Representing real numbers in Möbius number systems
Rescaling of Markov shifts.
Resolvent conditions and powers of operators
We discuss the relation between the growth of the resolvent near the unit circle and bounds for the powers of the operator. Resolvent conditions like those of Ritt and Kreiss are combined with growth conditions measuring the resolvent as a meromorphic function.
Results and open questions on some invariants measuring the dynamical complexity of a map
Let f be a continuous map on a compact connected Riemannian manifold M. There are several ways to measure the dynamical complexity of f and we discuss some of them. This survey contains some results and open questions about relationships between the topological entropy of f, the volume growth of f, the rate of growth of periodic points of f, some invariants related to exterior powers of the derivative of f, and several invariants measuring the topological complexity of f: the degree (for the case...
Return time statistics for unimodal maps
We prove that a non-flat S-unimodal map satisfying a weak summability condition has exponential return time statistics on intervals around a.e. point. Moreover we prove that the convergence to the entropy in the Ornstein-Weiss formula enjoys normal fluctuations.
Right closing almost conjugacy for G-shifts of finite type
A G-shift of finite type (G-SFT) is a shift of finite type which commutes with the continuous action of a finite group G. For irreducible G-SFTs we classify right closing almost conjugacy, answering a question of Bill Parry.
Rigidity of projective conjugacy for quasiperiodic flows of Koch type
For quasiperiodic flows of Koch type, we exploit an algebraic rigidity of an equivalence relation on flows, called projective conjugacy, to algebraically characterize the deviations from completeness of an absolute invariant of projective conjugacy, called the multiplier group, which describes the generalized symmetries of the flow. We then describe three ways by which two quasiperiodic flows with the same Koch field are projectively conjugate when their multiplier groups are identical. The first...
Robust dynamic output feedback fault-tolerant control for Takagi-Sugeno fuzzy systems with interval time-varying delay via improved delay partitioning approach
This paper addresses the problem of robust fault-tolerant control design scheme for a class of Takagi-Sugeno fuzzy systems subject to interval time-varying delay and external disturbances. First, by using improved delay partitioning approach, a novel n-steps iterative learning fault estimation observer under H ∞ constraint is constructed to achieve estimation of actuator fault. Then, based on the online estimation information, a fuzzy dynamic output feedback fault-tolerant controller considered...
Roots of continuous piecewise monotone maps of an interval.
Sand pile automata
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.
Semi-étale groupoids and applications
We associate a -algebra to a locally compact Hausdorff groupoid with the property that the range map is locally injective. The construction generalizes J. Renault’s reduced groupoid -algebra of an étale groupoid and has the advantage that it works for the groupoid arising from a locally injective dynamical system by the method introduced in increasing generality by Renault, Deaconu and Anantharaman-Delaroche. We study the -algebras of such groupoids and give necessary and sufficient conditions...
Sequence entropy pairs and complexity pairs for a measure
In this paper we explore topological factors in between the Kronecker factor and the maximal equicontinuous factor of a system. For this purpose we introduce the concept of sequence entropy -tuple for a measure and we show that the set of sequence entropy tuples for a measure is contained in the set of topological sequence entropy tuples [H- Y]. The reciprocal is not true. In addition, following topological ideas in [BHM], we introduce a weak notion and a strong notion of complexity pair for a...
Set-polynomials and polynomial extension of the Hales-Jewett theorem.
Sets of -recurrence but not -recurrence
For every , we produce a set of integers which is -recurrent but not -recurrent. This extends a result of Furstenberg who produced a -recurrent set which is not -recurrent. We discuss a similar result for convergence of multiple ergodic averages. We also point out a combinatorial consequence related to Szemerédi’s theorem.
Shadow trees of Mandelbrot sets
The topology and combinatorial structure of the Mandelbrot set (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in . Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, . In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence is realized....
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.
Shadowing and internal chain transitivity
The main result of this paper is that a map f: X → X which has shadowing and for which the space of ω-limits sets is closed in the Hausdorff topology has the property that a set A ⊆ X is an ω-limit set if and only if it is closed and internally chain transitive. Moreover, a map which has the property that every closed internally chain transitive set is an ω-limit set must also have the property that the space of ω-limit sets is closed. As consequences of this result, we show that interval maps with...
Shadowing in multi-dimensional shift spaces
We show that the class of expansive actions with P.O.T.P. is wider than the class of actions topologically hyperbolic in some direction . Our main tool is an extension of a result by Walters to the multi-dimensional symbolic dynamics case.
Shape index and other indices of Conley type for local maps on locally compact Hausdorff spaces
We present a scheme for constructing various Conley indices for locally defined maps. In particular, we extend the shape index of Robbin and Salamon to the case of a locally defined map in a locally compact Hausdorff space. We compare the shape index with the cohomological Conley index for maps. We also prove the commutativity property of the Conley index, which is analogous to the commutativity property of the fixed point index.