Cocyclic <Emphasis Type="Italic">n</Emphasis>-groups

We describe all cocyclic n-groups and the structure of (n, 2)-rings of endomorphisms of cocyclic n-groups. We prove that a cocyclic n-group is defined uniquely by its (n, 2)-ring of endomorphisms.     

Mapping tori of endomorphisms of free groups

For al large class of endomorphisms of finitely generated free groups we prove that their mapping tori groups are word-hyperbolic if and only if they don’t contain Baumslag-Solitar subgroups.     

The dynamics of expansive invertible onesided cellular automata

Using textile systems, we prove the conjecture of Boyle and Maass that the dynamical system defined by an expansive invertible onesided cellular automaton is topologically conjugate to a topological Markov shift. We also study expansive leftmost-permutive onesided cellular automata and bipermutive endomorphisms of mixing topological Markov shifts.

     

On One Class of Systems of Differential Equations and on Retarded Equations

We establish a connection between solutions to a broad class of large systems of ordinary differential equations and solutions to retarded differential equations. We prove that solving the Cauchy problem for systems of ordinary differential equations reduces to solving the initial value problem for a retarded differential equation as the number of equations increases unboundedly. In particular, the class of systems under consideration contains a system of differential equations which arises in modeling of multiphase synthesis.     

On some coding and mixing properties for a class of chaotic systems

We study certain ergodic properties of equilibrium measures of hyperbolic non-invertible maps f on basic sets with overlaps Λ. We prove that if the equilibrium measure \({\mu_\phi}\) of a Holder potential \({\phi}\) , is 1-sided Bernoulli, then f is expanding from the point of view of a pointwise section dimension of \({\mu_\phi}\) . If the measure of maximal entropy μ ₀ is 1-sided Bernoulli, then f is shown to be distance expanding on Λ; and if \({\mu_\phi}\) is 1-sided Bernoulli for f expanding, then \({\mu_\phi}\) must be the measure of maximal entropy. These properties are very different from the case of hyperbolic diffeomorphisms. Another result is about the non 1-sided Bernoullicity for certain equilibrium measures for hyperbolic toral endomorphisms. We also prove the non-existence of generating Rokhlin partitions for measure-preserving endomorphisms in several cases, among which the case of hyperbolic non-expanding toral endomorphisms with Haar measure. Nevertheless the system \({(\Lambda, f, \mu_\phi)}\) is shown to have always exponential decay of correlations on Holder observables and to be mixing of any order.     

Accessibility of partially hyperbolic endomorphisms with 1D center-bundles

We prove that partially hyperbolic endomorphisms with one dimensional center-bundles and non-trivial unstable bundles are stably accessible. And there is residual subset $\Res$ of partially hyperbolic volume preserving endomorphisms with one dimensional center-bundles such that every $f \in \Res$ is stably accessible. In the end, we prove the accessibility of Gan''s example.     

Fixed points of endomorphisms over special confluent rewriting systems

Cayley graphs of monoids defined through special confluent rewriting systems are known to be hyperbolic metric spaces which admit a compact completion given by irreducible finite and infinite words. In this paper, we prove that the fixed point submonoids for endomorphisms of these monoids which are boundary injective (or have bounded length decrease) are rational, with similar results holding for infinite fixed points. Decidability of these properties is proved, and constructibility is proved for the case of bounded length decrease. These results are applied to free products of cyclic groups, providing a new generalization for the case of infinite fixed points.     

Dimension polynomials in the generalized difference case

The aim of this paper is to generalize results on dimension polynomials of difference modules over difference rings for a wider class of rings of difference operators. We introduce the notion of quasi-commutativity, which generalizes the notion of commutativity and enables one to consider wider classes of monoids and groups of endomorphisms. Some properties of quasi-commutative monoids and groups are established; these properties allow us to apply some methods that are almost similar to the ones used in working with free commutative monoids and groups. Also we prove the theorem of existence of the dimension polynomial of generalized difference modules in the cases where the submonoid of endomorphisms is free quasi-commutative. Also the existence of its analog for the case of a direct product of a free quasi-commutative monoid and a finite cyclic group is established.     

Nondegenerate -biresolving textile systems and expansive automorphisms of onesided full shifts

We study nondegenerate, -biresolving textile systems and using properties of them, we prove a conjecture of Boyle and Maass on arithmetic constraints for expansive automorphisms of onesided full shifts and positively expansive endomorphisms of mixing topological Markov shifts. A similar result is also obtained for expansive leftmost-permutive endomorphisms of onesided full shifts.

     

Fixed points of endomorphisms over special confluent rewriting systems

Cayley graphs of monoids defined through special confluent rewriting systems are known to be hyperbolic metric spaces which admit a compact completion given by irreducible finite and infinite words. In this paper, we prove that the fixed point submonoids for endomorphisms of these monoids which are boundary injective (or have bounded length decrease) are rational, with similar results holding for infinite fixed points. Decidability of these properties is proved, and constructibility is proved for the case of bounded length decrease. These results are applied to free products of cyclic groups, providing a new generalization for the case of infinite fixed points.     

Multilinear Equations in Free Inverse Monoids

We prove that finding solutions to the class of single multilinear equations in FIM(A) is decidable; while finding solutions to the class of systems of multilinear equations in FIM(A) is undecidable.     

The Theory of Linear G-Difference Equations

We introduce the notion of difference equations defined on a structured set. The symmetry group of the structure determines the set of difference operators. All main notions in the theory of difference equations are introduced as invariants of the action of the symmetry group. Linear equations are modules over the skew group algebra, solutions are morphisms relating a given equation to other equations, symmetries of an equation are module endomorphisms, and conserved structures are invariants in the tensor algebra of the given equation.We show that the equations and their solutions can be described through representations of the isotropy group of the symmetry group of the underlying set. We relate our notion of difference equation and solutions to systems of classical difference equations and their solutions and show that out notions include these as a special case.     

Preservation of stability under discretization of systems of ordinary differential equations

We study the stability preservation problem while passing from ordinary differential to difference equations. Using the method of Lyapunov functions, we determine the conditions under which the asymptotic stability of the zero solutions to systems of differential equations implies that the zero solutions to the corresponding difference systems are asymptotically stable as well. We prove a theorem on the stability of perturbed systems, estimate the duration of transition processes for some class of systems of nonlinear difference equations, and study the conditions of the stability of complex systems in nonlinear approximation.     

Solution of one class of systems of stochastic differential equations

In this paper we prove that instead of solving a class of systems of stochastic differential equations with a multidimensional Wiener process one can solve a system of total differential equations. The latter system admits an application of classical methods. This fact enables one to solve the initial system explicitly.     

The topological conjugacy problem for generalized Hénon mappings: some negative results

Mappings of the plane, introduced by M. Hénon and R. Lozi, are presented as perturbations of endomorphisms of the line. When some heteroclinic tangencies occur, which allow topological conjugacy between the corresponding endomorphisms, and arbitrarily close to the endomorphisms case, we prove the nonexistence of topological conjugacy between Hénon mapping and Lozi mapping. The proof, as well as related results, relies on the existence of a topological invariant previously introduced by J. Palis.     

Dynamic programming in stochastic control of systems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay (SDDE). We prove a version of Bellman's principle of optimality (the dynamic programming principle) for a general class of such problems. That the class in general means that both the dynamics and the cost depends on the past in a general way. As an application, we study systems where the value function depends on the past only through some weighted average. For such systems we obtain a Hamilton-Jacobi-Bellman partial differential equation that the value function must solve if it is smooth enough. The weak uniqueness of the SDDEs we consider is our main tool in proving the result. Notions of strong and weak uniqueness for SDDEs are introduced, and we prove that strong uniqueness implies weak uniqueness, just as for ordinary stochastic differential equations.     

Topology of complements of hyperplane arrangements and isomonodromic deformations of fuchsian systems

We consider families of systems of first-order linear differential equations on complex linear spaces that represent the Cherednik variant of the Knizhnik-Zamolodchikov equations. For these families of equations, we prove a rigidity property with respect to a certain class of isomonodromic deformations; i.e., we show the absence of nontrivial special isomonodromic deformations with a movable divisor of singularities.     

Calculating the Number of Functions with a Given Endomorphism

An iterative procedure is proposed for calculating the number of k-valued functions of n variables such that each one has an endomorphism different from any constant and permutation. Based on this procedure, formulas are found for the number of three-valued functions of n variables such that each one has nontrivial endomorphisms. For any arbitrary semigroup of endomorphisms, the power is found of the set of all three-valued functions of n variables such that each one has endomorphisms from a specified semigroup.     

On the properties of solutions to a class of nonlinear systems of differential equations of large dimension

We consider the Cauchy problem for a class of nonlinear systems of differential equations of large dimension, establish some properties of solutions, and prove that for a sufficiently large number of differential equations the last component of the solution is an approximate solution to the initial value problem for a delay differential equation.     

Growth bound of delay-differential algebraic equations

This paper deals with delay-differential algebraic equations, a large class of linear and finite-memory functional differential equations. We introduce several representations of delay operators that provide a simple definition for the concept of solutions of such systems. Then we study exponential solutions and prove that the rightmost zeros of a system characteristic function determine its growth bound.