445 research outputs found

    Symbolic dynamics for Lozi maps

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    In this paper we study the family of the Lozi maps La,b:R2R2L_{a,b} : {\mathbb R}^2 \to {\mathbb R}^2, La,b=(1+yax,bx)L_{a,b} = (1 + y - a|x|, bx), and their strange attractors Λa,b\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

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    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

    Affine actions of a free semigroup on the real line

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    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''

    Rotation sets of billiards with one obstacle

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    We investigate the rotation sets of billiards on the mm-dimensional torus with one small convex obstacle and in the square with one small convex obstacle. In the first case the displacement function, whose averages we consider, measures the change of the position of a point in the universal covering of the torus (that is, in the Euclidean space), in the second case it measures the rotation around the obstacle. A substantial part of the rotation set has usual strong properties of rotation sets

    A toral diffeomorphism with a non-polygonal rotation set

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    We construct a diffeomorphism of the two-dimensional torus which is isotopic to the identity and whose rotation set is not a polygon

    Periodic points of latitudinal sphere maps

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    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