Press ? for presentation controls.
Best viewed in Firefox, Chrome and Opera.

Backward asymptotics
in Logistic-like maps

Roberto De Leo

roberto.deleo@howard.edu

Howard University

Washington, DC (USA)

ICDEA 2021, Sarajevo, 26 July 2021

Motivation

Complex rational maps

While $\omega$-limits have been thoroughly studied in literature,

much less has been done until 2020s for $\alpha$-limits.

The following is one of the strongest results known:

THEOREM [Hawkins, Taylor (2003)].
Let $f:\Bbb C\to\Bbb C$ be a rational map of degree $\geq2$.
Recall that, in this case, $f$ has a unique repellor,
its "Julia set" $J_f$.
Then, for any $z_0\in\Bbb C$ (except at most two points) and for almost all bitrajectories $t$ based at $z_0$, $s\alpha(t)=J_f$.

Example: the Newton map of $z^3-1$


From R. De Leo, CONJECTURES ABOUT SIMPLE DYNAMICS FOR SOME REAL NEWTON MAPS ON $\Bbb R^2$, Fractals, 27:6 (2019)

A Natural Question

What can be said about

backward limits of bitrajectories

in 1-dimensional real dynamical systems?

While working at the paper above,

I posed this question to Jim Yorke.

This started our collaboration on this topic,

leading to the recent publication of the following two works:

R. De Leo, J.A. Yorke, The graph of the logistic map is a tower, Discr. & Cont. DS (2021)

R. De Leo, J.A. Yorke, Infinite towers in the graphs of many dynamical systems, Nonlin. Dyn. (2021)

A few Definitions

Def 1: Trajectories of a discrete DS

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 sequence

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

such that:

  1. $x_0=x$;
  2. $x_{i+1}=f(x_i)$  for all  $i\in\Bbb Z$.

Def 2: $s\alpha$ 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

$\omega_f(x)=\omega(t_x)$,

where $t_x$ is any bitrajectory based at $x$.

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 $x$.

Def 3: Chain-recurrent points

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

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

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

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

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

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

Def 4: 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 $\epsilon>0$ there are $\epsilon$-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,
that is 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.

The Logistic map

Fundamental properties

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

$x\in[0,1]$,  $\mu\in(1,4)$

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

Bifurcation diagram of the logistic map

Types of attracting nodes:
  1. Periodic orbit
  2. Feigenbaum attractor
  3. Cycle of intervals (chaotic)
  4. Beginning of a window
  5. End of a window
This picture shows the attracting nodes for each $\mu$.

Bifurcation diagram of the logistic map

Types of repelling nodes:
  1. Periodic orbit
  2. Cantor set
    (subshift of finite type)
This picture shows all (visible) nodes for each $\mu$.

Conley's thm and the graph of a DS

THEOREM [Conley (1972), Norton (1995)]
Let $f:M\to M$ be a continuous map.
Outside the chain-recurrent set $\cal R_f\subset M$,
the dynamics of $f$ is gradient-like.
In other words, for every bitrajectory $t_x$

based at a non-chain-recurrent point $x\in M$,

there are nodes $N\neq M$ of $\cal R_f$ such that

$\alpha(t_x)\subset N$ and $\omega(t_x)\subset M$.

Conley's thm and the graph of a DS

A corollary of Conley's theorem is that

it is possible to associate a graph $\Gamma_f$ to a dynamical system $f$

in the following way:

  1. The nodes of $\Gamma_f$ are the equivalence classes of $\cal R_f$.

  2. There is an edge from node $N$ to node $M$ of $\Gamma_f$

    if there is a bitrajectory $t$ of $f$ such that

    $\alpha(t)\subset N$ and $\omega(t)\subset M$.

 

Important Remark: the graph $\Gamma_f$ does not have loops!

The graph of the Logistic map is a tower

In R. De Leo, J.A. Yorke, The graph of the logistic map is a tower, Discr. & Cont. DS (2021),
we proved that all nodes of the Logistic map
can be sorted in a linear order $N_0,\dots,N_p$ so that:

  1. $N_0$ is the repelling fixed point $x=0$;

  2. $N_p$ is the unique attracting node;

    Remark: $p$ can be $\infty$, which is the case when $N_p$ is a Cantor set.

  3. the distance from node $N_i$ to the critical point $c=0.5$

    decreases monotonically as $i$ increases;

  4. there is an edge from $N_i$ to $N_j$ if and only if  $i < j$.

We call such graph a tower.

The graph of the Logistic map is a tower

In particular this means that, arbitrarily close to each node $N_i$,

there are points whose trajectory asymptote

to each $N_j$ "below it", namely with $j > i$.

 

Similarly, arbitrarily close to each node $N_i$,

there are points having bitrajectories that asymptote backward

to each $N_j$ "above it", namely such that $j < i$.

A few examples of towers

A few examples of towers

$s\alpha$-limits
under the Logistic map

Trapping regions

THEOREM [RdL, Yorke (2021)]
Each repelling node $N$ has a periodic point $p_1(N)$ s.t.

$d(N,c)=d(p_1(N),c)$

and comes with a cyclic trapping region ${\cal T}(N)$,

that is a forward-invariant cycle of closed intervals

$J_1=J_1(N),\ell_\mu(J_1),\dots,\ell_\mu^{\ k-1}(J_1)$

such that:

  1. $J_1=[p_1(N),1-p_1(N)]$

     
  2. $\ell_\mu^{\ k}(J_1)\subset J_1$.

Trapping regions

Trapping regions

Trapping regions

Trapping regions

Trapping regions

Trapping regions

Remark 1
Trapping regions are nested one into the other.

Remark 2
Node $N_{i+1}$ is the set of all chain-recurrent points of ${\cal T}(N_i)$ that never fall in the interior of ${\cal T}(N_{i+1})$ under $\ell_\mu$.

Main result

We say that $x\in[0,1]$ is a level-$k$ point for $\ell_\mu$

if $x$ is inside ${\cal T}(N_{k-1})$ but is not inside ${\cal T}(N_{k})$

Notice that all points of $N_k$ are level-$k$ points.
THEOREM [RdL (2021)].
Let $x$ be a level-$k$ point.  If $x\in N_k$, then

$s\alpha(x)=N_0\cup N_1\cup\dots\cup N_k\;$

Otherwise, for $k < p$,

$s\alpha(x)=N_0\cup N_1\cup\dots\cup N_{r}$ for $r=k$ or $r=k-1$ or $r=k-2\;$;

for $k = p$,

$s\alpha(x)=N_0\cup N_1\cup\dots\cup N_{r}$ for $r=p-1$ or $r=p-2\;$.

Example 1

Example 2

Example 3

Thank you!

Then:

If $k=p$ If $k < p$
  1. if $x\in N_p$, then

    $s\alpha(x)=N_0\cup N_1\cup\dots\cup N_p\;$;

  2. otherwise,

    $s\alpha(x)=N_0\cup N_1\cup\dots\cup N_{r}$
    where $r=p-1$ or $r=p-2$.
  1. if $x\in N_k$, then

    $s\alpha(x)=N_0\cup N_1\cup\dots\cup N_k\;$;

  2. otherwise,


    where $r$ can be either $k$ or $k-1$ or $k-2$.