z)$ is a hyperbolic
critically finite rational map all of whose postcritical points are periodic,
then there exists a cycle of Fatou components whose boundaries are Jordan
curves. We give examples of critically finite hyperbolic rational maps $f$ with
the property that on the closure of a Fatou component $\Omega$ satisfying
$f(\Omega)=\Omega$, $f|_{\bdry \Omega}$ is not topologically conjugate to the
dynamics of any polynomial on its Julia set.
|
math
|
969 |
Laminations in holomorphic dynamics
|
math.DS
|
We suggest a way to associate to a rational map of the Riemann sphere a three
dimensional object called a hyperbolic orbifold 3-lamination. The relation of
this object to the map is analogous to the relation of a hyperbolic 3-manifold
to a Kleinian group. In order to construct the 3-lamination we analyze the
natural extension of a rational map and the complex affine structure on the
canonical 2-dimensional leaf space contained in it. In this paper the
construction is carried out in full for post-critically finite maps. We show
that the corresponding laminations have a compact convex core. As a first
application we give a three-dimensional proof of Thurston's rigidity for
post-critically finite mappings, via the "lamination extension" of the proofs
of the Mostow and Marden rigidity and isomorphism theorems for hyperbolic
3-manifolds. An Ahlfors-type argument for zero measure of the Julia set is
applied along the way. This approach also provides a new point of view on the
Lattes deformable examples.
|
math
|
970 |
Chaos in the Lorenz equations: a computer-assisted proof
|
math.DS
|
A new technique for obtaining rigorous results concerning the global dynamics
of nonlinear systems is described. The technique combines abstract existence
results based on the Conley index theory with computer- assisted computations.
As an application of these methods it is proven that for an explicit parameter
value the Lorenz equations exhibit chaotic dynamics.
|
math
|
971 |
Further travels with my ant
|
math.DS
|
We discuss some properties of a class of cellular automata sometimes called a
"generalized ant". This system is perhaps most easily understood by thinking of
an ant which moves about a lattice in the plane. At each vertex (or "cell"),
the ant turns right or left, depending on the the state of the cell, and then
changes the state of the cell according to certain prescribed rule strings.
(This system has been the subject of several Mathematical Entertainments
columns in the Mathematical Intelligencer; this article will be a future such
column). At various times, the distributions of the states of the cells for
certain ants is bilaterally symmetric; we categorize a class of ants for which
this is the case and give a proof using Truchet tiles.
|
math
|
972 |
Non-accessible critical points of Cremer polynomials
|
math.DS
|
It is shown that a polynomial with a Cremer periodic point has a
non-accessible critical point in its Julia set provided that the Cremer
periodic point is approximated by small cycles.
|
math
|
973 |
Dynamics of quadratic polynomials, I: Combinatorics and geometry of the Yoccoz puzzle
|
math.DS
|
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.
|
math
|
974 |
Dynamics of quadratic polynomials: Complex bounds for real maps
|
math.DS
|
We extend Sullivan's complex a priori bounds to real quadratic polynomials
with essentially bounded combinatorics. Combined with the previous results of
the first author, this yields complex bounds for all real quadratics. Local
connectivity of the corresponding Julia sets follows.
|
math
|
975 |
Commuting polynomials and polynomials with same Julia set
|
math.DS
|
It has been known since Julia that polynomials commuting under composition
have the same Julia set. More recently in the works of Baker and Eremenko,
Fern\'andez, and Beardon, results were given on the converse question: When do
two polynomials have the same Julia set? We give a complete answer to this
question and show the exact relation between the two problems of polynomials
with the same Julia set and commuting pairs.
|
math
|
976 |
Local connectivity of the Julia set of real polynomials
|
math.DS
|
One of the main questions in the field of complex dynamics is the question
whether the Mandelbrot set is locally connected, and related to this, for which
maps the Julia set is locally connected. In this paper we shall prove the
following
Main Theorem: Let $f$ be a polynomial of the form $f(z)=z^d +c$ with $d$ an
even integer and $c$ real. Then the Julia set of $f$ is either totally
disconnected or locally connected. In particular, the Julia set of $z^2+c$ is
locally connected if $c \in [-2,1/4]$ and totally disconnected otherwise.
|
math
|
977 |
A volume-preserving counterexample to the Seifert conjecture
|
math.DS
|
We prove that every 3-manifold possesses a $C^1$, volume-preserving flow with
no fixed points and no closed trajectories. The main construction is a
volume-preserving version of the Schweitzer plug. We also prove that every
3-manifold possesses a volume-preserving, $C^\infty$ flow with discrete closed
trajectories and no fixed points (as well as a PL flow with the same geometry),
which is needed for the first result. The proof uses a Dehn-twisted Wilson-type
plug which also preserves volume.
|
math
|
978 |
Homeomorphisms between Limbs of the Mandelbrot Set
|
math.DS
|
Given $p/q$ and $p'/q$ both irreducible, we construct homeomorphisms between
the $p/q$ and the $p'/q$ limbs of the Mandelbrot set. This homeomorphisms are
not compatible with the dynamics. Moreover, the filled Julia sets of
corresponding parameter values are also homeomorphic. All the homeomorphisms
above have counterparts on the combinatorial level relating corresponding
external arguments, in the dynamical planes as well as in the parameter spaces.
Assuming local connectivity of $M$ we may conclude that the constructed
homeomorphisms between limbs are compatible with the embeddings of the limbs in
the plane.
|
math
|
979 |
Topological conjugacy of circle diffeomorphisms
|
math.DS
|
The classical criterion for a circle diffeomorphism to be topologically
conjugate to an irrational rigid rotation was given by A. Denjoy. In 1985, one
of us (Sullivan) gave a new criterion. There is an example satisfying Denjoy's
bounded variation condition rather than Sullivan's Zygmund condition and vice
versa. This paper will give the third criterion which is implied by either of
the above criteria.
|
math
|
980 |
Necessity and Chance: deterministic chaos in ecology and evolution
|
math.DS
|
This is an outline of my Gibbs Lecture to the American Mathematical Society
in January 1994; it is essentially a sign-posted guide to a still-developing
literature.
|
math
|
981 |
Bizarre topology is natural in dynamical systems
|
math.DS
|
We describe an example of a $C^\infty$ diffeomorphism on a 7--manifold which
has a compact invariant set such that uncountably many of its connected
components are pseudocircles. (Any 7--manifold will suffice.) Furthermore, any
diffeomorphism which is sufficiently close (in the $C^1$ metric) to the
constructed map has a similar invariant set, and the dynamics of the map on the
invariant set are chaotic.
|
math
|
982 |
Critical points on the boundaries of Siegel disks
|
math.DS
|
Let $f$ be a polynomial map of the Riemann sphere of degree at least two. We
prove that if $f$ has a Siegel disk $G$ on which the rotation number satisfies
a diophantine condition, then the boundary of $G$ contains a critical point.
|
math
|
983 |
Dynamics of the family lambda tan z
|
math.DS
|
We study the dynamics of the tangent family z -> lambda tan z for lambda
complex and give a complete classification of their stable behavior. We also
characterize the the hyperbolic components and give a combinatorial description
their deployment in the parameter plane.
|
math
|
984 |
Local connectivity of the Mandelbrot set at certain infinitely renormalizable points
|
math.DS
|
We construct a subset of the Mandelbrot set which is dense on the boundary of
the Mandelbrot set and which consists of only infinitely renormalizable points
such that the Mandelbrot set is locally connected at every point of this
subset. We prove the local connectivity by finding bases of connected
neighborhoods directly.
|
math
|
985 |
On measure and Hausdorff dimension of Julia sets for holomorphic Collet--Eckmann maps
|
math.DS
|
Let $f:\bar\bold C\to\bar\bold C$ be a rational map on the Riemann sphere ,
such that for every $f$-critical point $c\in J$ which forward trajectory does
not contain any other critical point, $|(f^n)'(f(c))|$ grows exponentially fast
(Collet--Eckmann condition), there are no parabolic periodic points, and else
such that Julia set is not the whole sphere. Then smooth (Riemann) measure of
the Julia set is 0.
For $f$ satisfying additionally Masato Tsujii's condition that the average
distance of $f^n(c)$ from the set of critical points is not too small, we prove
that Hausdorff dimension of Julia set is less than 2. This is the case for
$f(z)=z^2+c$ with $c$ real, $0\in J$, for a positive line measure set of
parameters $c$.
|
math
|
986 |
Complex bounds for critical circle maps
|
math.DS
|
We use the methods developed with M. Lyubich for proving complex bounds for
real quadratics to extend E. De Faria's complex a priori bounds to all critical
circle maps with an irrational rotation number. The contracting property for
renormalizations of critical circle maps follows. In the Appendix we give an
application of the complex bounds for proving local connectivity of some Julia
sets.
|
math
|
987 |
The Renormalization Method and Quadratic-Like Maps
|
math.DS
|
The renormalization of a quadratic-like map is studied. The three-dimensional
Yoccoz puzzle for an infinitely renormalizable quadratic-like map is discussed.
For an unbranched quadratic-like map having the {\sl a priori} complex bounds,
the local connectivity of its Julia set is proved by using the
three-dimensional Yoccoz puzzle. The generalized version of Sullivan's sector
theorem is discussed and is used to prove his result that the Feigenbaum
quadratic polynomial has the {\sl a priori} complex bounds and is unbranched. A
dense subset on the boundary of the Mandelbrot set is constructed so that for
every point of the subset, the corresponding quadratic polynomial is unbranched
and has the {\sl a priori} complex bounds.
|
math
|
988 |
Period doubling, entropy, and renormalization
|
math.DS
|
We show that in any family of stunted sawtooth maps, the set of maps whose
set of periods is the set of all powers of 2 has no interior point, i.e., the
combinatorial description of the boundary of chaos coincides with the
topological description. We also show that, under mild assumptions, smooth
multimodal maps whose set of periods is the set of all powers of 2 are
infinitely renormalizable.
|
math
|
989 |
Surgery on postcritically finite rational maps by blowing up an arc
|
math.DS
|
Using Thurston's characterization of postcritically finite rational functions
as branched coverings of the sphere to itself, we give a new method of
constructing new conformal dynamical systems out of old ones. Let $f(z)$ be a
rational map and suppose that the postcritical set $P(f)$ is finite. Let
$\alpha$ be an embedded closed arc in the sphere and suppose that $f|{\alpha}$
is a homeomorphism. Define a branched covering $g$ as follows. Cut the sphere
open along $\alpha$. Glue in a closed disc $D$. Map $S^{2} - \Int (D)$ via $f$
and $\Int (D)$ by a homeomorphism to the complement of $f(\alpha)$. We prove
theorems which give combinatorial conditions on $f$ and $\alpha$ for $g$ to be
equivalent in the sense of Thurston to a rational map. The main idea in our
proofs is a general theorem which forces a possible obstruction for $g$ away
from the disc $D$ on which the new dynamics is defined.
|
math
|
990 |
Dynamics of quadratic polynomials II: rigidity
|
math.DS
|
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.
|
math
|
991 |
Lagrangian systems on hyperbolic manifolds
|
math.DS
|
This paper gives two results that show that the dynamics of a time-periodic
Lagrangian system on a hyperbolic manifold are at least as complicated as the
geodesic flow of a hyperbolic metric. Given a hyperbolic geodesic in the
Poincar\'e ball, Theorem A asserts that there are minimizers of the lift of the
Lagrangian system that are a bounded distance away and have a variety of
approximate speeds. Theorem B gives the existence of a collection of compact
invariant sets of the Euler-Lagrange flow that are semiconjugate to the
geodesic flow of a hyperbolic metric. These results can be viewed as a
generalization of the Aubry-Mather theory of twist maps and the
Hedlund-Morse-Gromov theory of minimal geodesics on closed surfaces and
hyperbolic manifolds.
|
math
|
992 |
Dynamical stability in Lagrangian systems
|
math.DS
|
This paper surveys various results concerning stability for the dynamics of
Lagrangian (or Hamiltonian) systems on compact manifolds. The main, positive
results state, roughly, that if the configuration manifold carries a hyperbolic
metric, \ie a metric of constant negative curvature, then the dynamics of the
geodesic flow persists in the Euler-Lagrange flows of a large class of
time-periodic Lagrangian systems. This class contains all time-periodic
mechanical systems on such manifolds. Many of the results on Lagrangian systems
also hold for twist maps on the cotangent bundle of hyperbolic manifolds. We
also present a new stability result for autonomous Lagrangian systems on the
two torus which shows, among other things, that there are minimizers of all
rotation directions. However, in contrast to the previously known
\cite{hedlund} case of just a metric, the result allows the possibility of gaps
in the speed spectrum of minimizers. Our negative result is an example of an
autonomous mechanical Lagrangian system on the two-torus in which this gap
actually occurs. The same system also gives us an example of a Euler-Lagrange
minimizer which is not a Jacobi minimizer on its energy level.
|
math
|
993 |
Teichmuller distance for some polynomial-like maps
|
math.DS
|
In this work we will show that the Teichm\"{u}ller distance for all elements
of a certain class of generalized polynomial-like maps (the class of
off-critically hyperbolic generalized polynomial-like maps) is actually a
distance, as in the case of real polynomials with connected Julia set, as
studied by Sullivan. This class contains several important classes of
generalized polynomial-like maps, namely: Yoccoz, Lyubich, Sullivan and
Fibonacci. In our proof we can not use external arguments (like external
classes). Instead we use hyperbolic sets inside the Julia sets of our maps.
Those hyperbolic sets will allow us to use our main analytic tool, namely
Sullivan's rigidity Theorem for non-linear analytic hyperbolic systems. Lyubich
has constructed a measure of maximal entropy measure $m$ on the Julia set of
any rational function $f$. Zdunik classified exactly when the Hausdorff
dimension of $m$ equals the Hausdorff dimension of the Julia set. We show that
the strict inequality holds if $f$ is off-crititcally hyperbolic, except for
Chebyshev polynomials. This result is a particular case of Zdunik's result if
we consider $f$ as a polynomial, but is an extension of Zdunik's result if $f$
is a generalized polynomial-like map. The proof follows from the non-existence
of invariant affine structure.
|
math
|
994 |
Organization of parameter space for simple circle maps: the Farey web
|
math.DS
|
We define the Farey web --- a collection of loci in the parameter plane of
families of simple non-invertible maps of the circle. We prove some results
about the arrangement of these loci and their relationships with other
dynamically significant features of the parameter plane. The results enable us
to provide short proofs for a number of theorems about the organization of
frequency-locking.
|
math
|
995 |
Heteroclinic orbits and transport in a perturbed integrable standard map
|
math.DS
|
Explicit formulae are given for the saddle connection for an integrable
family of standard maps studied by Suris. A generalization of Melnikov's method
shows that, upon perturbation, this connection is destroyed. We give explicit
formula for the first order approximation of the area of the lobes of the
resultant turnstile. It is shown that the lobe area is exponentially small in
the limit when the Suris map approaches the trivial twist map.
|
math
|
996 |
The Boltzmann-Sinai Ergodic Hypothesis for Hard Ball Systems
|
math.DS
|
This paper has been withdrawn by the authors, due a crucial error.
|
math
|
997 |
The characteristic exponents of the falling ball model
|
math.DS
|
We study the characteristic exponents of the Hamiltonian system of $n$ ($\ge
2$) point masses $m_1,\dots,m_n$ freely falling in the vertical half line
$\{q|\, q\ge 0\}$ under constant gravitation and colliding with each other and
the solid floor $q=0$ elastically. This model was introduced and first studied
by M. Wojtkowski. Hereby we prove his conjecture: All relevant characteristic
(Lyapunov) exponents of the above dynamical system are nonzero, provided that
$m_1\ge\dots\ge m_n$ (i. e. the masses do not increase as we go up) and $m_1\ne
m_2$.
|
math
|
998 |
Porosity of Collet-Eckmann Julia sets
|
math.DS
|
We prove that the Julia set of a rational map of the Riemann sphere
satisfying the Collet-Eckmann condition and having no parabolic periodic point
is mean porous, if it is not the whole sphere. It follows that the Minkowski
dimension of the Julia set is less than 2.
|
math
|
999 |
The periodic points of renormalization
|
math.DS
|
It will be shown that the renormalization operator, acting on the space of
smooth unimodal maps with critical exponent greater than 1, has periodic points
of any combinatorial type.
|
math
|
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