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We extend the shape index, introduced by Robbin and Salamon and Mrozek, to locally defined maps in metric spaces. We show that this index is additive. Thus our construction answers in the affirmative two questions posed by Mrozek in [12]. We also prove that the shape index cannot be arbitrarily complicated: the shapes of q-adic solenoids appear as shape indices in natural modifications of Smale's horseshoes but there is not any compact isolated invariant set for any locally defined map in a locally...
We introduce simple connection matrices. We prove the existence of simple connection matrices for filtered differential vector spaces and Morse decompositions of compact metric spaces.
We study 1) the slopes of central branches of iterates of S-unimodal maps, comparing them to the derivatives on the critical trajectory, 2) the hyperbolic structure of Collet-Eckmann maps estimating the exponents, and under a summability condition 3) the images of the density one under the iterates of the Perron-Frobenius operator, 4) the density of the absolutely continuous invariant measure.
Let be a compact quasi self-similar set in a complete metric space and let denote the space of all probability measures on , endowed with the Fortet-Mourier metric. We will show that for a typical (in the sense of Baire category) measure in the lower concentration dimension is equal to , while the upper concentration dimension is equal to the Hausdorff dimension of .
We study the action of G = SL(2,ℝ), viewed as a group definable in the structure M = (ℝ,+,×), on its type space . We identify a minimal closed G-flow I and an idempotent r ∈ I (with respect to the Ellis semigroup structure * on ). We also show that the “Ellis group” (r*I,*) is nontrivial, in fact it is the group with two elements, yielding a negative answer to a question of Newelski.
We study recurrence and non-recurrence sets for dynamical systems on compact spaces, in particular for products of rotations on the unit circle . A set of integers is called -Bohr if it is recurrent for all products of rotations on , and Bohr if it is recurrent for all products of rotations on . It is a result due to Katznelson that for each there exist sets of integers which are -Bohr but not -Bohr. We present new examples of -Bohr sets which are not Bohr, thanks to a construction which...
It is known that a set of positive integers is a Poincaré set (also called intersective set, see I. Ruzsa (1982)) if and only if , where
and denotes the Hausdorff dimension (see C. Bishop, Y. Peres (2017), Theorem 2.5.5). In this paper we study the set by replacing with . It is surprising that there are some new phenomena to be worthy of studying. We study them and give several examples to explain our results.
Let (X,f) be a dynamical system. In general the set of all ω-limit sets of f is not closed in the hyperspace of closed subsets of X. In this paper we study the case when X is a graph, and show that the family of ω-limit sets of a graph map is closed with respect to the Hausdorff metric.
By means of the fixed-point methods and the properties of the -pseudo almost periodic functions, we prove the existence, uniqueness, and exponential stability of the -pseudo almost periodic solutions for some models of recurrent neural networks with mixed delays and time-varying coefficients, where is a positive measure. A numerical example is given to illustrate our main results.
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