Round-off errors and p-adic numbers

We explore some connections between round-off errors in linear planar rotations and algebraic number theory. We discretize a map on a lattice in such a way as to retain invertibility, restricting the system parameter (the trace) to rational values with power-prime denominator pn . We show that this system can be embedded into a smooth expansive dynamical system over the p -adic integers, consisting of multiplication by a unit composed with a Bernoulli shift. In this representation, the original round-off system corresponds to restriction to a dense subset of the p -adic integers. These constructs are based on symbolic dynamics and on the representation of the discrete phase space as a ring of integers in a quadratic number field.

[1]  Harvey Cohn,et al.  A second course in number theory , 1962 .

[2]  Jean-Pierre Serre.  A Course in Arithmetic , 1973 .

[3]  Harvey Cohn,et al.  Introduction to the construction of class fields , 1985 .

[4]  E. Thiran,et al.  p-adic dynamics , 1989 .

[5]  F. Vivaldi.  Dynamics over irreducible polynomials , 1992 .

[6]  David K. Arrowsmith,et al.  Some p-adic representations of the Smale horseshoe , 1993 .

[7]  M. Blank.  Pathologies generated by round-off in dynamical systems , 1994 .

[8]  D. Arrowsmith,et al.  Geometry of p -adic Siegel discs , 1994 .

[9]  Franco Vivaldi,et al.  Periodicity and Transport from Round-Off Errors , 1994, Exp. Math..

[10]  T. Pezda.  Polynomial cycles in certain local domains , 1994 .

[11]  Hua-Chieh Li.  $p$-adic dynamical systems and formal groups , 1996 .

[12]  A weak Néron model with applications to $p$-adic dynamical systems , 1996 .

[13]  p-adic periodic points and Sen's theorem , 1996 .

[14]  Franco Vivaldi,et al.  Quasi-periodicity, global stability and scaling in a model of Hamiltonian round-off. , 1997, Chaos.

[15]  A KAM type theorem for systems with round-off errors , 1997, chao-dyn/9706005.

[16]  Fatou Components in p-adic Dynamics , 1998 .

[17]  F. Vivaldi,et al.  ANOMALOUS TRANSPORT IN A MODEL OF HAMILTONIAN ROUND-OFF , 1998 .

[18]  Phil Diamond,et al.  Boundedness and Dissipativity of Truncated Rotations on Uniform Planar Lattices , 1998 .

[19]  Nigel P. Smart,et al.  p-adic Chaos and Random Number Generation , 1998, Exp. Math..

[20]  F. Vivaldi,et al.  An algorithmic view of pseudochaos , 1999 .

[21]  Closure of Periodic Points Over a Non‐Archimedean Field , 2000 .

[22]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.