Šarkovského věta a diferenciální rovnice
Šarkovského věta a diferenciální rovnice. II
Šarkovského věta a diferenciální rovnice III
Článek je pokračováním našich dřívějších stejnojmenných příspěvků, v nichž jsme vyšetřovali možnosti aplikace různých variant Šarkovského věty o koexistenci periodických bodů a orbit pro intervalová zobrazení na diferenciální rovnice a inkluze. I tentokrát se budeme zabývat stejným problémem, avšak pro zobrazení na kružnici. Na rozdíl od intervalových zobrazení zde totiž mj. nemusí periodické orbity implikovat existenci pevných bodů, což představuje největší překážku. Na druhé straně lze takto rozšířit...
Scaling in a map of the two-torus.
Self-similar centralizers of circle maps.
Self-Similar Sets 3. Constructions with Sofic Systems.
Semiconjugacy to a map of a constant slope
It is well known that any continuous piecewise monotone interval map f with positive topological entropy is semiconjugate to some piecewise affine map with constant slope . We prove this result for a class of Markov countably piecewise monotone continuous interval maps.
Semidefinite characterisation of invariant measures for one-dimensional discrete dynamical systems
Using recent results on measure theory and algebraic geometry, we show how semidefinite programming can be used to construct invariant measures of one-dimensional discrete dynamical systems (iterated maps on a real interval). In particular we show that both discrete measures (corresponding to finite cycles) and continuous measures (corresponding to chaotic behavior) can be recovered using standard software.
Sequences of fixed point indices of iterations in dimension 2.
Shadowing and expansivity in subspaces
We address various notions of shadowing and expansivity for continuous maps restricted to a proper subset of their domain. We prove new equivalences of shadowing and expansive properties, we demonstrate under what conditions certain expanding maps have shadowing, and generalize some known results in this area. We also investigate the impact of our theory on maps of the interval.
Sharkovskiĭ's theorem holds for some discontinuous functions
We show that the Sharkovskiĭ ordering of periods of a continuous real function is also valid for every function with connected graph. In particular, it is valid for every DB₁ function and therefore for every derivative. As a tool we apply an Itinerary Lemma for functions with connected graph.
Sharp determinants.
Shift spaces and attractors in noninvertible horseshoes
As is well known, a horseshoe map, i.e. a special injective reimbedding of the unit square in (or more generally, of the cube in ) as considered first by S. Smale [5], defines a shift dynamics on the maximal invariant subset of (or ). It is shown that this remains true almost surely for noninjective maps provided the contraction rate of the mapping in the stable direction is sufficiently strong, and bounds for this rate are given.
Simple and complex dynamics for circle maps.
The continuous self maps of a closed interval of the real line with zero topological entropy can be characterized in terms of the dynamics of the map on its chain recurrent set. In this paper we extend this characterization to continuous self maps of the circle. We show that, for these maps, the chain recurrent set can exhibit a new dynamic behaviour which is specific of the circle maps of degree one.
Sinai-Ruelle-Bowen measures for contracting Lorenz maps and flows
Small perturbations of a discrete twist map
Smooth conjugacy classes of circle diffeomorphisms with irrational rotation number
We prove the C¹-density of every -conjugacy class in the closed subset of diffeomorphisms of the circle with a given irrational rotation number.
Solution to the gradient problem of C.E. Weil.
In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set G ⊂ R2 we construct a differentiable function f: G → R for which there exists an open set Ω1 ⊂ R2 such that ∇f(p) ∈ Ω1 for a p ∈ G but ∇f(q) ∉ Ω1 for almost every q ∈ G. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.
Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains
We consider a large class of piecewise expanding maps T of [0, 1] with a neutral fixed point, and their associated Markov chains Yi whose transition kernel is the Perron–Frobenius operator of T with respect to the absolutely continuous invariant probability measure. We give a large class of unbounded functions f for which the partial sums of f○Ti satisfy both a central limit theorem and a bounded law of the iterated logarithm. For the same class, we prove that the partial sums of f(Yi) satisfy a...
Some limit ratio theorem related to a real endomorphism in case of a neutral fixed point