Decompositions of factor maps involving bi-closing maps

We describe all possible decompositions of a finite-to-one factor map : _A S, from an irreducible shift of finite type onto a sofic shift, into two maps =, such that the range of is a shift of finite type, and is bi-closing. We also give necessary and sufficient conditions for to be almost topologically conjugate overS to a bi-closing map.     

Flow equivalence of sofic shifts

We classify certain sofic shifts (the irreducible Point Extension Type, or PET, sofic shifts) up to flow equivalence, using invariants of the canonical Fischer cover. There are two main ingredients.

1. (1)
   An extension theorem, for extending flow equivalences of subshifts to flow equivalent irreducible shifts of finite type which contain them.
    
2. (2)
   The classification of certain constant to one maps from SFTs via algebraic invariants of associated G-SFTs.
    

     

Recurrence and the asymptotic pseudo-orbit tracing property

In this paper, asymptotic pseudo-orbits and the asymptotic pseudo-orbit tracing property are considered for continuous maps from a compact metric space into itself. We investigate transitive properties and recurrent properties for a map with the asymptotic pseudo-orbit tracing property. In particular, a condition for which a homeomorphism is a non-wandering homeomorphism is obtained.     

Rigidity of critical circle mappings I

We prove that two C ³ critical circle maps with the same rotation number in a special set ? are C ^1+α conjugate for some α>0 provided their successive renormalizations converge together at an exponential rate in the C ⁰ sense. The set ? has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of C ^∞ critical circle maps with the same rotation number that are not C ^1+β conjugate for any β>0. The class of rotation numbers for which such examples exist contains Diophantine numbers. Received November 1, 1998 / final version received July 7, 1999     

Subsystem entropy for ? sofic shifts

We prove that any ?^d shift of finite type with positive topological entropy has a family of subsystems of finite type whose entropies are dense in the interval from zero to the entropy of the original shift. We show a similar result for ?^d sofic shifts, and also show every ?^d sofic shift can be covered by a ?^d shift of finite type arbitrarily close in entropy.     

Shadowing orbits and Kantorovich's theorem

Summary. Some recent results on the existence of shadowing orbits are related to Kantorovich's theorems on convergence of Newton's method. This connection leads to short proofs and to better understanding of some dilatation measures and existence results. Received March 24, 1994 / Revised version received May 15, 1995     

Global random attractors are uniquely determined by attracting deterministic compact sets

It is shown that for continuous dynamical systems an analogue of the Poincaré recurrence theorem holds for Ω-limit sets. A similar result is proved for Ω-limit sets of random dynamical systems (RDS) on Polish spaces. This is used to derive that a random set which attracts every (deterministic) compact set has full measure with respect to every invariant probability measure for theRDS. Then we show that a random attractor coincides with the Ω-limit set of a (nonrandom) compact set with probability arbitrarily close to one, and even almost surely in case the base flow is ergodic. This is used to derive uniqueness of attractors, even in case the base flow is not ergodic. Entrata in Redazione il 10 marzo 1997.     

The eskolemization of universal quantifiers

This paper is a sequel to the papers Baaz and Iemhoff (2006, 2009) [4] and [6] in which an alternative skolemization method called eskolemization was introduced that, when restricted to strong existential quantifiers, is sound and complete for constructive theories. In this paper we extend the method to universal quantifiers and show that for theories satisfying the witness property it is sound and complete for all formulas. We obtain a Herbrand theorem from this, and apply the method to the intuitionistic theory of equality and the intuitionistic theory of monadic predicates.     

Arithmetic isomorphism of hyperbolic toral automorphisms and sofic shifts

St.-Petersburg State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 26, No. 3, pp. 22–27, July–September, 1992.     

Ergodic properties of the Erdös measure, the entropy of the goldenshift, and related problems

We define a two-sided analog of the Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on that is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdös measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.With 11 PiguresTo the memory of Paul ErdösSupported in part by the INTAS grant 93-0570. The first author was supported by the French foundation PRO MATHEMATICA. The first author expresses his gratitude to l'Institut de Mathématiques de Luminy for support during his stay in Marseille in 1996-97. The second author is grateful to the University of Stony Brook for support during his visit in February–March 1996 and to the Institute for Advanced studies of Hebrew University for support during his being there in 1997     

Elimination of Skolem functions for monotone formulas in analysis

In this paper a new method, elimination of Skolem functions for monotone formulas, is developed which makes it possible to determine precisely the arithmetical strength of instances of various non-constructive function existence principles. This is achieved by reducing the use of such instances in a given proof to instances of certain arithmetical principles. Our framework are systems -qf , where (GA is a hierarchy of (weak) subsystems of arithmetic in all finite types (introduced in [14]), AC-qf is the schema of quantifier-free choice in all types and is a set of certain analytical principles which e.g. includes the binary K?nig's lemma. We apply this method to show that the arithmetical closures of single instances of -comprehension and -choice contribute to the growth of extractable bounds from proofs relative to only by a primitive recursive functional in the sense of Kleene. In subsequent papers these results are widely generalized and the method is used to determine the arithmetical content of single sequences of instances of the Bolzano-Weierstra? principle for bounded sequences in , the Ascoli-lemma and others. February 14, 1996     

Reducibility of covers of AFT shifts

In this paper we show that the reducibility structure of several covers of sofic shifts is a flow invariant. In addition, we prove that for an irreducible subshift of almost finite type the left Krieger cover and the past set cover are reducible. We provide an example which shows that there are non almost finite type shifts which have reducible left Krieger covers. As an application we show that the Matsumoto algebra of an irreducible, strictly sofic shift of almost finite type is not simple.     

On the rules of intermediate logics

If the Visser rules are admissible for an intermediate logic, they form a basis for the admissible rules of the logic. How to characterize the admissible rules of intermediate logics for which not all of the Visser rules are admissible is not known. In this paper we give a brief overview of results on admissible rules in the context of intermediate logics. We apply these results to some well-known intermediate logics. We provide natural examples of logics for which the Visser rule are derivable, admissible but nonderivable, or not admissible. Supported by the Austrian Science Fund FWF under projects P16264 and P16539.     

Stability of orbit spaces of endomorphisms

In this paper it is first proved that, for a hyperbolic set of aC ¹ (non-invertible) endomorphism of a compact manifold, the dynamical structure of its orbit space (inverse limit space) is stable underC ¹-small perturbations and is semi-stable underC ⁰-small perturbations. It is then proved that if an Axiom A endomorphism satisfies no-cycle condition then its orbit space is Θ-stable andR-stable underC ¹-small perturbations and is semi-Θ-stable and semi-R-stable underC ⁰-small perturbations. This research is supported by the National Natural Science Foundation of China     

Independence results in formal topology

We analyze a family of games by using formal topology as a tool. In order to win any game in the family one has to find a sequence of moves leading to one of the final states for that game. Thus, two results are relevant to the topic: to find terminating strategies and/or to show that every strategy is terminating. We will show that the language of formal topology can be useful to represent in a topological framework both of the problems, and in particular that the property of termination of all the strategies for a game is equivalent to the discreteness of a suitable formal space. Finally, we will provide some examples of games which are terminating according to any strategy, that is, such that the associated formal spaces are discrete, but the first order formulas expressing such a discreteness cannot be proved in Peano Arithmetic.     

Epsilon substitution for transfinite induction

We apply Mints technique for proving the termination of the epsilon substitution method via cut-elimination to the system of Peano Arithmetic with Transfinite Induction given by Arai.