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Backward dynamics
in
S-unimodal maps

Joint work with Jim Yorke

Howard University
Washington, DC (USA)

Motivations

Qualitative Behavior

A fundamental goal of Dynamical Systems theory:

Describe the qualitative behavior
of all possible forward/backward trajectories
under a given (discrete or continuous) flow.

In this talk I will illustrate the qualitative
forward and backward behavior of S-unimodal maps.

Since every S-unimodal map is top. conjugated to some logistic map,
I will use the logistic map family as my source of examples.

Finally, I will encode the information about the qualitative dynamics
in a graph, as Ana nicely illustrated in the previous talk.

A few Definitions

Bitrajectories

A discrete dynamical system on a topological space $M$

is a continuous map $f:M\to M$.

A bitrajectory based at $x\in M$ is a bi-infinite sequence

$\dots x_{-2},x_{-1},x_0,x_1,x_2\dots$

such that:

$x_0=x$;

$x_{i+1}=f(x_i)$ for all $i\in\Bbb Z$.

$s\alpha$ and $\omega$ limits

Given a bitrajectory $t=\{\dots,x_{-2},x_{-1},x_0,x_1,x_2,\dots\}$,

we denote by $\omega(t)$ the set of the limit points of $t$ for $i\to+\infty$

and by $\alpha(t)$ the set of the limit points of $t$ for $i\to-\infty$.

Given a point $x\in M$,
its $\omega$-limit is the set of the limit points of its forward trajectory.

Its $s\alpha$-limit (M. Hero, 1992), in turn, is the set

$s\alpha_f(x)=\cup_{t_x}\alpha(t_x)$,

where the union is over all bitrajectories $t_x$ based at $x$.

Chain-recurrent points

For each bitraj. $t$, all points of $\omega(t)$ and $\alpha(t)$ are chain-recurrent.

An $\varepsilon$-chain is a sequence $x_0,x_1,x_2,\dots$ such that

$d( f(x_i), x_{i+1} ) < \varepsilon$

for all $i=0,1,2,\dots$

A point $x$ is chain-recurrent if, for every $\varepsilon > 0$, there is
an $\varepsilon$-chain from $x$ to itself.

We denote by ${\cal R}_f\subset M$ the set of all chain-recurrent points.

Attracting and repelling Nodes

A main feature of chain-recurrence is the following:

within the set $\cal R_f$ of all chain-recurrent points, the relation

$x\sim y$ if for all $\varepsilon>0$ there are $\varepsilon$-chains from $x$ to $y$ and from $y$ to $x$

is an equivalence relation.

We call nodes the equivalence classes of $\cal R_f$.

From now on we assume that $M$ is also a measure space and we say that a node $N\subset\cal R_f$ is attracting if it contains an attractor,
namely a closed invariant set $A\subset M$ such that

$\omega_f(x)\subset A$ for a set of points $x\in M$ of positive measure.

If $N$ is not attracting, we say that $N$ is repelling.

Graph
of the
logistic map

The logistic map

$\ell_\mu(x)=\mu x(1-x)$,

$x\in[0,1]$, $\mu\in(1,4)$

$\ell_\mu([0,1])\subset [0,1]$;
$\ell_\mu$ has a unique attractor $A_\mu$ and almost all points asymptote to it;
the fixed point $x=0$ is repelling;
$\ell_\mu$ is not surjective.
Hence, $\ell_\mu$ has at least two nodes
(the fixed point 0 and the attractor).

Bifurcation diagram of the logistic map

Nodes coloring:
Black:	Attractor
White:	Fixed endpoint
Green:	Periodic orbits
Red:	Cantor sets

The period-3 window of the logistic map