Alexander Blokh

The “Spectral” Decomposition for One-Dimensional Maps

We construct the “spectral” decomposition of the sets \(\overline {Per\,f}\), ω(f) = ∪ω(x) and Ω(f) for a continuous map f: [0,1] → [0,1]. Several corollaries are obtained; the main ones describe the generic properties of f-invariant measures, the structure of the set Ω(f)\\(\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 space of -limit sets of a continuous map of the interval

We first give a geometric characterization of ω-limit sets. We then use this characterization to prove that the family of ω-limit sets of a continuous interval map is closed with respect to the Hausdorff metric. Finally, we apply this latter result to other dynamical systems.

An inequality for laminations, Julia sets and `growing trees'

For a closed lamination on the unit circle invariant under z\mapsto z^d we prove an inequality relating the number of points in the ‘gaps’ with infinite pairwise disjoint orbits to the degree; in particular, this gives estimates on the cardinality of any such ‘gap’ as well as on the number of distinct grand orbits of such ‘gaps’. As a tool, we introduce and study a dynamically defined growing tree in the quotient space. We also use our techniques to obtain for laminations an analog of Sullivans no wandering domain theorem. Then we apply these results to Julia sets of polynomials.

Non-existence of wandering intervals and structure of topological attractors of one dimensional dynamical systems 2. The smooth case

We prove that an arbitrary one dimensional smooth dynamical system with non-degenerate critical points has no wandering intervals.

NEW ORDER FOR PERIODIC ORBITS OF INTERVAL MAPS

We propose a new classification of periodic orbits of interval maps via over-rotation pairs. We prove for them a theorem similar to the Sharkovskĭi Theorem.

ROTATION NUMBERS, TWISTS AND A SHARKOVSKII-MISIUREWICZ-TYPE ORDERING FOR PATTERNS ON THE INTERVAL

We introduce rotation numbers and pairs characterizing cyclic patterns on an interval and a special order among them; then we prove the theorem which specializes the Sharkovskii theorem in this setting.

Measure and Dimension of Solenoidal Attractors of One Dimensional Dynamical Systems

AbstractLetf:M→M be aC∞-map of the interval or the circle with non-flat critical points. A closed invariant subsetA⊂M is called a solenoidal attractor off if it has the following structure:

On minimal maps of 2-manifolds

A\mathop \cap \limits_{n = 1}^\infty \mathop \cup \limits_{k = 0}^{P_n - 1} l_k^{(n)}