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A panoramic view of modern ergodic theory On the occasion of the Abel prize to H. Furstenberg and G. A. Margulis Anish Ghosh, Tata Institute July 1, 2020 1 / 132 Hillel Furstenberg From the Abel prize website When Hillel Furstenberg published one of his early papers, a rumor circulated that he was not an individual but instead a pseudonym for a group of mathematicians. The paper contained ideas from so many different areas, surely it could not possibly be the work of one man? 2 / 132 Gregory Margulis Jacques Tits on Margulis It is not exaggerated to say that, on several occasions, he has bewildered the experts by solving questions which appeared to be completely out of reach at the time. 3 / 132 Armand Borel on Margulis On more than one occasion, Borel mentioned that Margulis was the first person who caused confusion between two Borels in the Borel measure and the Borel subgroups, by using both Lie theory (or rather algebraic group theory) and ergodic theory simultaneously. He also declared that he was not related to the other Borel. Ji Lizhen, A Summary of the Work of Gregory Margulis Pure and Applied Mathematics Quarterly Volume 4, Number 1 (Special Issue: In honor of Gregory Margulis, Part 2 of 2) 1{69, 2008 4 / 132 Nostalgia 5 / 132 Early beginnings Furstenberg, Harry On the infinitude of primes. Amer. Math. Monthly 62 (1955), 353. Kazhdan, D. A.; Margulis, G. A. A proof of Selberg's hypothesis. (Russian) Mat. Sb. (N.S.) 75 (117) 1968 163{168. 6 / 132 Rigidity Connections to number theory Connections to probability Abel prize citation \for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics." Diverse achievements, common threads There are (at least) three major common threads running through the work of Furstenberg and Margulis 7 / 132 Connections to number theory Connections to probability Abel prize citation \for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics." Diverse achievements, common threads There are (at least) three major common threads running through the work of Furstenberg and Margulis Rigidity 8 / 132 Connections to probability Abel prize citation \for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics." Diverse achievements, common threads There are (at least) three major common threads running through the work of Furstenberg and Margulis Rigidity Connections to number theory 9 / 132 Diverse achievements, common threads There are (at least) three major common threads running through the work of Furstenberg and Margulis Rigidity Connections to number theory Connections to probability Abel prize citation \for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics." 10 / 132 Non-commutative methods, products of random matrices Unique ergodicity of horocycle flow Disjointness Fractal methods in ergodic theory,.. Hillel Furstenberg Introduced Ergodic theory in combinatorics (Szemer´edi'sTheorem) 11 / 132 Unique ergodicity of horocycle flow Disjointness Fractal methods in ergodic theory,.. Hillel Furstenberg Introduced Ergodic theory in combinatorics (Szemer´edi'sTheorem) Non-commutative methods, products of random matrices 12 / 132 Disjointness Fractal methods in ergodic theory,.. Hillel Furstenberg Introduced Ergodic theory in combinatorics (Szemer´edi'sTheorem) Non-commutative methods, products of random matrices Unique ergodicity of horocycle flow 13 / 132 Fractal methods in ergodic theory,.. Hillel Furstenberg Introduced Ergodic theory in combinatorics (Szemer´edi'sTheorem) Non-commutative methods, products of random matrices Unique ergodicity of horocycle flow Disjointness 14 / 132 Hillel Furstenberg Introduced Ergodic theory in combinatorics (Szemer´edi'sTheorem) Non-commutative methods, products of random matrices Unique ergodicity of horocycle flow Disjointness Fractal methods in ergodic theory,.. 15 / 132 Oppenheim's conjecture Construction of expander graphs Baker-Sprindˇzukconjectures in Diophantine approximation Normal subgroup theorem, Margulis Lemma, Bowen-Margulis measure, Random walks on homogeneous spaces.. Gregory Margulis Arithmeticity and Superrigidity theorems 16 / 132 Construction of expander graphs Baker-Sprindˇzukconjectures in Diophantine approximation Normal subgroup theorem, Margulis Lemma, Bowen-Margulis measure, Random walks on homogeneous spaces.. Gregory Margulis Arithmeticity and Superrigidity theorems Oppenheim's conjecture 17 / 132 Baker-Sprindˇzukconjectures in Diophantine approximation Normal subgroup theorem, Margulis Lemma, Bowen-Margulis measure, Random walks on homogeneous spaces.. Gregory Margulis Arithmeticity and Superrigidity theorems Oppenheim's conjecture Construction of expander graphs 18 / 132 Normal subgroup theorem, Margulis Lemma, Bowen-Margulis measure, Random walks on homogeneous spaces.. Gregory Margulis Arithmeticity and Superrigidity theorems Oppenheim's conjecture Construction of expander graphs Baker-Sprindˇzukconjectures in Diophantine approximation 19 / 132 Gregory Margulis Arithmeticity and Superrigidity theorems Oppenheim's conjecture Construction of expander graphs Baker-Sprindˇzukconjectures in Diophantine approximation Normal subgroup theorem, Margulis Lemma, Bowen-Margulis measure, Random walks on homogeneous spaces.. 20 / 132 ×2 : x ! 2x mod 1 gt acting on SL(2; R)= SL(2; Z) et 0 g = t 0 e−t Geodesic flow on the modular surface 1 t Horocycle flow: u = t 0 1 Dynamical systems ×2 : T ! T 21 / 132 gt acting on SL(2; R)= SL(2; Z) et 0 g = t 0 e−t Geodesic flow on the modular surface 1 t Horocycle flow: u = t 0 1 Dynamical systems ×2 : T ! T ×2 : x ! 2x mod 1 22 / 132 et 0 g = t 0 e−t Geodesic flow on the modular surface 1 t Horocycle flow: u = t 0 1 Dynamical systems ×2 : T ! T ×2 : x ! 2x mod 1 gt acting on SL(2; R)= SL(2; Z) 23 / 132 Geodesic flow on the modular surface 1 t Horocycle flow: u = t 0 1 Dynamical systems ×2 : T ! T ×2 : x ! 2x mod 1 gt acting on SL(2; R)= SL(2; Z) et 0 g = t 0 e−t 24 / 132 1 t Horocycle flow: u = t 0 1 Dynamical systems ×2 : T ! T ×2 : x ! 2x mod 1 gt acting on SL(2; R)= SL(2; Z) et 0 g = t 0 e−t Geodesic flow on the modular surface 25 / 132 Dynamical systems ×2 : T ! T ×2 : x ! 2x mod 1 gt acting on SL(2; R)= SL(2; Z) et 0 g = t 0 e−t Geodesic flow on the modular surface 1 t Horocycle flow: u = t 0 1 26 / 132 27 / 132 X ⊂ T is invariant under ×p if ×p(X ) ⊂ X There are many invariant closed subsets for ×p Example: the circle is invariant. So is the middle third Cantor set (for ×3) ×2; ×3 ×p : x ! px mod 1 28 / 132 There are many invariant closed subsets for ×p Example: the circle is invariant. So is the middle third Cantor set (for ×3) ×2; ×3 ×p : x ! px mod 1 X ⊂ T is invariant under ×p if ×p(X ) ⊂ X 29 / 132 Example: the circle is invariant. So is the middle third Cantor set (for ×3) ×2; ×3 ×p : x ! px mod 1 X ⊂ T is invariant under ×p if ×p(X ) ⊂ X There are many invariant closed subsets for ×p 30 / 132 ×2; ×3 ×p : x ! px mod 1 X ⊂ T is invariant under ×p if ×p(X ) ⊂ X There are many invariant closed subsets for ×p Example: the circle is invariant. So is the middle third Cantor set (for ×3) 31 / 132 Every x 2 [0; 1] has a base p expansion P1 −i x = 0:x1x2 ··· = i=1 xi p ; xi 2 f0; 1;:::; p − 1g This expansion is unique (except for rational numbers which have two) A number is rational if and only if its expansion is eventually periodic The point is that ×p is (semi-)conjugated to a Bernoulli shift with p symbols and any sub-shift gives an invariant Cantor set. 32 / 132 P1 −i x = 0:x1x2 ··· = i=1 xi p ; xi 2 f0; 1;:::; p − 1g This expansion is unique (except for rational numbers which have two) A number is rational if and only if its expansion is eventually periodic The point is that ×p is (semi-)conjugated to a Bernoulli shift with p symbols and any sub-shift gives an invariant Cantor set. Every x 2 [0; 1] has a base p expansion 33 / 132 This expansion is unique (except for rational numbers which have two) A number is rational if and only if its expansion is eventually periodic The point is that ×p is (semi-)conjugated to a Bernoulli shift with p symbols and any sub-shift gives an invariant Cantor set. Every x 2 [0; 1] has a base p expansion P1 −i x = 0:x1x2 ··· = i=1 xi p ; xi 2 f0; 1;:::; p − 1g 34 / 132 A number is rational if and only if its expansion is eventually periodic The point is that ×p is (semi-)conjugated to a Bernoulli shift with p symbols and any sub-shift gives an invariant Cantor set. Every x 2 [0; 1] has a base p expansion P1 −i x = 0:x1x2 ··· = i=1 xi p ; xi 2 f0; 1;:::; p − 1g This expansion is unique (except for rational numbers which have two) 35 / 132 The point is that ×p is (semi-)conjugated to a Bernoulli shift with p symbols and any sub-shift gives an invariant Cantor set. Every x 2 [0; 1] has a base p expansion P1 −i x = 0:x1x2 ··· = i=1 xi p ; xi 2 f0; 1;:::; p − 1g This expansion is unique (except for rational numbers which have two) A number is rational if and only if its expansion is eventually periodic 36 / 132 ×p shifts the expansion of a number by 1 ×p(0:x1x2 ::: ) = 0:x2x3 ::: For p ≥ 2 let ×p be as before the multiplication by p on the circle 37 / 132 ×p(0:x1x2 ::: ) = 0:x2x3 ::: For p ≥ 2 let ×p be as before the multiplication by p on the circle ×p shifts the expansion of a number by 1 38 / 132 For p ≥ 2 let ×p be as before the multiplication by p on the circle ×p shifts the expansion of a number by 1 ×p(0:x1x2 ::: ) = 0:x2x3 ::: 39 / 132 Are there many joint
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