Tent map

The full tent map on the unit interval isIts two affine branches expand lengths by two, and it preserves Lebesgue measure.

For $T_s(x)=1-s|x|$ with $1<s\le 2$, the largest closed interval mapped onto itself is $A=[1-s,1]=[T_s^2(0),T_s(0)]$. It equals the image of $[-1,1]$. Every point outside $A$ enters $A$ after one iterate, because $T_s([-1,1])=A$. It subsequently remains in $A$.

// Target: dynamical-systems.bigb

If $\sqrt{2}<s\le 2$, every nondegenerate subinterval of $A=[1-s,1]$ eventually maps onto $A$. Away from the critical point, lengths multiply by $s$; one fold loses at most a factor of two. If an interval and its image both contain zero, its second image is $A$. Otherwise every two steps increase length by at least $s^2/2>1$, which cannot continue inside bounded $A$. Applying the intermediate value theorem to an interval that covers itself then gives a periodic point in every subinterval.

// Target: dynamical-systems.bigb

For $1<\mu \le \sqrt{2}$, the interval $J=[1/(\mu +1),\mu /(\mu +1)]$ is invariant under the second iterate of the tent map. The affine coordinate $h(x)=(\mu /(\mu +1)-x)/((\mu -1)/(\mu +1))$ conjugates that restriction to the tent map with parameter ${\mu }^2$. Its two linear pieces give the identity directly. This turns the two-iterate horseshoe threshold into a four-iterate threshold and explains why chaotic invariant sets need not have periodic points near the original nonzero fixed point.

For $1<s\le \sqrt{2}$, set $h_s(x)=-(s-1)x$ and $J_s=[-(s-1),s-1]$. The tent map sends $J_s$ to $[1+s-s^2,1]$, with disjoint interiors, and $T_s^2(J_s)\subseteq J_s$. Directly,Successive squaring eventually brings every $s>1$ into $(\sqrt{2},2]$. A horseshoe for an interval map of an iterate in the renormalized coordinate lifts to a horseshoe of an iterate of the original map.

// Target: dynamical-systems.bigb

Given a finite measurable partition of an interval, the itinerary of $x$ records which partition element contains each iterate $T^n(x)$. Prescribing a finite initial word defines an itinerary cylinder. For the full tent map, every length-$N$ binary cylinder has Lebesgue measure $2^{-N}$ up to endpoint conventions, so its itinerary process is a fair i.i.d. Bernoulli process.

An itinerary cylinder is the set of points whose first finitely many itinerary symbols equal a prescribed finite word. Such cylinders generate the symbolic σ-algebra and pull back to finite intersections of inverse images of partition elements.