regularity for p harmonic equations with right hand side in orlicz zygmund classes abstract we establish regularity results for solutions of some degenerate elliptic pdes with right hand side in a suitable orlicz zygmund class the nonnegative function which measures the degree of degeneracy of the ellipticity bounds is assumed to be exponentially integrable we find that the scale of improved regularity is logarithmic and we indicate its exact dependence on the degree of the degeneracy of the problem

new maximal functions and multiple weights for the multilinear calderon zygmund theory a multi sub linear maximal operator that acts on the product of m lebesgue spaces and is smaller than the m fold product of the hardy littlewood maximal function is studied the operator is used to obtain a precise control on multilinear singular integral operators of calderon zygmund type and to build a theory of weights adapted to the multilinear setting a natural variant of the operator which is useful to control certain commutators of multilinear calderon zygmund operators with bmo functions is then considered the optimal range of strong type estimates a sharp end point estimate and weighted norm inequalities involving both the classical muckenhoupt weights and the new multilinear ones are also obtained for the commutators

an exact renormalisation group for bloch electrons in a magnetic field bloch electrons in a magnetic field can be modelled by a one dimensional hamiltonian h h x p which is periodic in x and p when beta the ratio of planck s constant h to the area of a unit cell is a rational number p q the spectrum is a set of q bloch bands if beta is perturbed away from this rational value the spectrum becomes a cantor set but it is still possible to define a set of generalised bloch waves which form a complete basis set for a given band projecting these states into the band and taking matrix elements of the hamiltonian leads to a new effective hamiltonian h1v x p and planck constant h v describing the spectrum of the vth band since degrees of freedom have been eliminated by projecting into a band this is a renormalisation group rg transformation the rg transformation can be iterated indefinitely if the measure of the spectrum is zero or if beta is a liouville number the measure of the spectrum vanishes if h x p has centres of threefold or fourfold symmetry in the phase plane and the rg transformation explains the hierarchical structure of the spectrum observed in these cases

geometrically finite groups patterson sullivan measures and ratner s ridigity theorem the rigidity properties of the horospherical foliations of geometrically finite hyperbolic manifolds are investigated ratner s theorem generalizes to these foliations with respect to the patterson sullivan measure in the spirit of mostow we prove the nonexistence of invariant measurable distributions on the boundary of hyperbolic space for geometrically finite groups finally we show that the frame flow on geometrically finite hyperbolic manifolds is bernoulli

critical properties of electron eigenstates in incommensurate systems this paper describes some properties of the eigenvalue equation n 1 n 1 2 cos 2 n n e n this is an example of the more general problem of a hermitian eigenvalue equation in the form of a difference equation with periodic coefficients these equations arise in solid state physics they occur in connection with tight binding models for electrons in one dimensional solids with an incommensurate modulation of the structure and in models for the energy bands of bloch electrons moving in a plane with a perpendicular magnetic field the model studied has a critical point when 1 following some earlier work by azbel azbel m ya phys rev lett 43 1954 1979 an approximate renormalization group transformation is derived this predicts that the spectrum and eigenstates have a remarkable recursive structure at the critical point which is dependent on the expansion of as a continued fraction also when is an irrational number there is a localization transition from extended states to localized states as increases through the critical point this localization transition which was previously discovered by aubry andre aubry s andre g ann israel phys soc 3 133 1979 using the thouless formula for the localization length is explained by the renormalization group transformation derived here

trudinger s exponential integrability for riesz potentials of functions in generalized grand morrey spaces abstract our aim in this paper is to discuss trudinger s exponential integrability for riesz potentials of functions in generalized grand morrey spaces our result will imply the boundedness of the riesz potential operator from a grand morrey space to a morrey space

calderon zygmund estimates and non uniformly elliptic operators abstract we consider a class of non uniformly nonlinear elliptic equations whose model is given by div d u p 2 d u a x d u q 2 d u div f p 2 f a x f q 2 f where p q and a x 0 and establish the related nonlinear calderon zygmund theory in particular we provide sharp conditions under which the natural and optimal calderon zygmund type result f p a x f q l loc d u p a x d u q l loc holds for every 1 these problems naturally emerge as euler lagrange equations of some variational integrals introduced and studied by marcellini 41 and zhikov 53 in the framework of homogenisation and lavrentiev phenomenon

bi sobolev mappings and elliptic equations in the plane abstract suppose that f u v is a homeomorphism in the plane of the sobolev class w loc 1 1 such that its inverse is of the same sobolev class we prove that u and v have the same set of critical points as an application we show that u and v are distributional solutions to the same non trivial degenerate elliptic equation in divergence form we study similar properties also in higher dimensions

very weak solutions of degenerate parabolic systems with non standard p x t growth abstract we study higher integrability of very weak solutions to certain degenerate parabolic systems whose model is the parabolic p x t laplacian system t u div d u p x t 2 d u div f p x t 2 f under natural assumptions on the exponent function p 0 t 2 we prove that any very weak solution u 0 t r n with d u p 1 e l 1 belongs to the natural energy space i e d u p l loc 1 provided e 0 is small enough

the coexistence problem for the discrete mathieu operator we solve the coexistence problem for the periodic discrete mathieu operator in all parametric cases the main tool in the proof will be bezout s theorem for projective plane curves as an additional result we obtain the gap opening and gap growth powers for this operator

an exact effective hamiltonian for a perturbed landau level considers the effect of a scalar potential v x y on a landau level in two dimensions an exact effective hamiltonian is derived which describes the effect of the potential on a single landau level expressed as a power series in v ec where ec is the cyclotron energy the effective hamiltonian can be represented as a function h x p in a one dimensional phase space the function h x p resembles the potential v x y when the area of a flux quantum is much smaller than the square of the characteristic length scale of v then h approximately v also h x p retains the translational and rotational symmetries of v x y exactly but reflection symmetries are not retained beyond the lowest order of the perturbation expansion

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

local regularity for solutions of anisotropic obstacle problems abstract this paper deals with anisotropic obstacle problems for the a harmonic equation div a x u x 0 a local regularity result is given under suitable assumptions

rigidity of measures invariant under semisimple groups in positive characteristic m ratner has conjectured a positive characteristic version of her seminal results classifying orbit closures and invariant measures of unipotent flows on homogeneous spaces in this paper we provide a partial answer by establishing a positive characteristic version of her classification result for measures invariant under semisimple groups

nonlinear cauchy riemann operators in ℝⁿ this paper has arisen from an effort to provide a comprehensive and unifying development of the l p theory of quasiconformal mappings in r n the governing equations for these mappings form nonlinear differential systems of the first order analogous in many respects to the cauchy riemann equations in the complex plane this approach demands that one must work out certain variational integrals involving the jacobian determinant guided by such integrals we introduce two nonlinear differential operators denoted by d and d which act on weakly differentiable deformations f r n of a domain c r n solutions to the so called cauchy riemann equations d f 0 and d f 0 are simply conformal deformations preserving and reversing orientation respectively these operators though genuinely nonlinear possess the important feature of being rank one convex among the many desirable properties we give the fundamental l p estimate d f p a p n d f p in quest of the best constant a p n we are faced with fascinating problems regarding quasiconvexity of some related variational functionals applications to quasiconformal mappings are indicated

classification of joinings for kleinian groups we classify all locally finite joinings of a horospherical subgroup action on g when g psl2 r or psl2 c and is a geometrically finite zariski dense subgroup of g this generalizes ratner s 1983 joining theorem for the case when is a lattice in g

existence results for some nonuniformly elliptic equations with irregular data abstract in this paper we prove the existence of solutions of some nonlinear elliptic equations with degenerate coercivity of the type formula when the datum f belongs to the space lr where r 2 and 0 1

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

the maximum principle of alexandrov for very weak solutions abstract we extend the classical maximal principle of alexandrov to very weak solutions of the elliptic equation i j 1 n a i j x 2 u x i x j f

effective equidistribution for closed orbits of semisimple groups on homogeneous spaces we prove effective equidistribution with polynomial rate for large closed orbits of semisimple groups on homogeneous spaces under certain technical restrictions notably the acting group should have finite centralizer in the ambient group the proofs make extensive use of spectral gaps and also of a closing lemma for such actions

total bandwidth for the harper equation iii corrections to scaling earlier work on the spectral measure for the harper equation discrete mathieu equation showed that if there is a large common period p for the lattice and the sinusoidal potential the spectral measure scaled as c p where c is a constant independent of the number of oscillations q of the potential in p lattice spacings the corrections to this scaling law are found for q 1 and q 2 for q 1 it is shown that the corrections to scaling are logarithmic functions of p while for q 2 the leading corrections are reduced by a factor p 2 analytical and numerical support is given for the assertion that this difference between odd and even values of q persists for higher values of q provided they are substantially less than p

t minima and application to the convergence of some integral functionals with infinite energy minima abstract we consider integral functionals j as in 1 we study the calderon zygmund theory for infinite energy minima u t minima of j and the stability of the t minima with respect to the convergence in 22 with v w 0 1 p

on measures invariant under tori on quotients of semisimple groups we classify invariant and ergodic probability measures on arithmetic homogeneous quotients of semisimple s algebraic groups invariant under a maximal split torus in at least one simple local factor and show that the algebraic support of such a measure splits into the product of four homogeneous spaces a torus a homogeneous space on which the measure is up to nite index the haar measure a product of homogeneous spaces on each of which the action degenerates to a rank one action and a homogeneous space in which every element of the action acts with zero entropy

on very weak solutions to a class of double obstacle problems abstract we prove the uniqueness of very weak solutions by hodge decomposition and obtain the existence by an approximation argument moreover we prove that the solution quasiminimizes the r dirichlet integral in k r and finally the stability of very weak solutions is established for varying obstacle functions

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

cantor spectrum for the almost mathieu equation for a dense g of pairs in r2 we prove that the operator hu n u n 1 u n 1 cos 2 n u n has a nowhere dense spectrum along the way we prove several interesting results about the case pq of which we mention a if q is not an integral multiple of then all gaps are open and b if q is even and is chosen suitably then the middle gap is closed for all

on the total bandwidth for the rational harper s equation in the last years several contributions have been done around the total bandwidth of the spectrum for the harper s operator in particular an interesting conjecture has been proposed by thouless which gives also strong convincing arguments for the proof in special cases on the other hand in the study of the cantor structure of the spectrum b helffer and j sjostrand have justified an heuristic semiclassical approach proposed by m wilkinson the aim of this article is to explain how one can use the first step of this approach to give a rigorous semi classical proof of the thouless formula in some of the simplest cases we shall also indicate how one can hope with more effort to prove rigorously recent results of last and wilkinson on the same conjecture

gauss polynomials and the rotation algebra newton s binomial theorem is extended to an interesting noncommutative setting as follows if in a ring ba ab with commuting witha andb then the generalized binomial coefficient left begin array 20 c n k end array right r arising in the expansion left a b right n sum limits k 0 n left begin array 20 c n k end array right gamma a n k b k resulting from these relations is equal to the value at of the gaussian polynomial left begin array 20 c n k end array right frac left n right left k right left n k right where m 1 x m 1 x m 1 1 x this is of course known in the case 1 from this it is deduced that in the universal c algebraa gq generated by unitariesu andv such thatvu e 2 i uv the spectrum of the self adjoint element u v u v has all the gaps that have been predicted to exist provided that either is rational or is a liouville number in the latter case the gaps are labelled in the natural way viak theory by the set of all non zero integers and the spectrum is a cantor set

scaling localization and bandwidths for equations with competing periods abstract the finite size scaling theory of the measure of the spectrum of harper s equation is reexamined for sequences of fractions tending to a rational limit a simple criterion is derived which determines whether the corrections to scaling behave as log p 2 or p 2 as the denominator p is increased the question of how special are the properties of the harper equation is studied it is shown that if the pure cosine term in the diagonal term is replaced by a distorted periodic function the different subbands undergo a transition from localized to extended at a value of the strength of the off diagonal term that depends on energy in contrast to the harper equation where the transition is energy independent this has a crucial effect on the measure of the spectrum

scaling for the discrete mathieu equation the scaling of the total width of the band for the discrete mathieu equation is studied in the critical region near the transition between localized and extended states for the special case in which there is one period of the modulation forp lattice spacings a general expression for the bandwidthw in the critical region is found at the critical point an analytic expression forpw is found which agrees to one part in 108 with the result deduced from numerical work

wkb analysis of energy band structure of modulated systems a discrete wkb approximation is applied to harper s equation describing the motion of an electron in a square lattice subject to a magnetic field the treatment is valid when the modulation parameter proportional to the magnetic field is a rational number m n in the limit of large n a uniform asymptotic expression is derived for the floquet wavenumber which characterizes the dynamics and spectral properties the case of energies near the classical separatrix crucial to the properties of the spectrum near the transition between localized and extended states is included an application is a detailed calculation of the total width of the bands in the critical region for m 1 and 2 the results are consistent with thouless conjecture that the bandwidth obeys a universal scaling form independent of m

translate of horospheres and counting problems let g be a semisimple lie group without compact factors gamma be an irreducible lattice in g in the first part of the article we give the necessary and sufficient condition under which a sequence of translates of probability horospherical measures is convergent and the limiting measure is also determined when it is convergent see theorems 1 and 2 for the precise statements in the second part two applications are presented the first one is of geometric nature and the second one gives an alternative way to count the number of rational points on a flag variety

invariant measures and the set of exceptions to littlewood s conjecture we classify the measures on sl k r sl k z which are invariant and ergodic under the action of the group a of positive diagonal matrices with positive entropy we apply this to prove that the set of exceptions to littlewood s conjecture has hausdorff dimension zero

an example of phase holonomy in wkb theory discusses the application of wkb theory to harper s equation psi n 1 psi n 1 2 alpha cos 2 pi beta n delta psi n e psi m in the case in which beta is very close to a rational number p q the wkb wavefunction for this system is a vector valued quantity proportional to an eigenvector u of a matrix h x p which is parametrised by the phase space coordinates x and p the complex phase of u is determined by a non holonomic connection rule when transported around a cycle and in phase space u is multiplied by a phase factor ei gamma c this phase change manifests itself as a modification of the bohr sommerfeld quantisation conditions

gradient regularity for elliptic equations in the heisenberg group abstract we give dimension free regularity conditions for a class of possibly degenerate sub elliptic equations in the heisenberg group exhibiting super quadratic growth in the horizontal gradient this solves an issue raised in j j manfredi g mingione regularity results for quasilinear elliptic equations in the heisenberg group math ann 339 2007 485 544 where only dimension dependent bounds for the growth exponent are given we also obtain explicit a priori local regularity estimates and cover the case of the horizontal p laplacean operator extending some regularity proven in a domokos j j manfredi c 1 regularity for p harmonic functions in the heisenberg group for p near 2 in contemp math vol 370 2005 pp 17 23 in turn using some recent techniques of caffarelli and peral l caffarelli i peral on w 1 p estimates for elliptic equations in divergence form comm pure appl math 51 1998 1 21 the a priori estimates found are shown to imply the suitable local calderon zygmund theory for the related class of non homogeneous possibly degenerate equations involving discontinuous coefficients these last results extend to the sub elliptic setting a few classical non linear euclidean results t iwaniec projections onto gradient fields and l p estimates for degenerated elliptic operators studia math 75 1983 293 312 e dibenedetto j j manfredi on the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems amer j math 115 1993 1107 1134 and to the non linear case estimates of the same nature that were available in the sub elliptic setting only for solutions to linear equations

smallness and cancellation in some elliptic systems with measure data abstract in a bounded open subset r n we study dirichlet problems with elliptic systems involving a finite radon measure on r n with values into r n defined by div a x u x d u x in u 0 on where a i x y 1 n j 1 n a i j x y j with 1 n the equation index we prove the existence of a distributional solution u r n obtained as the limit of approximations by assuming i that coefficients a i j are bounded caratheodory functions ii ellipticity of the diagonal coefficients a i j and iii smallness of the quadratic form associated to the off diagonal coefficients a i j i e verifying a r staircase support condition with r 0 such a smallness condition is satisfied for instance in each one of these cases a a i j a j i skew symmetry b a i j is small c a i j may be decomposed into two parts the first enjoying skew symmetry and the second being small in absolute value we give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works a brezis s type nonexistence result is also given for general smooth elliptic hyperbolic systems

quasilinear elliptic systems with measure data abstract we study the existence of solutions of quasilinear elliptic systems involving n equations and a measure on the right hand side with the form i 1 n x i 1 n j 1 n a i j x u x j u in u 0 on where 1 n is the equation index is an open bounded subset of r n u r n and is a finite randon measure on r n with values in r n existence of a solution is proved for two different sets of assumptions on a examples are provided that satisfy our conditions but do not satisfy conditions required on previous works on this matter

mobius disjointness for homogeneous dynamics we prove sarnak s mobius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected lie groups namely we show that if g is any such group g a lattice and u g an ad unipotent element then for every x g and every function f continuous on the 1 point compactification of g the sequence f xun cannot correlate with the mobius function on average

stability for p laplace type equation in a borderline case abstract we study the dirichlet problem for a p laplacian type operator in the setting of the orlicz zygmund space q log r n q 1 and 0 more precisely our aim is to establish under which assumptions on 0 existence and uniqueness of the solution are assured and to prove continuity and stability of the associated nonlinear operator

a sum rule for the dispersion relations of the rational harper equation the authors derive an exact sum rule satisfied by the dispersion relations of the commensurable case of harper s equation 1955 they use this result to derive a lower bound for the total bandwidth of the spectrum and to provide a stronger analytical justification for a result due to thouless 1990 concerning the total bandwidth when the commensurability is a high order rational

measurable rigidity of actions on infinite measure homogeneous spaces ii we consider the problems of measurable isomorphisms and join ings measurable centralizers and quotients for certain classes of ergodic group actions on infinite measure spaces our main focus is on systems of algebraic origin actions of lattices and other discrete subgroups g on homoge neous spaces g h where h is a sufficiently rich unimodular subgroup in a semi simple group g we also consider actions of discrete groups of isometries isom x of a pinched negative curvature space x acting on the space of horospheres hor x for such systems we prove that the only measurable isomorphisms joinings quotients etc are the obvious algebraic or geometric ones this work was inspired by the previous work of shalom and steger but uses completely different techniques which lead to more general results 1 introduction and statement of the main results

the work of einsiedler katok and lindenstrauss on the littlewood conjecture this document is intended as a slightly expanded writeup of my anticipated talk at the ams current events bulletin in new orleans january 2007 it is a brief report on the work of einsiedler katok and lindenstrauss on the littlewood conjecture 5 it is not intended in any sense for specialists and is indeed aimed at readers without any specific background either in measure theory dynamics or number theory any reader with any background in ergodic theory will be better served by consulting either the original paper or one of the surveys written by those authors see 7 and 12

on non linear elliptic equation with weight abstract in this paper we prove the existence and uniqueness theorem of the dirichlet problem for certain non linear elliptic operators with muckenhoupt weight

measurable quotients of unipotent translations on homogeneous spaces let u be a nilpotent unipotent subgroup of a lie group g and let f be a closed subgroup of g marina ratner showed that every ergodic u invariant probability measure on the homogeneous space f g is of a simple algebraic form we use this fundamental new result to show that every measurable quotient of the u action on j g is of a simple algebraic form roughly speaking any quotient is a double coset space a g k

invariant distributions and time averages for horocycle flows there are infinitely many obstructions to existence of smooth solutions of the cohomological equation uu f where u is the vector field generating the horocycle flow on the unit tangent bundle sm of a riemann m surface of finite area and f is a given function on sm we study the sobolev regularity of these obstructions construct smooth solutions of the cohomological equation and derive asymptotics for the ergodic averages of horocycle flows

multiple mixing for a class of conservative surface flows arnold and kochergin mixing conservative flows on surfaces stand as the main and almost only natural class of mixing transformations for which higher order mixing has not been established nor disproved under suitable arithmetic conditions on their unique rotation vector of full lebesgue measure in the first case and of full hausdorff dimension in the second we show that these flows are mixing of any order for this we show that they display a generalization of the so called ratner property on slow divergence of nearby orbits that implies strong restrictions on their joinings which in turn yields higher order mixing this is the first case in which the ratner property is used to prove multiple mixing outside its original context of horocycle flows and we expect our approach will have further applications

positive hausdorff dimensional spectrum for the critical almost mathieu operator we show that there exists a dense set of frequencies with positive hausdorff dimension for which the hausdorff dimension of the spectrum of the critical almost mathieu operator is positive

joinings of higher rank diagonalizable actions on locally homogeneous spaces we classify joinings between a fairly general class of higher rank diagonalizable actions on locally homogeneous spaces in particular we classify joinings of the action of a maximal r split torus on g with g a simple lie group of r rank 2 and g a lattice we deduce from this a classification of measurable factors of such actions as well as certain equidistribution properties

regularity of solutions of degenerate a harmonic equations we study the regularity of solutions to degenerate a harmonic equations under suitable integrability assumptions on the ellipticity and growth coefficients in particular we show a self improving property of the gradient of the solutions extending previous results by gehring and iwaniec sbordone valid in the uniformly elliptic setting

a distance between orbits that controls commutator estimates and invertibility of operators abstract a distance between orbit spaces generated by a single element is introduced and it is shown that if an operator is invertible in one orbit it is also invertible in nearby orbits thus proving a version of shneiberg s theorem for orbital methods the same machinery is used to extend the celebrated rochberg weiss commutator theorem to the setting of orbital methods it is shown that these results apply to the real and complex methods of interpolation by proving that these methods can be suitably obtained as orbits of a single element

nonlinear commutators for the fractional p laplacian and applications we prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p laplacian this implies that solutions to certain degenerate nonlocal equations are higher differentiable also weakly fractional p harmonic functions which a priori are less regular than variational solutions are in fact classical as an application we show that sequences of uniformly bounded frac n s harmonic maps converge strongly outside at most finitely many points

convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations abstract we study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation u t div a x u f x u 0 on a bounded domain subject to dirichlet boundary and to initial conditions the data are supposed to satisfy suitable regularity and growth conditions our approach to the convergence result and decay estimate is based on the ojasiewicz simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates the abstract results and their applications are discussed also in the framework of orlicz sobolev spaces

rigidity of group actions on homogeneous spaces iii consider homogeneous g h and g f for an s algebraic group g a lattice acts on the left strictly conservatively the following rigidity results are obtained morphisms factors and joinings defined apriori only in the measurable category are in fact algebraically constrained arguing in an elementary fashion we manage to classify all the measurable commuting with the action assuming ergodicity we find they are algebraically defined

on the measure of the spectrum for the almost mathieu operator we obtain partial results on the conjecture that for the almost mathieu operator at irrational frequency the measure of the spectrum s 4 2 for 2 we show that if n is rational and alpha n to alpha irrational then s alpha n lambda theta to 4 2 lambda

on coercive variational integrals abstract it is well known that sequential weak lower semicontinuity of a variational integral f u f u x d x on the sobolev space w 1 p r n under a p growth condition on the integrand f is equivalent to quasiconvexity in the sense of morrey we show that coercivity on dirichlet classes likewise is equivalent to a quasiconvexity condition we also discuss some examples and extend a sequential weak lower semicontinuity result to the case of signed integrands

on disjointness properties of some parabolic flows the ratner property a quantitative form of divergence of nearby trajectories is a central feature in the study of parabolic homogeneous flows discovered by marina ratner and used in her 1980th seminal works on horocycle flows it pushed forward the disjointness theory of such systems in this paper exploiting a recent variation of the ratner property we prove new disjointness phenomena for smooth parabolic flows beyond the homogeneous world in particular we establish a general disjointness criterion based on the switchable ratner property we then apply this new criterion to study disjointness properties of smooth time changes of horocycle flows and smooth arnol d flows on mathbb t 2 focusing in particular on disjointness of distinct flow rescalings as a consequence we answer a question by marina ratner on the mobius orthogonality of time changes of horocycle flows in fact we prove mobius orthogonality for all smooth time changes of horocycle flows and uniquely ergodic realizations of arnol d flows considered

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians