Periodic solutions of fully nonlinear autonomous equations of Benjamin–Ono type

We prove the existence of time-periodic, small amplitude solutions of autonomous quasi-linear or fully nonlinear completely resonant pseudo-PDEs of Benjamin–Ono type in Sobolev class. The result holds for frequencies in a Cantor set that has asymptotically full measure as the amplitude goes to zero.     

A chaotic function with a distributively scrambled set of full Lebesgue measure

Misiurewicz proved that there exists a continuous map of the interval [0, 1] onto itself for which there exists a scrambled set of full Lebesgue measure. In this paper, we form a continuous interval map which has a distributively scrambled set of full Lebesgue measure in which each point has dense orbit. This contains Misiurewicz’s result, since any distributively scrambled set must be scrambled but the converse is not generally true.     

Markov extensions for dynamical systems with holes: An application to expanding maps of the interval

We introduce the Markov extension, represented schematically as a tower, to the study of dynamical systems with holes. For tower maps with small holes, we prove the existence of conditionally invariant probability measures which are absolutely continuous with respect to Lebesgue measure (abbreviated a.c.c.i.m.). We develop restrictions on the Lebesgue measure of the holes and simple conditions on the dynamics of the tower which ensure existence and uniqueness in a class of Holder continuous densities. We then use these results to study the existence and properties of a.c.c.i.m. forC ^1+α expanding maps of the interval with holes. We obtain the convergence of the a.c.c.i.m. to the SRB measure of the corresponding closed system as the measure of the hole shrinks to zero.     

Disguised toric dynamical systems

We study families of polynomial dynamical systems inspired by biochemical reaction networks. We focus on complex balanced mass-action systems, which have also been called toric. They are known or conjectured to enjoy very strong dynamical properties, such as existence and uniqueness of positive steady states, local and global stability, persistence, and permanence. We consider the class of disguised toric dynamical systems, which contains toric dynamical systems, and to which all dynamical properties mentioned above extend naturally. By means of (real) algebraic geometry we show that some reaction networks have an empty toric locus or a toric locus of Lebesgue measure zero in parameter space, while their disguised toric locus is of positive measure. We also propose some algorithms one can use to detect the disguised toric locus.     

Existence and uniqueness results for impulsive functional differential equations with scalar multiple delay and infinite delay

In this paper, we discuss local and global existence and uniqueness results for first order impulsive functional differential equations with multiple delay. We shall rely on a nonlinear alternative of Leray–Schauder. For the global existence and uniqueness we apply a recent nonlinear alternative of Leray–Schauder type in Fréchet spaces, due to M. Frigon and A. Granas [Résultats de type Leray–Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22 (2) (1998) 161–168]. The goal of this paper is to extend the problems considered by A. Ouahab [Local and global existence and uniqueness results for impulsive differential equations with multiple delay, J. Math. Anal. Appl. 323 (2006) 456–472].     

Grand Bochner–Lebesgue space and its associate space

Our aim is to introduce the grand Bochner–Lebesgue space in the spirit of Iwaniec–Sbordone spaces, also known as grand Lebesgue spaces, and prove some of its properties. We will also deal with the associate space for grand Bochner–Lebesgue spaces.     

Bernoulli automorphisms of finitely generated free MV-algebras

MV-algebras can be viewed either as the Lindenbaum algebras of ?ukasiewicz infinite-valued logic, or as unit intervals [0,u] of lattice-ordered abelian groups in which a strong order unit u>0 has been fixed. They form an equational class, and the free n-generated free MV-algebra is representable as an algebra of piecewise-linear continuous functions with integer coefficients over the unit n-dimensional cube. In this paper we show that the automorphism group of such a free algebra contains elements having strongly chaotic behaviour, in the sense that their duals are measure-theoretically isomorphic to a Bernoulli shift. This fact is noteworthy from the viewpoint of algebraic logic, since it gives a distinguished status to Lebesgue measure as an averaging measure on the space of valuations. As an ergodic theory fact, it provides explicit examples of volume-preserving homeomorphisms of the unit cube which are piecewise-linear with integer coefficients, preserve the denominators of rational points, and enjoy the Bernoulli property.     

Existence and uniqueness of solutions of differential equations with respect to non-additive measures

By taking Sugeno-derivative into account, first, we investigate the existence of solutions to the initial value problems (IVP) of first-order differential equations with respect to non-additive measure (more precisely, distorted Lebesgue measure). It particularly occurs in the mathematical modeling of biology. We begin by expressing the differential equation in terms of ordinary derivative and the derivative with respect to the distorted Lebesgue measure. Then, by using the fixed point theorem on cones, we construct an operator and prove the existence of positive non-decreasing solutions on cones in semi-order Banach spaces. In addition, we also use Picard–Lindelöf theorem to prove the existence and uniqueness of the solution of the equation. Second, we investigate the existence of a solution to the boundary value problem (BVP) on cones with integral boundary conditions of a mix-order differential equation with respect to non-additive measures. Moreover, the Krasnoselskii fixed point theorem is also applied to both BVP and IVP and obtains at least one positive non-decreasing solution. Examples with graphs are provided to validate the results.     

Continuous transformation of Baire measures into Lebesgue measure

A recent result by Wulbert on the existence of continuous functions with measure zero level sets is slightly extended and its proof is considerably simplified. As a by-product, a criterion is established for a Baire measure to allow a continuous transformation into Lebesgue measure.

     

Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains,with applications to quasilinear Riccati type equations

A weighted norm inequality of Muckenhoupt–Wheeden type is obtained for gradients of solutions to a class of quasilinear equations with measure data on Reifenberg flat domains. This essentially leads to a resolution of an existence problem for quasilinear Riccati type equations with a gradient source term of arbitrary power law growth.     

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szeg? polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szeg? polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.     

Invariant measures exist under a summability condition for unimodal maps

Summary For unimodal maps with negative Schwarzian derivative a sufficient condition for the existence of an invariant measure, absolutely continuous with respect to Lebesgue measure, is given. Namely the derivatives of the iterations of the map in the (unique) critical value must be so large that the sum of (some root of) the inverses is finite.Oblatum 7-V-1990 & 19-XI-1990Partially supported by the NWO grant.     

A Boundary-Value Problem for Weakly Nonlinear Hyperbolic Equations with Data on the Entire Boundary of a Domain

In a domain that is the Cartesian product of a segment and a p-dimensional torus, we investigate a boundary-value problem for weakly nonlinear hyperbolic equations of higher order. For almost all (with respect to Lebesgue measure) parameters of the domain, we establish conditions for the existence of a unique solution of the problem.     

Finite fractal dimension of pullback attractors for non-autonomous 2D Navier–Stokes equations in some unbounded domains

We study the asymptotic behaviour of non-autonomous 2D Navier–Stokes equations in unbounded domains for which a Poincaré inequality holds. In particular, we give sufficient conditions for their pullback attractor to have finite fractal dimension. The existence of pullback attractors in this framework comes from the existence of bounded absorbing sets of pullback asymptotically compact processes [T. Caraballo, G. ?ukaszewicz, J. Real, Pullback attractors for asymptotically compact nonautonomous dynamical systems, Nonlinear Anal. 64 (3) (2006) 484–498]. We show that, under suitable conditions, the method of Lyapunov exponents in [P. Constantin, C. Foias, R. Temam, Attractors representing turbulent flows, Mem. Amer. Math. Soc. 53 (1984) [5]] for the dimension of attractors can be developed in this new context.     

How good is lebesgue measure?

So, what is the answer to the question “How good is Lebesgue measure?” In the class of invariant measures, Lebesgue measure seems to be the best candidate to be a canonical measure. In the class of countably additive not necessarily invariant measures, to find a universal measure we have to use a strong additional set-theoretical assumption and this seems to be too high a price. Thus the best improvement of Lebesgue measure seems to be the Banach construction of a finitely additive isometrically invariant extension of Lebesgue measure on the plane and line. However, such a measure does not exist on Rⁿ for n ≤ 3, and to keep the theory of measures uniform for all dimensions we cannot accept the Banach measure on the plane as the best solution to the measure problem. From this discussion it seems clear that there is no reason to depose Lebesgue measure from the place it has in modern mathematics. Lebesgue measure also has a nice topological property called regularity: for every EL and every ɛ > 0, there exists an open set V⊃E and closed set F ⊂ E such that m(V/F) < ɛ. It is not difficult to prove that Lebesgue measure is the richest countably additive measure having this property (see [Ru], Thm. 2.20, p. 50).     

Note on the Dirichlet problem with L2-boundary data

The paper studies the existence of non-tangential limit for solutions of the Dirichlet problem with L²-boundary data. We also prove the mutual absolute continuity of the associated harmonic measure and the Lebesgue surface measure.     

Strong solutions for two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems

A well-known diffuse interface model for incompressible isothermal mixtures of two immiscible fluids consists of the Navier–Stokes system coupled with a convective Cahn–Hilliard equation. In some recent contributions the standard Cahn–Hilliard equation has been replaced by its nonlocal version. The corresponding system is physically more relevant and mathematically more challenging. Indeed, the only known results are essentially the existence of a global weak solution and the existence of a suitable notion of global attractor for the corresponding dynamical system defined without uniqueness. In fact, even in the two-dimensional case, uniqueness of weak solutions is still an open problem. Here we take a step forward in the case of regular potentials. First we prove the existence of a (unique) strong solution in two dimensions. Then we show that any weak solution regularizes in finite time uniformly with respect to bounded sets of initial data. This result allows us to deduce that the global attractor is the union of all the bounded complete trajectories which are strong solutions. We also demonstrate that each trajectory converges to a single equilibrium, provided that the potential is real analytic and the external forces vanish.     

Recurrence and non‐ergodicity in generalized wind‐tree models

In this paper, we consider generalized wind‐tree models and ‐covers over compact translation surfaces. Under suitable hypothesis, we prove the recurrence of the linear flow in a generic direction and the non‐ergodicity of Lebesgue measure.     

State-dependent Foster–Lyapunov criteria for subgeometric convergence of Markov chains

We consider a form of state-dependent drift condition for a general Markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric Foster–Lyapunov condition. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a question posed in Connor and Kendall (2007) [2] concerning the existence of so-called ‘tame’ Markov chains. Furthermore, we show that our ‘subsampled drift condition’ implies the existence of finite moments for the return time to a small set.     

Transversality family of expanding rational semigroups

We study finitely generated expanding semigroups of rational maps with overlaps on the Riemann sphere. We show that if a d$d$-parameter family of such semigroups satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the Julia set is the minimum of 2 and the zero of the pressure function. Moreover, the Hausdorff dimension of the exceptional set of parameters is estimated. We also show that if the zero of the pressure function is greater than 2$2$, then typically the 2-dimensional Lebesgue measure of the Julia set is positive. Some sufficient conditions for a family to satisfy the transversality conditions are given. We give non-trivial examples of families of semigroups of non-linear polynomials with the transversality condition for which the Hausdorff dimension of the Julia set is typically equal to the zero of the pressure function and is less than 2$2$. We also show that a family of small perturbations of the Sierpinski gasket system satisfies that for a typical parameter value, the Hausdorff dimension of the Julia set (limit set) is equal to the zero of the pressure function, which is equal to the similarity dimension. Combining the arguments on the transversality condition, thermodynamical formalisms and potential theory, we show that for each a∈C$a\in \mathbb{C}$ with |a|≠0,1$\left|a\right|\ne 0,1$, the family of small perturbations of the semigroup generated by {z²,az²}$\{z^2,a{z}^2\}$ satisfies that for a typical parameter value, the 2-dimensional Lebesgue measure of the Julia set is positive.