Steven P. Lalley

Renewal theorems in symbolic dynamics, with applications to geodesic flows, noneuclidean tessellations and their fractal limits

0. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I. Renewal theorems in symbolic dynamics . . . . . . . . . . . . 1. Background: Shifts, suspension flows, thermodynamic formalism . 2. 3. 4. 5. 6. 7. 1 5 5 Renewal measures and renewal theorems . . . . . . . . . . . . . . 7 A modification for finite sequences . . . . . . . . . . . . . . . . . 10 Equidistribution theorems . . . . . . . . . . . . . . . . . . . . . . 13 Periodic orbits of suspension flows . . . . . . . . . . . . . . . . . 17 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . 20 Perturbation theory for Perron-Frobenius operators . . . . . . . . 23 8. Fourier analysis of the renewal equation . . . . . . . . . . . . . . 26 Part II. Applications to discrete groups . . . . . . . . . . . . . . . . . 32 9. Symbolic dynamics for Schottky groups . . . . . . . . . . . . . . 32 10. Symbolic dynamics for Fuchsian groups . . . . . . . . . . . . . . 34 11. The geodesic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 12. Distribution of noneuclidean lattice points and fundamental polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 13. Packing and covering functions of the limit set . . . . . . . . . . . 43 14. Random walk and Hausdorff measure . . . . . . . . . . . . . . . . 53 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

Random Series in Powers of Algebraic Integers: Hausdorff Dimension of the Limit Distribution

We study the distributions F θ, p of the random sums [sum ] ∞ 1 e n θ n , where e 1 , e 2 , … are i.i.d. Bernoulli- p and θ is the inverse of a Pisot number (an algebraic integer β whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p =.5, F θ, p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain β of small degree, simulation gives the Hausdorff dimension to several decimal places.

Distribution of periodic orbits of symbolic and Axiom A flows

Indeed, Margulis [lo] announced that for the geodesic flow on a d-dimensional compact manifold of curvature - 1 the number of periodic orbits 7 with (minimal) period r(1) I x is asymptotic to ecd-‘jx/(d - 1)x. This result bears a striking resemblance to the prime number theorem. Parry and Pollicott [12], following earlier work by Bowen [2, 41, generalized Margulis’ theorem to weakly mixing Axiom A flows, proving that # { 7: r(1) I x} eh*/hx, where h is the topological entropy of the flow. Sarnak [15] has related results for the horocycle flow. Bowen [3] and Parry [ll] proved analogues of the Dirichlet density theorem for mixing Axiom A flows, e.g., if 7(G) represents the integral of the continuous function G over one period of 7, then Z roj s ,7(G)/~(l) - (eh”/hx)lG dji, where ji is the invariant probability measure of maximum entropy. This paper pursues an altogether different analogy, this between the distribution problems for periodic orbits of Axiom A and symbolic flows and those of classical probability theory. This analogy leads to theorems which apparently have no counterparts in number theory. Moreover, it leads to techniques quite different from those commonly used in studying periodic orbits: in particular, there is no use of zeta functions or any of the attendant Tauberian theorems. We do not believe that the main results of this paper can be obtained by analyzing zeta functions. These results do, however, make use of the groundwork done by Bowen in [4], which reduces

Falconer's formula for the Hausdorff dimension of a self-affine set in R 2

Let A 1 , A 2 ,…, A k be a finite set of contractive, affine, invertible self-mappings of R 2 . A compact subset Λ of R 2 is said to be self-affine with affinities A 1 , A 2 ,…, A k if It is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A 1 , A 2 ,…, A k are similarity transformations, the set Λ is said to be self-similar . Self-similar sets are well understood, at least when the images A i (Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [ 12, 10 ]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

Percolation on Fuchsian groups

It is shown that, for site percolation on the dual Dirichlet tiling graph of a co-compact Fuchsian group of genus ≥ 2, infinitely many infinite connected clusters exist almost surely for certain values of the parameter p = P{site is open}. In such cases, the set of limit points at ∞ of an infinite cluster is shown to be a perfect, nowhere dense set of Lebesgue measure 0. These results are also shown to hold for a class of hyperbolic triangle groups.

Renewal theorem for a class of stationary sequences

SummaryA renewal theorem is obtained for stationary sequences of the form ξn=ξ(...,Xn-1,Xn,Xn+1...), whereXn,

Stochastic search in a convex region

n \in \mathbb{Z}

Statistically self-affine sets: Hausdorff and box dimensions

, are i.i.d. r.v.s. valued in a Polish space. This class of processes is sufficiently broad to encompass functionals of recurrent Markov chains, functionals of stationary Gaussian processes, and functionals of one-dimensional Gibbs states. The theorem is proved by a new coupling construction.