dimension of the boundary of quasiconformal siegel disks we study quasiconformal siegel disks with critical points in their boundaries the main result asserts that every subarc of the boundary of the siegel disk has the hausdorff dimension strictly larger than 1 and that the boundary does not have a tangent at any point

space time dynamics from algebra representations we present a model for introducing dynamics into a space time geometry this space time structure is constructed from a c algebra defined in terms of the generators of an irreducible unitary representation of a finite dimensional lie algebra this algebra is included as a subalgebra in a bigger algebra the generators of which mix the representations of in a way that relates different space times and creates the dynamics this construction can be considered eventually as a model for two dimensional quantum gravity

a phase transition for circle maps and cherry flows we study c2 weakly order preserving circle maps with a flat interval the main result of the paper is about a sharp transition from degenerate geometry to bounded geometry depending on the degree of the singularities at the boundary of the flat interval we prove that the non wandering set has zero hausdorff dimension in the case of degenerate geometry and it has hausdorff dimension strictly greater than zero in the case of bounded geometry our results about circle maps allow to establish a sharp phase transition in the dynamics of cherry flows

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

map dependence of the fractal dimension deduced from iterations of circle maps every orientation preserving circle mapg with inflection points including the maps proposed to describe the transition to chaos in phase locking systems gives occasion for a canonical fractal dimensiond namely that of the associated set of for whichf g has irrational rotation number we discuss how this dimension depends on the orderr of the inflection points in particular in the smooth case we find numerically thatd r d r 1 r 1 8

limit sets of s unimodal maps with zero entropy one dimensional mappings at the limit of period doubling are studied in this paper without the use of the renormalization theory of feigenbaum and others the principal result is that the attracting part of the nonwandering set is a cantor set of measure zero under the additional assumption that the map has negative schwarzian derivative

critical circle maps near bifurcation we estimate harmonic scalings in the parameter space of a one parameter family of critical circle maps these estimates lead to the conclusion that the hausdorff dimension of the complement of the frequency locking set is less than 1 but not less than 1 3 moreover the rotation number is a holder continuous function of the parameter

differentiable circle maps with a flat interval we study weakly order preserving circle maps with a flat interval which are differentiable even on the boundary of the flat interval we obtain estimates on the lebesgue measure and the hausdorff dimension of the non wandering set also a sharp transition is found from degenerate geometry to bounded geometry depending on the degree of the singularities at the boundary of the flat interval

metric attractors for smooth unimodal maps we classify the measure theoretic attractors of general c 3 unimodal maps with quadratic critical points the main ingredient is the decay of geometry

noncrystallographic coxeter group h4 in e8 the e8 lattice is constructed in terms of icosians by matching two sets of f4 lattices described by quaternions embedding the noncrystallographic group h4 into the weyl group w e8 has been described using matrix generators with an emphasis on the relevant coxeter elements the conjugacy classes of h4 in terms of quaternions and the characters of the two four dimensional irreducible representations are explicitly calculated

differentiability at the tip of arnold tongues for diophantine rotations numerical studies and renormalization group explanations we study numerically the regularity of arnold tongues corresponding to diophantine rotation numbers of circle maps at the edge of validity of kam theorem this serves as a good test for the numerical stability of two different algorithms we find empirically that arnold tongues are only finitely differentiable at the tip we also find several scaling properties of the sobolev norms of the conjugacy near the breakdown we also provide a renormalization group explanation of the regularity at the tip and the scaling behaviors of the sobolev regularity we also uncover empirically some other patterns which require explanation

infinite dimensional generalizations of finite dimensional symmetries infinite dimensional algebras associated with simple finite dimensional lie algebra g are considered generalizations of sl2 are studied in details a manifest expression for the virasoro like generalization of d 3 conformal algebra so 3 2 is also presented applications in higher spin theory and in conformal qft are discussed

quasicrystals and icosians a family of quasicrystals of dimensions 1 2 3 4 governed by the root lattice e8 is constructed the use of the icosian ring found in the quaternions with coefficients in q square root 5 allows simultaneous interpretation of the construction both in physical space and as a result of the standard cut and projection method in double dimension icosians are seen to provide a natural co ordination scheme for these quasicrystals nested sequences of quasicrystals form systems whose symmetries are all derivable from inflational and reflective symmetries directly related to the arithmetic of the icosians the use of coxeter diagrams clarifies the amazing relationship of e8 and quasicrystal symmetries and leads to the fundamental chain e8 contains implies d6 contains implies a4 contains implies a1 a1 that underlies five fold symmetry in quasicrystals decomposition of quasicrystals into concentric shells and a counting formula for the cardinalities of these shells is discussed

deformations and renormalisations ofw deformations of the infiniten limit of the zamolodchikovwn algebra are discussed a recent one due to pope romans and shen with non zero central extensions for every conformal spin is shown to be formally renormalisable to one representable in moyal bracket form another deformation is discovered which like the algebra of pope et al possesses automatic closure but has non zero central extension only in the virasoro subalgebra

the scaling of arnol d tongues for differentiable homeomorphisms of the circle we show both for diffeomorphisms of the circle and for differentiable homeomorphisms that are not diffeomorphisms that the widths of the arnol d tongues in a one parameter family scale asq 3 whenq is the denominator of the rotation number

torus maps and the problem of a one dimensional optical resonator with a quasiperiodically moving wall abstract we study the problem of the asymptotic behavior of the electromagnetic field in an optical resonator one of whose walls is at rest and the other is moving quasiperiodically with d 2 incommensurate frequencies we show that this problem can be reduced to a problem about the behavior of the iterates of a map of the d dimensional torus that preserves a foliation by irrational straight lines in particular the jacobian of this map has d 1 eigenvalues equal to 1 we present rigorous and numerical results about several dynamical features of such maps we also show how these dynamical features translate into properties for the field in the cavity in particular we show that when the torus map satisfies a kam theorem which happens for a cantor set of positive measure of parameters the energy of the electromagnetic field remains bounded when the torus map is in a resonant region which happens in open sets of parameters inside the gaps of the previous cantor set the energy grows exponentially

e8 lattice with icosians and z5 symmetry a simple method for the construction of the e8 root system with icosians is suggested it is confronted with the root system of e8 obtained by integral octonions embeddings of the maximal subgroups su 5 su 5 and su 3 e6 in e8 with respective five fold and three fold symmetries are discussed

theory of circle maps and the problem of one dimensional optical resonator with a periodically moving wall we consider the electromagnetic field in a cavity with a periodically oscillating perfectly reflecting boundary and show that the mathematical theory of circle maps leads to several physical predictions notably well known results in the theory of circle maps which we review briefly imply that there are intervals of parameters where the waves in the cavity get concentrated in wave packets whose energy grows exponentially even if these intervals are dense for typical motions of the reflecting boundary in the complement there is a positive measure set of parameters where the energy remains bounded

on hausdorff dimension of unimodal attractors there exists a universal constant 1 such that every attractor of every c 4 unimodal map with a non degenerate critical point is an analytic manifold or its hausdorff dimension is equal to or less than

structure constants for new infinite dimensional lie algebras of u n n tensor operators and applications the structure constants for moyal brackets of an infinite basis of functions on the algebraic manifolds m of pseudo unitary groups u n n are provided they generalize the virasoro and algebras to higher dimensions the connection with volume preserving diffeomorphisms on m higher generalized spin and tensor operator algebras of u n n is discussed these centrally extended infinite dimensional lie algebras also provide the arena for nonlinear integrable field theories in higher dimensions residual gauge symmetries of higher extended objects in the light cone gauge and c algebras for tractable non commutative versions of symmetric curved spaces

rigidity of critical circle mappings ii we prove that two c3 critical circle maps with the same rotation number in a special set are c1 conjugate for some 0 provided their successive renormalizations converge together at an exponential rate in the c0 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 c1 conjugate for any 0 the class of rotation numbers for which such examples exist contains diophantine numbers

fractal properties of critical invariant curves we examine the dimension of the invariant measure for some singular circle homeomorphisms for a variety of rotation numbers through both the thermodynamic formalism and numerical computation the maps we consider include those induced by the action of the standard map on an invariant curve at the critical parameter value beyond which the curve is destroyed our results indicate that the dimension is universal for a given type of singularity and rotation number and that among all rotation numbers the golden mean produces the largest dimension