Hubbard and Papadantonakis. The first loop whose associated monodromy automorphism is convincing. The following tables give the permutation on the critical points down to depth 10; the labelling is by itinerary. Generated using SaddleDrop 0.5.3 Level 9 A 0.111001000 0.101001000 * 0.111001001 0.101001001 * 0.111001011 0.101001011 * 0.111001010 0.101001010 * 0.111001110 0.101001110 * 0.111001111 0.101001111 * 0.111001101 0.101001101 * 0.111001100 0.101001100 * 0.111000100 0.101000100 * 0.111000101 0.101000101 * 0.111000111 0.101000111 * 0.111000110 0.101000110 * 0.111000010 0.101000010 * 0.111000011 0.101000011 * 0.111000001 0.101000001 * 0.111000000 0.101000000 * 0.101000000 0.111000000 * 0.101000001 0.111000001 * 0.101000011 0.111000011 * 0.101000010 0.111000010 * 0.101000110 0.111000110 * 0.101000111 0.111000111 * 0.101000101 0.111000101 * 0.101000100 0.111000100 * 0.101001100 0.111001100 * 0.101001101 0.111001101 * 0.101001111 0.111001111 * 0.101001110 0.111001110 * 0.101001010 0.111001010 * 0.101001011 0.111001011 * 0.101001001 0.111001001 * 0.101001000 0.111001000 * B 0.011100100 0.010100100 * 0.011100101 0.010100101 * 0.011100111 0.010100111 * 0.011100110 0.010100110 * 0.011100010 0.010100010 * 0.011100011 0.010100011 * 0.011100001 0.010100001 * 0.011100000 0.010100000 * 0.010100000 0.011100000 * 0.010100001 0.011100001 * 0.010100011 0.011100011 * 0.010100010 0.011100010 * 0.010100110 0.011100110 * 0.010100111 0.011100111 * 0.010100101 0.011100101 * 0.010100100 0.011100100 * C 0.001110010 0.001010010 * 0.001110011 0.001010011 * 0.001110001 0.001010001 * 0.001110000 0.001010000 * 0.001010000 0.001110000 * 0.001010001 0.001110001 * 0.001010011 0.001110011 * 0.001010010 0.001110010 * D 0.000111001 0.000101001 * 0.000111000 0.000101000 * 0.000101000 0.000111000 * 0.000101001 0.000111001 * 0.100101001 0.100111001 * 0.100101000 0.100111000 * 0.100111000 0.100101000 * 0.100111001 0.100101001 * E 0.000011100 0.000010100 * 0.000010100 0.000011100 * 0.010010100 0.010011100 * 0.010011100 0.010010100 * 0.110011100 0.110010100 * 0.110010100 0.110011100 * 0.100010100 0.100011100 * 0.100011100 0.100010100 * A 00.1x100hhhh 16 x2 B 0.01x100hhh 8 x2 C 0.001x100hh 4 x2 D 0.h001x100h 4 x2 E 0.hh001x100 4 x2 Level 10: A 0.1110010000 0.1010010000 * 0.1110010001 0.1010010001 * 0.1110010011 0.1010010011 * 0.1110010010 0.1010010010 * 0.1110010110 0.1010010110 * 0.1110010111 0.1010010111 * 0.1110010101 0.1010010101 * 0.1110010100 0.1010011100 * A' 0.1110011100 0.1010010100 * A' 0.1110011101 0.1010011101 * 0.1110011111 0.1010011111 * 0.1110011110 0.1010011110 * 0.1110011010 0.1010011010 * 0.1110011011 0.1010011011 * 0.1110011001 0.1010011001 * 0.1110011000 0.1010011000 * 0.1110001000 0.1010001000 * 0.1110001001 0.1010001001 * 0.1110001011 0.1010001011 * 0.1110001010 0.1010001010 * 0.1110001110 0.1010001110 * 0.1110001111 0.1010001111 * 0.1110001101 0.1010001101 * 0.1110001100 0.1010001100 * 0.1110000100 0.1010000100 * 0.1110000101 0.1010000101 * 0.1110000111 0.1010000111 * 0.1110000110 0.1010000110 * 0.1110000010 0.1010000010 * 0.1110000011 0.1010000011 * 0.1110000001 0.1010000001 * 0.1110000000 0.1010000000 * 0.1010000000 0.1110000000 * 0.1010000001 0.1110000001 * 0.1010000011 0.1110000011 * 0.1010000010 0.1110000010 * 0.1010000110 0.1110000110 * 0.1010000111 0.1110000111 * 0.1010000101 0.1110000101 * 0.1010000100 0.1110000100 * 0.1010001100 0.1110001100 * 0.1010001101 0.1110001101 * 0.1010001111 0.1110001111 * 0.1010001110 0.1110001110 * 0.1010001010 0.1110001010 * 0.1010001011 0.1110001011 * 0.1010001001 0.1110001001 * 0.1010001000 0.1110001000 * 0.1010011000 0.1110011000 * 0.1010011001 0.1110011001 * 0.1010011011 0.1110011011 * 0.1010011010 0.1110011010 * 0.1010011110 0.1110011110 * 0.1010011111 0.1110011111 * 0.1010011101 0.1110011101 * 0.1010011100 0.1110010100 * A' 0.1010010100 0.1110011100 * A' 0.1010010101 0.1110010101 * 0.1010010111 0.1110010111 * 0.1010010110 0.1110010110 * 0.1010010010 0.1110010010 * 0.1010010011 0.1110010011 * 0.1010010001 0.1110010001 * 0.1010010000 0.1110010000 * B 0.0111001000 0.0101001000 * 0.0111001001 0.0101001001 * 0.0111001011 0.0101001011 * 0.0111001010 0.0101001010 * 0.0111001110 0.0101001110 * 0.0111001111 0.0101001111 * 0.0111001101 0.0101001101 * 0.0111001100 0.0101001100 * 0.0111000100 0.0101000100 * 0.0111000101 0.0101000101 * 0.0111000111 0.0101000111 * 0.0111000110 0.0101000110 * 0.0111000010 0.0101000010 * 0.0111000011 0.0101000011 * 0.0111000001 0.0101000001 * 0.0111000000 0.0101000000 * 0.0101000000 0.0111000000 * 0.0101000001 0.0111000001 * 0.0101000011 0.0111000011 * 0.0101000010 0.0111000010 * 0.0101000110 0.0111000110 * 0.0101000111 0.0111000111 * 0.0101000101 0.0111000101 * 0.0101000100 0.0111000100 * 0.0101001100 0.0111001100 * 0.0101001101 0.0111001101 * 0.0101001111 0.0111001111 * 0.0101001110 0.0111001110 * 0.0101001010 0.0111001010 * 0.0101001011 0.0111001011 * 0.0101001001 0.0111001001 * 0.0101001000 0.0111001000 * C 0.0011100100 0.0010100100 * 0.0011100101 0.0010100101 * 0.0011100111 0.0010100111 * 0.0011100110 0.0010100110 * 0.0011100010 0.0010100010 * 0.0011100011 0.0010100011 * 0.0011100001 0.0010100001 * 0.0011100000 0.0010100000 * 0.0010100000 0.0011100000 * 0.0010100001 0.0011100001 * 0.0010100011 0.0011100011 * 0.0010100010 0.0011100010 * 0.0010100110 0.0011100110 * 0.0010100111 0.0011100111 * 0.0010100101 0.0011100101 * 0.0010100100 0.0011100100 * D 0.0001110010 0.0001010010 * 0.0001110011 0.0001010011 * 0.0001110001 0.0001010001 * 0.0001110000 0.0001010000 * 0.0001010000 0.0001110000 * 0.0001010001 0.0001110001 * 0.0001010011 0.0001110011 * 0.0001010010 0.0001110010 * 0.1001010010 0.1001110010 * 0.1001010011 0.1001110011 * 0.1001010001 0.1001110001 * 0.1001010000 0.1001110000 * 0.1001110000 0.1001010000 * 0.1001110001 0.1001010001 * 0.1001110011 0.1001010011 * 0.1001110010 0.1001010010 * E 0.0000111001 0.0000101001 * 0.0000111000 0.0000101000 * 0.0000101000 0.0000111000 * 0.0000101001 0.0000111001 * 0.0100101001 0.0100111001 * 0.0100101000 0.0100111000 * 0.0100111000 0.0100101000 * 0.0100111001 0.0100101001 * 0.1100111001 0.1100101001 * 0.1100111000 0.1100101000 * 0.1100101000 0.1100111000 * 0.1100101001 0.1100111001 * 0.1000101001 0.1000111001 * 0.1000101000 0.1000111000 * 0.1000111000 0.1000101000 * 0.1000111001 0.1000101001 * F 0.0000011100 0.0000010100 * 0.0000010100 0.0000011100 * 0.0010010100 0.0010011100 * 0.0010011100 0.0010010100 * 0.0110011100 0.0110010100 * 0.0110010100 0.0110011100 * 0.0100010100 0.0100011100 * 0.0100011100 0.0100010100 * 0.1100011100 0.1100010100 * 0.1100010100 0.1100011100 * 0.1000010100 0.1000011100 * 0.1000011100 0.1000010100 * A 00.1x100hhhhh 30 x2 (exc A') A' 00.1x1001y100 2 x2 B 0.01x100hhhh 16 x2 C 0.001x100hhh 8 x2 D 0.h001x100hh 8 x2 E 0.hh001x100h 8 x2 F 0.hhh001x100 6 x2 (exc A')