Programs, logs and test functions for the computer-assisted bounds on the Levitt-Yoccoz gasket.

Programs, logs and test functions for the computer-assisted bounds on the Levitt-Yoccoz gasket Roberto De Leo, Department of Mathematics, Howard University, Washington, DC, USA This record contains the programs and the data of the computer-assisted proofs in R. De Leo, "Dynamics and Hausdorff dimension of the Levitt-Yoccoz gasket" R. De Leo, "The norm exponent of the Levitt-Yoccoz gasket: souls, entropy and diffusion" (also posted together as a single arXiv preprint), in particular of the bounds 1.6853 < s_A(Gamma_N) = dim_H N < 1.6854 (N = the Levitt-Yoccoz, or Rauzy, gasket) 2.4411 <= s_N(Gamma_N) <= 2.4465, 2.74 <= s_N(Gamma_G) <= 2.777. FILES README.txt this file levitt_yoccoz_programs.zip all the programs, scripts and logs, and the smaller test functions; see the README.txt files inside hF0.5_d4_L.bin the test function h- of the certificate of 1.6853 < dim_H N (241 089 280 bytes) hF0.5_d4_U.bin the test function h+ of the certificate of dim_H N < 1.6854 (241 089 280 bytes) SHA-256 checksums of the two test functions (also printed in the s_A paper): 8c587e7cd1461153b6f39916d0caa2b5913664966945a3fc8f93cff38721928b hF0.5_d4_L.bin da0e2e05008f9dbeaebdcae48a930ba48948c22913b2eb02b1ad40c0c284afcc hF0.5_d4_U.bin THE CERTIFICATE OF 1.6853 < dim_H N < 1.6854 The verifier p1verify.c (in levitt_yoccoz_programs/sA_certificate, SHA-256 87a7933882a409a5816e11322954cc9eb2543396a7dc76ad64eb9d4d4609f529) checks, in interval arithmetic, that K_kappa h- >= h- at kappa = kappa_- and K_kappa h+ <= 0.99999999 h+ at kappa = kappa_+ on the whole state space, where K_kappa is the transfer operator of the derivative cocycle defined in the s_A paper and kappa_- = the binary64 number nearest to 0.3147 (2 - kappa_- = 1.68530000000000002...) kappa_+ = the binary64 number nearest to 0.31460000000000005, (2 - kappa_+ = 1.68539999999999995...) the successor of the binary64 number nearest to 0.3146. The certificate consists of the runs CELLRANGE=lo,hi p1verify hF0.5_d4_L.bin 0.3147 64 1024 1024 1.0 1e300 18 CELLRANGE=lo,hi p1verify hF0.5_d4_U.bin 0.31460000000000005 64 1024 1024 0 0.99999999 18 over the 1499 ranges [lo,hi) = [10000 i, min(10000 (i+1), 14981400)), i = 0, ..., 1498, of the 14 981 400 cells of the tau_23-half of the mesh. Their logs are in levitt_yoccoz_programs/sA_certificate/logs/F0.5_d4; each begins with the SHA-256 checksum of the test function it checked. combine.py checks that the logs cover all the cells, come from these two files and all end with CERTIFIED. The full computation took 8 125 (h-) and 3 936 (h+) thread-hours, that is, 5.4 and 2.6 thread-hours per range on average; any single range can be re-run and compared with its log. FILE FORMAT OF THE TEST FUNCTIONS (little-endian) int32 M (= 174) int32 N (= 984) float64 ell[0..M] the nodes ell_m of the mesh, in [0,1] float64 theta[0..N-1] the nodes vartheta_t of the direction coordinate, in [1,2] u [3,4) float64 H[N][M+1][M+1] nodal values: H[t][a][b] is the value at the node with coordinates (ell_a, ell_b) and direction vartheta_t; only the entries with a + b <= M are used The mesh is the mesh F(1/2) of Section 6 of the s_A paper (notation as there). 15 153 600 nodal values are used. HOW THE TEST FUNCTIONS WERE COMPUTED (floating point; not part of the proof) p1solve2.c, starting from the test functions hL2.bin, hU2.bin of the coarser certificate (in the zip file), interpolated to the mesh F(1/2): twenty power iterations at the new kappa, followed by twelve max-min (h-) / min-max (h+) corrections of the nodal values. The verifier certifies whatever file it is given, so how the files were obtained plays no role in the proof. LICENSE Creative Commons Attribution 4.0 International (CC BY 4.0).