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

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

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

Teichm\"uller space of Fibonacci maps

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

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

The cohomological equations in nonsmooth categories

The functional equation $\varphi(Fx) - \varphi(x) = \gamma(x)$ is considered in topological, measurable and related categories from the point of view of functional analysis and general theory of dynamical systems. The material is presented in the form of a self-contained survey

On the Lebesgue measure of the Julia set of a quadratic polynomial

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

Examples of Feigenbaum Julia sets with small Hausdorff dimension

We give examples of infinitely renormalizable quadratic polynomials F_c: z\maps to z^2+c with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrar y close to 1. The combinatorics of the renormalization involved is close to the Chebyshev one . The argument is based upon a new tool, a ``Recursive Quadratic Estimate'' for the Poincar\'e series of an infinitely re normalizable map.Comment: 13 page

The Quasi-Additivity Law in Conformal Geometry

On a Riemann surface $S$ of finite type containing a family of $N$ disjoint disks $D_i$ (``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