Complex dynamics and arithmetic equidistribution

Laura DeMarco

April 2018

Northwestern University

Equidistribution

Goal: applications of arithmetic equidistribution theorems to complex dynamics and connections to arithmetic geometry

En = a finite set in X

The points of {En} are equidistributed with respectto a measure µ if the discrete measures

µn =1

|En|X

z2En

�z

converge (weak-⇤) to the measure µ.

X = compact metric space (e.g. X = C)

|En| ! 1 as n ! 1

f : X ! X

Complex dynamics

X = P1(C) = C= Riemann sphere

f(z) =P (z)

Q(z)2 C(z)

Study orbits of pointsx, f(x), f2(x), f3(x), . . .

is a rational function<latexit sha1_base64="DfK6tbqfCq370RYEcrAfnVBNhbk=">AAAB/nicbVA9SwNBEJ2LXzF+RQUbm8UgWIW7NGoXtLGM4JlAcoS5zV6yZG/v2N0TwpnCv2JjoWLr77Dz37j5KDTxwcDb92bYmRemgmvjut9OYWV1bX2juFna2t7Z3SvvH9zrJFOU+TQRiWqFqJngkvmGG8FaqWIYh4I1w+H1xG8+MKV5Iu/MKGVBjH3JI07RWKlbPuKaIFHTFwoSZZLOjIpbdacgy8SbkwrM0eiWvzq9hGYxk4YK1LrtuakJclSGU8HGpU6mWYp0iH3WtlRizHSQT/cfk1Or9EiUKFvSkKn6eyLHWOtRHNrOGM1AL3oT8T+vnZnoIsi5TDPDJJ19FGWCmIRMwiA9rhg1YmQJUsXtroQOUCE1NrKSDcFbPHmZ+LXqZdW7rVXqV/M0inAMJ3AGHpxDHW6gAT5QeIRneIU358l5cd6dj1lrwZnPHMIfOJ8/GJKVtg==</latexit><latexit sha1_base64="DfK6tbqfCq370RYEcrAfnVBNhbk=">AAAB/nicbVA9SwNBEJ2LXzF+RQUbm8UgWIW7NGoXtLGM4JlAcoS5zV6yZG/v2N0TwpnCv2JjoWLr77Dz37j5KDTxwcDb92bYmRemgmvjut9OYWV1bX2juFna2t7Z3SvvH9zrJFOU+TQRiWqFqJngkvmGG8FaqWIYh4I1w+H1xG8+MKV5Iu/MKGVBjH3JI07RWKlbPuKaIFHTFwoSZZLOjIpbdacgy8SbkwrM0eiWvzq9hGYxk4YK1LrtuakJclSGU8HGpU6mWYp0iH3WtlRizHSQT/cfk1Or9EiUKFvSkKn6eyLHWOtRHNrOGM1AL3oT8T+vnZnoIsi5TDPDJJ19FGWCmIRMwiA9rhg1YmQJUsXtroQOUCE1NrKSDcFbPHmZ+LXqZdW7rVXqV/M0inAMJ3AGHpxDHW6gAT5QeIRneIU358l5cd6dj1lrwZnPHMIfOJ8/GJKVtg==</latexit><latexit sha1_base64="DfK6tbqfCq370RYEcrAfnVBNhbk=">AAAB/nicbVA9SwNBEJ2LXzF+RQUbm8UgWIW7NGoXtLGM4JlAcoS5zV6yZG/v2N0TwpnCv2JjoWLr77Dz37j5KDTxwcDb92bYmRemgmvjut9OYWV1bX2juFna2t7Z3SvvH9zrJFOU+TQRiWqFqJngkvmGG8FaqWIYh4I1w+H1xG8+MKV5Iu/MKGVBjH3JI07RWKlbPuKaIFHTFwoSZZLOjIpbdacgy8SbkwrM0eiWvzq9hGYxk4YK1LrtuakJclSGU8HGpU6mWYp0iH3WtlRizHSQT/cfk1Or9EiUKFvSkKn6eyLHWOtRHNrOGM1AL3oT8T+vnZnoIsi5TDPDJJ19FGWCmIRMwiA9rhg1YmQJUsXtroQOUCE1NrKSDcFbPHmZ+LXqZdW7rVXqV/M0inAMJ3AGHpxDHW6gAT5QeIRneIU358l5cd6dj1lrwZnPHMIfOJ8/GJKVtg==</latexit><latexit sha1_base64="DfK6tbqfCq370RYEcrAfnVBNhbk=">AAAB/nicbVA9SwNBEJ2LXzF+RQUbm8UgWIW7NGoXtLGM4JlAcoS5zV6yZG/v2N0TwpnCv2JjoWLr77Dz37j5KDTxwcDb92bYmRemgmvjut9OYWV1bX2juFna2t7Z3SvvH9zrJFOU+TQRiWqFqJngkvmGG8FaqWIYh4I1w+H1xG8+MKV5Iu/MKGVBjH3JI07RWKlbPuKaIFHTFwoSZZLOjIpbdacgy8SbkwrM0eiWvzq9hGYxk4YK1LrtuakJclSGU8HGpU6mWYp0iH3WtlRizHSQT/cfk1Or9EiUKFvSkKn6eyLHWOtRHNrOGM1AL3oT8T+vnZnoIsi5TDPDJJ19FGWCmIRMwiA9rhg1YmQJUsXtroQOUCE1NrKSDcFbPHmZ+LXqZdW7rVXqV/M0inAMJ3AGHpxDHW6gAT5QeIRneIU358l5cd6dj1lrwZnPHMIfOJ8/GJKVtg==</latexit>

Goal: applications of arithmetic equidistribution theorems to complex dynamics and connections to arithmetic geometry

Dynamical equidistribution

f(z) =P (z)

Q(z)2 C(z)

Theorem. [Brolin (1960s), Lyubich, Freire-Lopes-Mane (1983)]

f : C ! CAssume deg f � 2.

For all a 2 C (with at most 2 exceptions), the preimages<latexit sha1_base64="Bzyaza437YX1AizITFDmLTwlCE4=">AAACPHicbVA9bxNBEN0LAYL5cqCkGcVBChKy7twAXZRIKGWQYhLJZ1lz6zlulb3d0+5cgnXyH0uT/0BHR5OCRLTU7DkuyMdUT29m3sx7WaWV5zj+Ga08WH346PHak87TZ89fvOyuv/rqbe0kDaXV1h1l6EkrQ0NWrOmocoRlpukwO95t+4cn5Lyy5oBnFY1L/GZUriRyoCbdg9RYZaZkuPPZOkCtYRNTZSAtkJu0RC6yrNmdzz dh61RxAchQWs8wAPouqWpV/Lv3wAVBOKyCPPlJtxf340XBXZAsQU8sa3/S/ZFOrazL8IbU6P0oiSseN+hYSU3zTlp7qlAeB/FRgAZL8uNm4X4ObwMzhTx8n1vDsGD/32iw9H5WZmGyteNv91ryvt6o5vzjuFGmqpmMvD6U1xrYQhslTJUjyXoWAEqnwq8gC3QoOQTeCSEkty3fBcNB/1M/+TLobe8s01gTb8SG2BKJ+CC2xZ7YF0MhxZn4JX6Ly+g8uoiuoj/XoyvRcue1uFHR33+Qka3f</latexit><latexit sha1_base64="Bzyaza437YX1AizITFDmLTwlCE4=">AAACPHicbVA9bxNBEN0LAYL5cqCkGcVBChKy7twAXZRIKGWQYhLJZ1lz6zlulb3d0+5cgnXyH0uT/0BHR5OCRLTU7DkuyMdUT29m3sx7WaWV5zj+Ga08WH346PHak87TZ89fvOyuv/rqbe0kDaXV1h1l6EkrQ0NWrOmocoRlpukwO95t+4cn5Lyy5oBnFY1L/GZUriRyoCbdg9RYZaZkuPPZOkCtYRNTZSAtkJu0RC6yrNmdzz dh61RxAchQWs8wAPouqWpV/Lv3wAVBOKyCPPlJtxf340XBXZAsQU8sa3/S/ZFOrazL8IbU6P0oiSseN+hYSU3zTlp7qlAeB/FRgAZL8uNm4X4ObwMzhTx8n1vDsGD/32iw9H5WZmGyteNv91ryvt6o5vzjuFGmqpmMvD6U1xrYQhslTJUjyXoWAEqnwq8gC3QoOQTeCSEkty3fBcNB/1M/+TLobe8s01gTb8SG2BKJ+CC2xZ7YF0MhxZn4JX6Ly+g8uoiuoj/XoyvRcue1uFHR33+Qka3f</latexit><latexit sha1_base64="Bzyaza437YX1AizITFDmLTwlCE4=">AAACPHicbVA9bxNBEN0LAYL5cqCkGcVBChKy7twAXZRIKGWQYhLJZ1lz6zlulb3d0+5cgnXyH0uT/0BHR5OCRLTU7DkuyMdUT29m3sx7WaWV5zj+Ga08WH346PHak87TZ89fvOyuv/rqbe0kDaXV1h1l6EkrQ0NWrOmocoRlpukwO95t+4cn5Lyy5oBnFY1L/GZUriRyoCbdg9RYZaZkuPPZOkCtYRNTZSAtkJu0RC6yrNmdzz dh61RxAchQWs8wAPouqWpV/Lv3wAVBOKyCPPlJtxf340XBXZAsQU8sa3/S/ZFOrazL8IbU6P0oiSseN+hYSU3zTlp7qlAeB/FRgAZL8uNm4X4ObwMzhTx8n1vDsGD/32iw9H5WZmGyteNv91ryvt6o5vzjuFGmqpmMvD6U1xrYQhslTJUjyXoWAEqnwq8gC3QoOQTeCSEkty3fBcNB/1M/+TLobe8s01gTb8SG2BKJ+CC2xZ7YF0MhxZn4JX6Ly+g8uoiuoj/XoyvRcue1uFHR33+Qka3f</latexit><latexit sha1_base64="Bzyaza437YX1AizITFDmLTwlCE4=">AAACPHicbVA9bxNBEN0LAYL5cqCkGcVBChKy7twAXZRIKGWQYhLJZ1lz6zlulb3d0+5cgnXyH0uT/0BHR5OCRLTU7DkuyMdUT29m3sx7WaWV5zj+Ga08WH346PHak87TZ89fvOyuv/rqbe0kDaXV1h1l6EkrQ0NWrOmocoRlpukwO95t+4cn5Lyy5oBnFY1L/GZUriRyoCbdg9RYZaZkuPPZOkCtYRNTZSAtkJu0RC6yrNmdzz dh61RxAchQWs8wAPouqWpV/Lv3wAVBOKyCPPlJtxf340XBXZAsQU8sa3/S/ZFOrazL8IbU6P0oiSseN+hYSU3zTlp7qlAeB/FRgAZL8uNm4X4ObwMzhTx8n1vDsGD/32iw9H5WZmGyteNv91ryvt6o5vzjuFGmqpmMvD6U1xrYQhslTJUjyXoWAEqnwq8gC3QoOQTeCSEkty3fBcNB/1M/+TLobe8s01gTb8SG2BKJ+CC2xZ7YF0MhxZn4JX6Ly+g8uoiuoj/XoyvRcue1uFHR33+Qka3f</latexit>

En = f�n(a)<latexit sha1_base64="lXINbkPUWMjw+fydv6T7xCo4+TY=">AAAB93icbVBNS8NAEJ3Ur1o/GvXoZWkRKmJJvKgHoSiCxwrWFtoYNttNu3SzCbsboYb+Ei89qHj1r3jz37j9OGj1wcDjvRlm5gUJZ0o7zpeVW1peWV3Lrxc2Nre2i/bO7r2KU0log8Q8lq0AK8qZoA3NNKetRFIcBZw2g8HVxG8+UqlYLO70MKFehHuChYxgbSTfLl77Al2g8CE7FqMKPvTtslN1pkB/iTsn5VqpczQGgLpvf3a6MUkjKjThWKm26yTay7DUjHA6KnRSRRNMBrhH24YKHFHlZdPDR+jAKF0UxtKU0Giq/pzIcKTUMApMZ4R1Xy16E/E/r53q8MzLmEhSTQWZLQpTjnSMJimgLpOUaD40BBPJzK2I9LHERJusCiYEd/Hlv6RxUj2vurcmjEuYIQ/7UIIKuHAKNbiBOjSAQArP8AKv1pM1tt6s91lrzprP7MEvWB/fPfqTMg==</latexit><latexit sha1_base64="8R4WH6bT1/krMB4w9uQl3iboCjE=">AAAB93icbVBNS8NAEJ3Ur1o/WvXoZWkRKmJJvKgHoSiCxwrWFtoYNttNu3SzCbsboYb+C29ePKh49a94679x+3HQ1gcDj/dmmJnnx5wpbdsjK7O0vLK6ll3PbWxubecLO7v3KkokoXUS8Ug2fawoZ4LWNdOcNmNJcehz2vD7V2O/8UilYpG404OYuiHuChYwgrWRvEL+2hPoAgUP6bEYlvGhVyjZFXsCtEicGSlVi+2j51F1UPMK3+1ORJKQCk04Vqrl2LF2Uyw1I5wOc+1E0RiTPu7SlqECh1S56eTwITowSgcFkTQlNJqovydSHCo1CH3TGWLdU/PeWPzPayU6OHNTJuJEU0Gmi4KEIx2hcQqowyQlmg8MwUQycysiPSwx0SarnAnBmX95kdRPKucV59aEcQlTZGEfilAGB06hCjdQgzoQSOAF3uDderJerQ/rc9qasWYze/AH1tcPRd6UuA==</latexit><latexit sha1_base64="8R4WH6bT1/krMB4w9uQl3iboCjE=">AAAB93icbVBNS8NAEJ3Ur1o/WvXoZWkRKmJJvKgHoSiCxwrWFtoYNttNu3SzCbsboYb+C29ePKh49a94679x+3HQ1gcDj/dmmJnnx5wpbdsjK7O0vLK6ll3PbWxubecLO7v3KkokoXUS8Ug2fawoZ4LWNdOcNmNJcehz2vD7V2O/8UilYpG404OYuiHuChYwgrWRvEL+2hPoAgUP6bEYlvGhVyjZFXsCtEicGSlVi+2j51F1UPMK3+1ORJKQCk04Vqrl2LF2Uyw1I5wOc+1E0RiTPu7SlqECh1S56eTwITowSgcFkTQlNJqovydSHCo1CH3TGWLdU/PeWPzPayU6OHNTJuJEU0Gmi4KEIx2hcQqowyQlmg8MwUQycysiPSwx0SarnAnBmX95kdRPKucV59aEcQlTZGEfilAGB06hCjdQgzoQSOAF3uDderJerQ/rc9qasWYze/AH1tcPRd6UuA==</latexit><latexit sha1_base64="qF3i9P3qHxZZvnlnm0sfmDYduwM=">AAAB93icbVBNS8NAEJ3Ur1o/GvXoZbEI9WBJvKgHoSiCxwrGFtpYNttNu3SzCbsboYb+Ei8eVLz6V7z5b9y2OWjrg4HHezPMzAsSzpR2nG+rsLS8srpWXC9tbG5tl+2d3XsVp5JQj8Q8lq0AK8qZoJ5mmtNWIimOAk6bwfBq4jcfqVQsFnd6lFA/wn3BQkawNlLXLl93BbpA4UN2LMZVfNS1K07NmQItEjcnFcjR6NpfnV5M0ogKTThWqu06ifYzLDUjnI5LnVTRBJMh7tO2oQJHVPnZ9PAxOjRKD4WxNCU0mqq/JzIcKTWKAtMZYT1Q895E/M9rpzo88zMmklRTQWaLwpQjHaNJCqjHJCWajwzBRDJzKyIDLDHRJquSCcGdf3mReCe185p761Tql3kaRdiHA6iCC6dQhxtogAcEUniGV3iznqwX6936mLUWrHxmD/7A+vwBLQSRqQ==</latexit>

|En| = (deg f)n<latexit sha1_base64="b3UR0cxU5XZqbm5d4KwzMOUCATg=">AAAB+3icbVBNS8NAEJ3Ur1q/oj16WVqEilASL+pBKIrgsYK1haaWzXbTLt1swu5GCG39K3rwoOLVP+LNf+P246CtDwYe780wM8+POVPacb6tzNLyyupadj23sbm1vWPv7t2pKJGE1kjEI9nwsaKcCVrTTHPaiCXFoc9p3e9fjv36A5WKReJWpzFthbgrWMAI1kZq2/nhVVsM0TkqeR3aRcHhvUBtu+iUnQnQInFnpFgpeEfPAFBt219eJyJJSIUmHCvVdJ1YtwZYakY4HeW8RNEYkz7u0qahAodUtQaT40fowCgdFETSlNBoov6eGOBQqTT0TWeIdU/Ne2PxP6+Z6OC0NWAiTjQVZLooSDjSERongTpMUqJ5aggmkplbEelhiYk2eeVMCO78y4ukdlw+K7s3JowLmCIL+1CAErhwAhW4hirUgEAKT/AKb9aj9WK9Wx/T1ow1m8nDH1ifP8qXlJg=</latexit><latexit sha1_base64="jragmAmz4u+0QyIUlkK37F4E+9Y=">AAAB+3icbVBNS8NAEN3Ur1q/oj16WVqEilASL+pBKIrgsYKxhSaGzXbTLt1swu5GCGn9Fd69eFDx6h/x1n/jtvWgrQ8GHu/NMDMvSBiVyrLGRmFpeWV1rbhe2tjc2t4xd/fuZJwKTBwcs1i0AyQJo5w4iipG2okgKAoYaQWDy4nfeiBC0pjfqiwhXoR6nIYUI6Ul3ywPr3w+hOew5nZJD4aH9xz6ZtWqW1PARWL/kGqj4h49jRtZ0ze/3G6M04hwhRmSsmNbifJyJBTFjIxKbipJgvAA9UhHU44iIr18evwIHmilC8NY6OIKTtXfEzmKpMyiQHdGSPXlvDcR//M6qQpPvZzyJFWE49miMGVQxXCSBOxSQbBimSYIC6pvhbiPBMJK51XSIdjzLy8S57h+VrdvdBgXYIYi2AcVUAM2OAENcA2awAEYZOAZvII349F4Md6Nj1lrwfiZKYM/MD6/AdJ7lh4=</latexit><latexit sha1_base64="jragmAmz4u+0QyIUlkK37F4E+9Y=">AAAB+3icbVBNS8NAEN3Ur1q/oj16WVqEilASL+pBKIrgsYKxhSaGzXbTLt1swu5GCGn9Fd69eFDx6h/x1n/jtvWgrQ8GHu/NMDMvSBiVyrLGRmFpeWV1rbhe2tjc2t4xd/fuZJwKTBwcs1i0AyQJo5w4iipG2okgKAoYaQWDy4nfeiBC0pjfqiwhXoR6nIYUI6Ul3ywPr3w+hOew5nZJD4aH9xz6ZtWqW1PARWL/kGqj4h49jRtZ0ze/3G6M04hwhRmSsmNbifJyJBTFjIxKbipJgvAA9UhHU44iIr18evwIHmilC8NY6OIKTtXfEzmKpMyiQHdGSPXlvDcR//M6qQpPvZzyJFWE49miMGVQxXCSBOxSQbBimSYIC6pvhbiPBMJK51XSIdjzLy8S57h+VrdvdBgXYIYi2AcVUAM2OAENcA2awAEYZOAZvII349F4Md6Nj1lrwfiZKYM/MD6/AdJ7lh4=</latexit><latexit sha1_base64="SBdnYX/3J1DrofIq2R3sFP0t6Fk=">AAAB+3icbVBNS8NAEJ34WetXtEcvi0Wol5J4UQ9CUQSPFYwttLFstpt26WYTdjdCSetf8eJBxat/xJv/xm2bg7Y+GHi8N8PMvCDhTGnH+baWlldW19YLG8XNre2dXXtv/17FqSTUIzGPZTPAinImqKeZ5rSZSIqjgNNGMLia+I1HKhWLxZ0eJtSPcE+wkBGsjdSxS6PrjhihC1Rpd2kPhccPAnXsslN1pkCLxM1JGXLUO/ZXuxuTNKJCE46VarlOov0MS80Ip+NiO1U0wWSAe7RlqMARVX42PX6MjozSRWEsTQmNpurviQxHSg2jwHRGWPfVvDcR//NaqQ7P/IyJJNVUkNmiMOVIx2iSBOoySYnmQ0MwkczcikgfS0y0yatoQnDnX14k3kn1vOreOuXaZZ5GAQ7gECrgwinU4Abq4AGBITzDK7xZT9aL9W59zFqXrHymBH9gff4AuaGTDw==</latexit>

are equidistributed with respect to a measure µf . The measureµf is the unique measure of maximal entropy, and

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suppµf = Jf = {repelling periodic points}<latexit sha1_base64="CNWC8ODxvAu2Hw6yONsKucJbyLg=">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</latexit><latexit sha1_base64="IbgZ4fff9ziSMhKHCy9setscn4E=">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</latexit><latexit sha1_base64="IbgZ4fff9ziSMhKHCy9setscn4E=">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</latexit><latexit sha1_base64="KrmkuJndsrE0CT2vGWiFXvXFwnY=">AAACNnicbVBNSysxFM2oz4/6nlZdugkWwYWUmbdRF4LoRlyIglWhKSWT3qnBfJFkxDLMv3Lj33CnGxcqbv0JZmoXfl1IOJx7z03OSY3gzsfxfTQ2PvFncmp6pjb799/cfH1h8dTp3DJoMS20PU+pA8EVtDz3As6NBSpTAWfp5V7VP7sC67hWJ35goCNpX/GMM+oD1a0fEkn9hZWFy40pMVnHRObdDG/jg+FNdFBXywtSEJnq68KCARGYPjZgue5xho3myruSlGW33oib8bDwT5CMQAON6qhbvyM9zXIJyjNBnWsnsfGdglrPmYCyRnIHhrJL2od2gIpKcJ1i6LvEq4Hp4UzbcJTHQ/azoqDSuYFMw2Tl0n3vVeRvvXbus81OwZXJPSj28VCWC+w1rkLEPW6BeTEIgDLLw18xu6CWMh+iroUQku+Wf4LW/+ZWMzmOGzu7ozSm0TJaQWsoQRtoB+2jI9RCDN2gB/SEnqPb6DF6iV4/RseikWYJfano7R3KUK1g</latexit>

Dynamical equidistribution

M = {c � C : the orbit of z0 = 0 is bounded}

Theorem (Levin, 1989)The c for which fn

c (0) = 0are equidistributed withrespect to the harmonicmeasure µM on M.

Fact. (Douady-Hubbard, Sibony 1981; Mane-Sad-Sullivan 1983; D. 2002)

The measure µM is the bifurcation measure of the family {fc}.

fc(z) = z2 + c

Arithmetic equidistribution

h : Q⇤ ! R

Fact. h(↵) = 0 if and only if ↵ is a root of unity.

Theorem. (Bilu, 1997) Let ↵n 2 Q⇤be a sequence with

h(↵n) ! 0 and deg↵n ! 1. Then the probability measures

µn =1

deg↵n

X

�2Gal(Q/Q)

��↵n

converge to the uniform distribution on S1.

Weil height function

h(↵) =1

deg↵

0

@log |a0| +X

P (↵i)=0

log+ |↵i|

1

A

For a/b 2 Q, h(a/b) = logmax{|a|, |b|}

Dynamical-arithmetic equidistribution

“good”() associated to an adelic measure with continuouspotentials (Baker-Rumely)() associated to an adelically metrized line bundlewith continuous, semipositive metric (Zhang)

Theorem.(Baker–Rumely, Chambert-Loir, Favre–Rivera-Letelier, 2006)Let K be a number field.Suppose h is a “good” height on P1(K).Then for any sequence with h(↵n) ! 0 and deg↵n ! 1,the Gal(K/K) conjugates of ↵n are equidistributedwith respect to the measure µh.

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extended to other varieties(Thuillier, Yuan, 2008)

Dynamical-arithmetic equidistribution

Key examples

Bilu’s theoremDynamical canonical heights, when f defined over K (Call-Silverman, 1994)Néron-Tate height on an elliptic curve (Néron, Tate, 1960)Mandelbrot-set height

Theorem.(Baker–Rumely, Chambert-Loir, Favre–Rivera-Letelier, 2006)Let K be a number field.Suppose h is a “good” height on P1(K).Then for any sequence with h(↵n) ! 0 and deg↵n ! 1,the Gal(K/K) conjugates of ↵n are equidistributedwith respect to the measure µh.

<latexit sha1_base64="B9cKpNyMVei9OnauvFPIAAV8w40=">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</latexit><latexit sha1_base64="B9cKpNyMVei9OnauvFPIAAV8w40=">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</latexit><latexit sha1_base64="B9cKpNyMVei9OnauvFPIAAV8w40=">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</latexit><latexit sha1_base64="B9cKpNyMVei9OnauvFPIAAV8w40=">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</latexit>

extended to other varieties(Thuillier, Yuan, 2008)

Iterating complex polynomials

A B

Fix points A and B in the complex plane. For how many parameters c will both A and B be preperiodic? Are there finitely many?

fc(z) = z2 + c

Application to complex dynamics: warm-up problem(D.-Baker 2011)

A B

Fix points A and B in the complex plane. For how many parameters c will both A and B be preperiodic? Are there finitely many?

Iterating complex polynomials fc(z) = z2 + c

Application to complex dynamics: warm-up problem(D.-Baker 2011)

A B

Fix points A and B in the complex plane. For how many parameters c will both A and B be preperiodic? Are there finitely many?

Iterating complex polynomials fc(z) = z2 + c

Application to complex dynamics: warm-up problem(D.-Baker 2011)

For example, the point A=0 is preperiodic for a dense set of c in the boundary of the Mandelbrot set.

Theorem. (Baker-D.) The set of c such that both A and Bare preperiodic for z2 + c is infinite if and only if A2 = B2.

One implication is easy:

For every A 2 C, the point Ais preperiodic for infinitelymany parameters c.

Find solutions to equationsfnc (A) = fm

c (A) for any n,m.

Iterating complex polynomials fc(z) = z2 + c

Application to complex dynamics: warm-up problem(D.-Baker 2011)

If A is algebraic, then so is B.

For each place v of k, define generalized Mandelbrot set:Mv(A) =

⇢c 2 Cv : sup

n|fn

c (A)|v <1�

k = Q(A) ⇢ Q

We want to show: If A and B are both preperiodic forinfinitely many polynomials fc(z) = z2 + c, then A2 = B2.

this step uses complex

analysisB=0

A=-1

Infinitely many c whereboth are preperiodic =) M(A) = M(B) =) A2 = B2

A is preperiodic for fc () hM(A)(c) = 0.

this step uses arithmeticequidistribution

When A and B are complex, non-algebraic:

k = Q(A) is a function field

A preperiodic for fc =) hM(A)(c) = 0

=) Mv(A) = Mv(B) at all places v

=) hM(A) = hM(B)

We want to show: If A and B are both preperiodic forinfinitely many polynomials fc(z) = z2 + c, then A2 = B2.

Infinitely many c whereboth are preperiodic

Define good height hM(A) on P1(k) such that<latexit sha1_base64="7k+2TO671dRbGJXnqchKuG2Hpj0=">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</latexit><latexit sha1_base64="7k+2TO671dRbGJXnqchKuG2Hpj0=">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</latexit><latexit sha1_base64="7k+2TO671dRbGJXnqchKuG2Hpj0=">AAACMXicbVBNbxMxFPS2BdrwFdojl6cmSOkl2u2lcCsfBzggBYnQStkQ2c7brBWvvbLfIkWr/U1c+ksqcaAHWvXKn8BJc4CWkSyNZt7I740otfIUxz+jjc2te/cfbO+0Hj56/ORp+9nuF28rJ3EorbbuVHCPWhkckiKNp6VDXgiNJ2L+dumffEPnlTWfaVHiuOAzozIlOQVp0v7wDrMQhZm1U8hRzXKCbj6p61Rk8LHpvT5oumANdNOCUy5EPWi+Jr1UcFfPm4Mu+ErmQDknmLQ7cT9eAe6SZE06bI3BpH2eTq2sCjQkNfd+lMQljWvuSEmNTSutPJZczvkMR4EaXqAf16uTG3gRlClk1oVnCFbq34maF94vChEml4v7295S/J83qih7Oa6VKStCI28+yioNZGHZH0yVQ0l6EQiXToVdQebccUmh5VYoIbl98l0yPOy/6iefDjvHb9ZtbLPnbJ/1WMKO2DF7zwZsyCT7zn6wX+wyOosuoqvo+mZ0I1pn9tg/iH7/AeVzqFU=</latexit><latexit sha1_base64="7k+2TO671dRbGJXnqchKuG2Hpj0=">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</latexit>

Via arithmetic equidistribution (on Berkovich P1v, for each place v of k):

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for c 2 C, c is preperiodic for Ac is preperiodic for B

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=) M(A) = M(B) =) A2 = B2using complex

analysis, as before

Plot c such that fnc (0) = 0 for some n � 0.

Theorem. (Baker-D.) The set of c such that both A and Bare preperiodic for z2 + c is infinite if and only if A2 = B2.

B=0

Plot c such that fnc (�1) = �1 for some n � 0.

Theorem. (Baker-D.) The set of c such that both A and Bare preperiodic for z2 + c is infinite if and only if A2 = B2.

A=-1

Mv(A) =⇢

c 2 Cv : supn

|fnc (A)|v <1

�

B=0

Theorem. (Baker-D.) The set of c such that both A and Bare preperiodic for z2 + c is infinite if and only if A2 = B2.

By digging into the proof of the equidistribution theorems:

Conjecturally, there are only 3 values, namely c = 0, -1, -2.

A=-1

Theorem. (Fili, 2017) If A = �1 and B = 0 are bothpreperiodic for fc, then deg c 108. (And there areat most 2108 such c.)

My real interest is to understandthe geometry and arithmetic of the

moduli space of all rational maps of a given degree

and relations between this and the arithmetic

of elliptic curves

Conjecture. Let V be an N -dimensional complex algebraicvariety in the moduli space Md of rational maps of degree d.Let (a0, a1, . . . , aN ) be an (N + 1)-tuple of marked points.Then the points are simultaneously preperiodic on a Zariski-densesubset of V if and only if the points are dynamically related.

Compare: Lang, Manin-Mumford (Raynaud), André-Oort (Pila, Tsimerman), Bombieri-Masser-Zannier, Pink, Zilber, … conjectures/theorems

Special case

C

'

f'

P 2 E is torsion if n · P = 0 for some n.

P 2 E is torsion

() P is preperiodic for '

() ⇡(P ) is preperiodic for f'

Take, for example, '(P ) = P + P = 2P .

(Schroder, 1871; Lattes, 1918)

elliptic curve

⇡ P ⇠ �P

P is preperiodic for ' if its orbit is finite.

Complex dynamics and elliptic curves

Joint work with N. Myrto Mavraki: Bogomolov conjecture in families

{t 2 B(Q) : ht(Pt) < c}is finite.

If A is isotrivial, this is a special case of the Bogomolov Conjecture, proved by Ullmo and Zhang, building on equidistribution theorem of Szpiro-Ullmo-Zhang, 1997.

Height zero (torsion) case is solved by Masser-Zannier,extending Manin-Mumford (Raynaud, 1983) to families.

A

Bt

At P

Theorem (D.-Mavraki, 2017)True for A isogenous to a fibered productE1 ⇥ · · ·⇥ Em of elliptic surfaces.

Conjecture (S-W Zhang, 1998).B = smooth curve defined over Q.A ! B a family of abelian varieties of rel dimension > 1.ht = Neron-Tate canonical height on At, for t 2 B(Q).For each “non-special” section P : B ! A, there is a constantc = c(P ) > 0 so that

Joint work with N. Myrto Mavraki: Bogomolov conjecture in families

Other key ingredients in proof: • Silverman (local) variation of canonical height, 1992-1994• Zhang intersection theory of adelically metrized line bundles, 1995• Equidistribution theorems of Chambert-Loir, Thuillier, Yuan, 2008• Masser-Zannier results for height 0 (torsion) points, 2014

Theorem (D.-Mavraki, 2017)True for A isogenous to a fibered productE1 ⇥ · · ·⇥ Em of elliptic surfaces.

{t 2 B(Q) : ht(Pt) < c}is finite.

Conjecture (S-W Zhang, 1998).B = smooth curve defined over Q.A ! B a family of abelian varieties of rel dimension > 1.ht = Neron-Tate canonical height on At, for t 2 B(Q).For each “non-special” section P : B ! A, there is a constantc = c(P ) > 0 so that