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**PREPRINTS IN THIS SERIES, IN PDF FORMAT.**

* Starred papers have appeared in the journal cited.

*Periods Implying Almost All Periods, Trees with Snowflakes, and Zero Entropy Maps*

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 $m \leq Edg(X)$ is an odd integer with prime divisors less than $End(X)+1$.

*The "spectral" decomposition for one-dimensional maps*

We construct the "spectral" decomposition of the sets $\overline{Per\,f},$ $\omega(f)=\cup\omega(x)$ and $\Omega(f)$ for a continuous map $f:[0,1]\rightarrow [0,1]$. Several corollaries are obtained; the main ones describe the generic properties of $f$-invariant measures, the structure of the set $\Omega(f)\setminus \overline{Per\,f}$ 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.

*The Fibonacci Unimodal Map*

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.

*Quasisymmetric Conjugacies Between Unimodal Maps*

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.

*A Comparison of Harmonic and Balanced Measures on Cantor Repellors*

Let *J* be a Cantor repellor of a conformal map *f*. Provided *f* is a polynomial-like or *R*-symmetric, we prove that harmonic measure on *J* is equivalent to the measure of maximal entropy if and only if *f* is conformally equivalent to a polynomial. We also show that this is not true for general Cantor repellors: there is a non-polynomial algebraic function generating a Cantor repellor on which above two measures coincide.

*Dynamics of Certain Non-Conformal Degree Two Maps on the Plane*

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 ($z \mapsto z^2 + c$), 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).

*On the Quasisymmetrical Classification of Infinitely Renormalizable Maps: I. Maps with Feigenbaum's Topology*

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.

*On the Quasisymmetrical Classification of Infinitely Renormalizable Maps: II. Remarks on Maps with a Bounded Type Topology*

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.

*On the Realization of Fixed Point Portraits (an addendum to Goldberg & Milnor: Fixed Point Portraits)*

We establish that every formal critical portrait (as defined by Goldberg and Milnor), can be realized by a postcritically finite polynomial.

*Periodic Orbits for Hamiltonian systems in Cotangent Bundles*

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.