convergence of spherical averages for actions of free groups let x v be a probability space and suppose a free group fm with m generators acts on x v by measure preserving transformations let ai am be a set of free generators for fm and let t1 tm x x be transformations corresponding to the generators write t i t 1 for i 1 m and set a m 1 1 m we also have the action fm on l1 x v defined by tg p o tg 1 g e fm consider the set wa of all finite words over the alphabet a

stationary measures and invariant subsets of homogeneous spaces iii let g be a real lie group be a lattice in g and g be a compactly generated closed subgroup of g if the zariski closure of the group ad g is semisimple with no compact factor we prove that every g orbit closure in g is a finite volume homogeneous space we also establish related equidistribution properties

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

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

summation of series in statistical mechanics by continued exponentials abstract continued exponentials ces are used to extend a series when a finite number of coefficients are known since the ce does not have singularities built in as pade approximants and the maximum entropy me method do it is a useful technique to estimate thermodynamic functions from partial series right up to the transition point radius of convergence for first order phase transitions the method is tested on fisher s cluster model an ideal mixture of clusters with no excluded volume that contains a first order gas liquid phase transition with a critical point the one dimensional ising model an example of a fluid above the critical point and the two dimensional ising model an example of a fluid below the critical point where the effect of excluded volume is included in addition we compare the use of the ce to estimate the equation of state for hard spheres from the known partial virial series through b 8 with the results from pade approximants and the me method the results from the three methods are comparable with the results from the ce approximation agreeing best with simulation data ces can also be used to approximate series that are asymptotic to a power law in the case where the exponent is known we use the case of diffusion in one dimension as an example with power law behavior and compare the use of ces and the me method to treat this system

random walks on homogeneous spaces and diophantine approximation on fractals we extend results of y benoist and j f quint concerning random walks on homogeneous spaces of simple lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily zariski dense but satisfies some expansion properties for the adjoint action using these dynamical results we study diophantine properties of typical points on some self similar fractals in mathbb r d as examples we show that for any self similar fractal mathcal k subseteq mathbb r d satisfying the open set condition for instance any translate or dilate of cantor s middle thirds set or of a koch snowflake almost every point with respect to the natural measure on mathcal k is not badly approximable furthermore almost every point on the fractal is of generic type which means in the one dimensional case that its continued fraction expansion contains all finite words with the frequencies predicted by the gauss measure we prove analogous results for matrix approximation and for the case of fractals defined by mobius transformations

stationary measures and invariant subsets of homogeneous spaces ii let g be a real lie group be a lattice in g and be a compactly generated closed subgroup of g if the zariski closure of the group ad is semisimple with no compact factor we prove that every orbit closure in g is a nite volume homogeneous space we also establish related equidistribution properties

generic stationary measures and actions let g be a countably infinite group and let be a generating probability measure on g we study the space of stationary borel probability measures on a topological g space and in particular on z g where z is any perfect polish space we also study the space of stationary measurable g actions on a standard nonatomic probability space equip the space of stationary measures with the weak topology when has finite entropy we show that a generic measure is an essentially free extension of the poisson boundary of g when z is compact this implies that the simplex of stationary measures on z g is a poulsen simplex we show that this is also the case for the simplex of stationary measures on 0 1 g we furthermore show that if the action of g on its poisson boundary is essentially free then a generic measure is isomorphic to the poisson boundary next we consider the space of stationary actions equipped with a standard topology known as the weak topology here we show that when g has property t the ergodic actions are meager we also construct a group g without property t such that the ergodic actions are not dense for some finally for a weaker topology on the set of actions which we call the very weak topology we show that a dynamical property e g ergodicity is topologically generic if and only if it is generic in the space of measures there we also show a glasner king type 0 1 law stating that every dynamical property is either meager or residual

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

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

random walks on random coset spaces with applications to furstenberg entropy we determine the range of furstenberg entropy for stationary ergodic actions of nonabelian free groups by an explicit construction involving random walks on random coset spaces

on raghunathan s measure conjecture this paper represents the last in our three part series on raghunathan s measure conjecture see r1 r2 for parts i and ii more specifically let g be a real lie group all groups in this paper are assumed to be second countable with the lie algebra r a discrete subgroup of g and ir g r g the projection lwg rg the group g acts by right translations on r g x x g x e r g g e g let gt be a borel probability measure on r g define

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

stationary measures and equidistribution for orbits of nonabelian semigroups on the torus let be a semigroup of d d nonsingular integer matrices and consider the action of on the torus t we assume throughout that the action is strongly irreducible there is no subtorus invariant under a finite index subsemigroup of the strong irreducibility assumption in particular implies that acts ergodically on t equipped with the lebesgue measure m therefore the orbit of lebesgue almost every x t is dense and in an appropriate sense even becomes equidistributed however when is cyclic there is a set of full hausdorff dimension of exceptional points x for which x fails to be dense when is bigger the distribution of individual orbits can be expected to be much more restrictive an important result in this direction is due to furstenberg who showed for d 1 in which case 1 an abelian semigroup never satisfies condition 1 indeed the group generated by a semigroup satisfying 1 is nonamenable assumption 2 is a technical condition which is in particular satisfied when is a zariski dense semigroup of sld z 16