Timeline for Continuous-time extension of a discrete dynamical system

8 events

when toggle format	what		by	license	comment
Jan 14, 2020 at 9:08	vote	accept	ABIM
Jan 9, 2020 at 14:37	history	edited	YCor	CC BY-SA 4.0	removed capitals from title
Jan 9, 2020 at 10:57	answer	added	user37691		timeline score: 3
Jan 9, 2020 at 10:40	comment	added	Mateusz Kwaśnicki		Then this is essentially a question whether there is a smooth simple curve passing through given $N$ points $x_1, \ldots, x_N$ in a given order. Or, more precisely, we require this curve to pass through $M$ points $x_1, \ldots, x_M$ in a given order and have no self-intersections except possibly at $x_M$, in which case it must be tangent to itself; here $M$ is the least number such that $x_M \in \{x_1, \ldots, x_{M-1}\}$, or $M = N$ if no such number exists. This seems to be fairly straightforward, no?
Jan 9, 2020 at 10:22	comment	added	ABIM		@MateuszKwaśnicki No, $x$ should be specified (so the "anti-discritization" is dependent on the choice of x); It would only need to hold for an (arbitrarily large but fixed) number of iterations.
Jan 9, 2020 at 10:20	history	edited	ABIM	CC BY-SA 4.0	added 50 characters in body
Jan 9, 2020 at 10:10	comment	added	Mateusz Kwaśnicki		If you mean "for all $x$", then no, it is not possible. For every $F$, the Jacobian determinant of the corresponding flow is non-zero, and hence it remains positive. Thus, $f(x) = (-x_1,x_2,x_3,\ldots,x_d)$ provides a simple counter-example. On the other hand, if $x$ is fixed, then I do not see the answer right away.
Jan 9, 2020 at 9:28	history	asked	ABIM	CC BY-SA 4.0