Derivation of invariant varieties of periodic points from singularity confinement in the case of Toda map

Abstract

We have shown in [1] that the invariant varieties of periodic points (IVPP) of all periods of some higher dimensional rational maps can be derived, iteratively, from the singularity confinement (SC). We generalize this algorithm, in this paper, to apply to any birational map, which has more invariants than the half of the dimension.

An analysis of indeterminate points in discrete integrable system

“…the third IVPP. This had been found in the previous work [7], however, not completely because it contains the extra factor which corresponds to Λ.…”

“…The 3-point Toda map has similar situation to the 3d Lotka-Volterra map, but more complicate and higher dimensional. This map has singularity confinement property also, but this sequence includes indeterminate points [7];…”

“…Birational maps have indeterminate points which are mapped to 0/0 symbolically. Moreover, they appear naturally in an extended singularity confinement sequence of integrable maps [7]. Singularity confinement is proposed as a test of integrability [1].…”

Degeneration of the Julia set to singular loci of algebraic curves

“…This is the way we studied the behavior of periodic points in our previous papers. 4,6,7 Now we want to know the paths of the periodic points along which they move as a changes and approaches in the limit a = 0. We can do it if we eliminate the parameter a from K (2) a (x) = 0 and y − L (2) a (x) = 0 of (5).…”

“…− (−486 + 713x 3 − 13x 10 + 2394x 5 + 68x 9 − 1568x 6 + 1044x − 220x 8 + 717x 2 − 1864x 4 + x 11 + 640x 7 )y 6 + (−2059x 4 + 344x 9 + 5x 11 + 1278x + 2493x 7 + 374x 7 + 9x)y 2 + 4x 3 (47x 4 − 9 + 5x 6 + 15x 2 − 26x 5 − 38x 3 − 6x)y − 8x 6 (−3x + x 2 + 3).…”