Entropy, Dimension and Uniform Distribution

The starting point for this work is an old question of H. Furstenberg. Recall that a Borel probability measure on the torus T = R=Z is called p-invariant if p = , where p(x) def = p x (mod 1) and p def = 1 p . Suppose that is non-atomic, and is simultaneously 2and 3-invariant. Assume also that is ergodic under the joint action of 2; 3. Does it follow that is necessarily Lebesgue measure? In his landmark article from 1967, Furstenberg established the corresponding topological assertion; namely, he showed that an in nite closed set in T, invariant under multiplication by two multiplicatively independent integers p and q (log p= log q = 2 Q), must be all of T. The measure-theoretic question remained open for a longer period, and in fact, in the form presented by Furstenberg is still unsolved, since the best positive result so far (Rudolph 1990, Johnson 1992) assumes in addition that has positive entropy. For more background, see section I.1. A new approach was recently introduced by B. Host (1995), who proved the following: for any p-invariant ergodic measure with positive entropy, if q is relatively prime to p, then for -almost every x 2 T, the sequence fqnx (mod 1)g is uniformly distributed. This pointwise result implies the result of Johnson and Rudolph, when p and q are relatively prime. The rst question we set to answer is: what are the combinatorial conditions on a sequence of integers fcng such that for every p-invariant and ergodic measure , the sequence fcnx (mod 1)g is uniformly distributed -almost surely? This question, and Host's method of proof, lead to the following de nition of the p-adic collision exponent : p(fcng) = lim sup n!1 log jf0 k; ` < pn : ck c` (mod pn)gj n log p : 5 Abstract 6 The collision exponent measures how well the numbers fck (mod pn)gpn 1 k=0 are distributed, as n ! 1. Surely, 1 p 2, and if p and q are relatively prime, then p(fqng) = 1. Sequences for which p = 1 will satisfy Host's conclusion. In fact, a more relaxed de nition su ces, leading to introducing the reduced p-adic collision exponent : 0p(fcng) = lim "#0 inf fc0ng p(fc0ng); where fc0ng ranges over sequences agreeing with fcng on a set of indices with density 1 ". Sequences for which 0p(fcng) = 1 will still satisfy Host's conclusion. (These are called sequences with almost-sub-exponential collisions in Chapter I.) In the rst chapter we prove su cient conditions on a sequence fcng guaranteeing that 0pfcng = 1. The condition is stated in terms of the existence of q-adic smooth interpolation with a nite number of critical points, for every prime qjp. This gives a class of sequences which includes, for instance, the sequence cn =P fi(n)qn i , where the numbers qi are distinct and are relatively prime to p and fi are any polynomials. More generally, recursion sequences for which the free coe cient of the recursion polynomial is relatively prime to p are in this class as well, provided they satisfy a simple irreducibility condition. Another example is f25ng, for p = 3. In the multi-dimensional setting we derive su cient conditions for a pair of endomorphisms A;B 2 End(Td) (with A diagonal) and an A-invariant and ergodic measure , such that B-orbits of the form fBn!g are uniformly distributed for -almost every ! 2 Td. This is done in Chapter I for the case d = 2, and in Chapter II for the general case. These results are used in the rst chapter to give analogs of Host's result for Cantor sets in T2. They also give a simple proof (for the case of relatively-prime multipliers) of a new result by B. Kra (1996), that extends Furstenberg's diophantine theorem, quoted before. In the second chapter we present some more multi-dimensional results. Let A;B be two diagonal endomorphisms of the d-dimensional torus with corresponding eigenvalues relatively prime. We show that for any A-invariant ergodic measure , there exists a projection onto a torus Tr of dimension r dim , that maps -almost every B-orbit to a uniformly distributed sequence in Tr . Consequently, the projections of Bn converge to Lebesgue measure on Tr (allowing an exceptional zero-density set of indices). As a corollary we obtain that the Hausdor dimension of any bi-invariant measure is an integer. In the one-dimensional case, entropy and dimension of measures are closely related: for6 The collision exponent measures how well the numbers fck (mod pn)gpn 1 k=0 are distributed, as n ! 1. Surely, 1 p 2, and if p and q are relatively prime, then p(fqng) = 1. Sequences for which p = 1 will satisfy Host's conclusion. In fact, a more relaxed de nition su ces, leading to introducing the reduced p-adic collision exponent : 0p(fcng) = lim "#0 inf fc0ng p(fc0ng); where fc0ng ranges over sequences agreeing with fcng on a set of indices with density 1 ". Sequences for which 0p(fcng) = 1 will still satisfy Host's conclusion. (These are called sequences with almost-sub-exponential collisions in Chapter I.) In the rst chapter we prove su cient conditions on a sequence fcng guaranteeing that 0pfcng = 1. The condition is stated in terms of the existence of q-adic smooth interpolation with a nite number of critical points, for every prime qjp. This gives a class of sequences which includes, for instance, the sequence cn =P fi(n)qn i , where the numbers qi are distinct and are relatively prime to p and fi are any polynomials. More generally, recursion sequences for which the free coe cient of the recursion polynomial is relatively prime to p are in this class as well, provided they satisfy a simple irreducibility condition. Another example is f25ng, for p = 3. In the multi-dimensional setting we derive su cient conditions for a pair of endomorphisms A;B 2 End(Td) (with A diagonal) and an A-invariant and ergodic measure , such that B-orbits of the form fBn!g are uniformly distributed for -almost every ! 2 Td. This is done in Chapter I for the case d = 2, and in Chapter II for the general case. These results are used in the rst chapter to give analogs of Host's result for Cantor sets in T2. They also give a simple proof (for the case of relatively-prime multipliers) of a new result by B. Kra (1996), that extends Furstenberg's diophantine theorem, quoted before. In the second chapter we present some more multi-dimensional results. Let A;B be two diagonal endomorphisms of the d-dimensional torus with corresponding eigenvalues relatively prime. We show that for any A-invariant ergodic measure , there exists a projection onto a torus Tr of dimension r dim , that maps -almost every B-orbit to a uniformly distributed sequence in Tr . Consequently, the projections of Bn converge to Lebesgue measure on Tr (allowing an exceptional zero-density set of indices). As a corollary we obtain that the Hausdor dimension of any bi-invariant measure is an integer. In the one-dimensional case, entropy and dimension of measures are closely related: for Abstract 7 a p-invariant ergodic measure , we have (by Billingsley, 1965) dim = h( ; p)= log p. In the d-dimensional non-conformal case a similar formula does not hold. In order to translate statements about entropy to statements about dimension, one needs to consider a weighted average of entropies of projections, called the Ledrappier-Young dimension. Invariant sets for a single toral endomorphism can have fractional Hausdor dimension. Using the above measure-theoretic result, we show that the situation is di erent for sets invariant under two diagonal endomorphisms: under the same conditions on A and B, any closed bi-invariant set has integer dimension. Moreover, given a closed A-invariant set S, there exists a projection onto a torus Tr of dimension r dimS such that the projections of fBnSg converge to Tr in the Hausdor metric (outside a negligible exceptional set of indices). Chapter III contains a part of a new work, that extends the results of the rst chapter. The basic tool used from this work is a theorem about entropies of convolutions: given ergodic p-invariant measures f ig on T whose normalized entropies hi = h( i; p)= log p satisfy X hi j log hij = 1, the entropy of the convolution 1 n converges to log p. A variant of this result gives a powerful tool we call the Bootstrap Lemma. We show how this result can be combined with Host's method of the rst chapter to prove the following. For every p-invariant ergodic with positive entropy, 1 N PN 1 n=0 cn converges weak to Lebesgue measure as N ! 1, for any integer sequence fcng satisfying 0p(fckg) < 2. This extends the main result of Johnson and Rudolph (1995), who considered the sequence ck = qk when p and q are multiplicatively independent. Note that while in Chapter I we had a stronger conclusion (fcnxg uniformly distributed -a.e.), the result there could only be proved for 0p(fckg) = 1. This chapter also contains an algorithm for computing the reduced p-adic exponent of any linear recursion sequence, for any integer p. The algorithm involves only solving linear equations. As in the previous two chapters, we obtain topological corollaries concerning Hausdor dimension from the measure-theoretic results. One of them is the following: for any sequence fSig of p-invariant closed subsets of T, if 1 Xi=1 dimH(Si) j log dimH(Si)j =1; then dimH(S1 + + Sn) ! 1.7 a p-invariant ergodic measure , we have (by Billingsley, 1965) dim = h( ; p)= log p. In the d-dimensional non-conformal case a similar formula does not hold. In order to translate statements about entropy to statements about dimension, one needs to consider a weighted average of entropies of projections, called the Ledrappier-Young dimension. Invariant sets for a single toral endomorphism can have fractional Hausdor dimension. Using the above measure-theoretic result, we show that the situation is di erent for sets invariant under two diagonal endomorphisms: under the same conditions on A and B, any closed bi-invariant set has integer dimension. Moreover, given a closed A-invariant set S, there exists a projection onto a torus Tr of dimension r dimS such that the projections of fBnSg converge to Tr in the Hausdor metric (outside a negligible exceptional set of indices). Chapter III contains a part of a new work, that extends the results of the rst chapter. The basic tool used from this work is a theorem about entropies of convolutions: given ergodic p-invariant measures f ig on T whose normalized entropies hi = h( i; p)= log p satisfy X hi j log hij = 1, the entropy of the convolution 1 n converges to log p. A variant of this result gives a powerful tool we call the Bootstrap Lemma. We show how this result can be combined with Host's method of the rst chapter to prove the following. For every p-invariant ergodic with positive entropy, 1 N PN 1 n=0 cn converges weak to Lebesgue measure as N ! 1, for any integer sequence fcng satisfying 0p(fckg) < 2. This extends the main result of Johnson and Rudolph (1995), who considered the sequence ck = qk when p and q are multiplicatively independent. Note that while in Chapter I we had a stronger conclusion (fcnxg uniformly distributed -a.e.), the result there could only be proved for 0p(fckg) = 1. This chapter also contains an algorithm for computing the reduced p-adic exponent of any linear recursion sequence, for any integer p. The algorithm involves only solving linear equations. As in the previous two chapters, we obtain topological corollaries concerning Hausdor dimension from the measure-theoretic results. One of them is the following: for any sequence fSig of p-invariant closed subsets of T, if 1 Xi=1 dimH(Si) j log dimH(Si)j =1; then dimH(S1 + + Sn) ! 1. Abstract 8 Note: This thesis is written in the format \Doctoral Thesis Composed of Articles". The three chapters of this work correspond to the following articles: 1. Meiri (1996), Entropy and uniform distribution of orbits in Td, to appear in Israel Journal of Mathematics. 2. Meiri and Peres (1996), Bi-invariant sets and measures have integer Hausdor dimension, submitted to Ergodic Theory and Dynamical Systems. 3. Lindenstrauss, Meiri and Peres (1997), Entropy of convolutions on the circle, preprint. The rst two articles appear in their entirety; Chapter III contains excerpts from the third one. Although each chapter can be read independently, I made an e ort to streamline their contents, by removing some repetitions and replacing cross references between the articles by chapter references. For instance, Theorem I.6.1 refers to Theorem 6.1 of the rst chapter/article. I also expanded some of the examples. Otherwise, the articles remain intact. Acknowledgments. The authors of the articles express their thanks to Hillel Furstenberg, Dan Rudolph, and Benjamin Weiss for many helpful and enlightening discussions.8 Note: This thesis is written in the format \Doctoral Thesis Composed of Articles". The three chapters of this work correspond to the following articles: 1. Meiri (1996), Entropy and uniform distribution of orbits in Td, to appear in Israel Journal of Mathematics. 2. Meiri and Peres (1996), Bi-invariant sets and measures have integer Hausdor dimension, submitted to Ergodic Theory and Dynamical Systems. 3. Lindenstrauss, Meiri and Peres (1997), Entropy of convolutions on the circle, preprint. The rst two articles appear in their entirety; Chapter III contains excerpts from the third one. Although each chapter can be read independently, I made an e ort to streamline their contents, by removing some repetitions and replacing cross references between the articles by chapter references. For instance, Theorem I.6.1 refers to Theorem 6.1 of the rst chapter/article. I also expanded some of the examples. Otherwise, the articles remain intact. Acknowledgments. The authors of the articles express their thanks to Hillel Furstenberg, Dan Rudolph, and Benjamin Weiss for many helpful and enlightening discussions. Chapter I Entropy and Uniform Distribution of Orbits in Td