36 research outputs found

### Symbolic dynamics for Lozi maps

In this paper we study the family of the Lozi maps $L_{a,b} : {\mathbb R}^2
\to {\mathbb R}^2$, $L_{a,b} = (1 + y - a|x|, bx)$, and their strange
attractors $\Lambda_{a,b}$. We introduce the set of kneading sequences for the
Lozi map and prove that it determines the symbolic dynamics for that map. We
also introduce two other equivalent approaches

### Lozi-like maps

We define a broad class of piecewise smooth plane homeomorphisms which have
properties similar to the properties of Lozi maps, including the existence of a
hyperbolic attractor. We call those maps Lozi-like. For those maps one can
apply our previous results on kneading theory for Lozi maps. We show a strong
numerical evidence that there exist Lozi-like maps that have kneading sequences
different than those of Lozi maps

### Periodic points of latitudinal sphere maps

For the maps of the two-dimensional sphere into itself that preserve the latitude foliation and are differentiable at the poles, lower estimates of the number of fixed points for the maps and their iterates are obtained. Those estimates also show that the growth rate of the number of fixed points of the iterates is larger than or equal to the logarithm of the absolute value of the degree of the map

### Affine actions of a free semigroup on the real line

We consider actions of the free semigroup with two generators on the real
line, where the generators act as affine maps, one contracting and one
expanding, with distinct fixed points. Then every orbit is dense in a
half-line, which leads to the question whether it is, in some sense, uniformly
distributed. We present answers to this question for various interpretations of
the phrase ``uniformly distributed''

### Counting Preimages

For non-invertible maps, subshifts that are mainly of finite type and piecewise monotone interval maps, we investigate what happens if we follow backward trajectories, which are random in the sense that, at each step, every preimage can be chosen with equal probability. In particular, we ask what happens if we try to compute the entropy this way. It turns out that, instead of the topological entropy, we get the metric entropy of a special measure, which we call the fair measure. In general, this entropy (the fair entropy) is smaller than the topological entropy. In such a way, for the systems that we consider, we get a new natural measure and a new invariant of topological conjugacy

### Topological entropy of Bunimovich stadium billiards

We estimate from below the topological entropy of the Bunimovich stadium
billiards. We do it for long billiard tables, and find the limit of estimates
as the length goes to infinity.Comment: 10 pages, 4 figure

### Expansion properties of Double Standard Maps

For the family of Double Standard Maps $f_{a,b}=2x+a+\frac{b}{\pi} \sin2\pi x
\quad\pmod{1}$ we investigate the structure of the space of parameters $a$ when
$b=1$ and when $b\in[0,1)$. In the first case the maps have a critical point,
but for a set of parameters $E_1$ of positive Lebesgue measure there is an
invariant absolutely continuous measure for $f_{a,1}$. In the second case there
is an open nonempty set $E_b$ of parameters for which the map $f_{a,b}$ is
expanding. We show that as $b\nearrow 1$, the set $E_b$ accumulates on many
points of $E_1$ in a regular way from the measure point of view