Fixed and Periodic Points | Nathan Dalaklis

1. Audio
   Observation

   The speaker says “a function f” while introducing fixed points and periodic points.

2. Formula
   Observation

   Chalkboard text uses $f$ in $f(x)=x$, $f^m(p)=p$, and the least-period condition.

f

A function whose iterates are being considered.

The video does not state a specific domain or codomain for $f$.

1. Audio
   Observation

   The speaker says “a fixed point x of a function f satisfies the property that f of x is equal to x.”

2. Formula
   Observation

   Chalkboard writes “A FIXED POINT x OF A FUNCTION f SATISFIES THE PROPERTY” and below it $f(x)=x$.

x

A point that is unchanged by one application of $f$.

The video does not specify the ambient set containing $x$.

1. Audio
   Observation

   The speaker says “a periodic point p of period m of a function f has the property that f to the m of p is equal to p.”

2. Formula
   Observation

   Chalkboard writes “A PERIODIC POINT p OF PERIOD m OF A FUNCTION f SATISFIES THE PROPERTY” and below it $f^m(p)=p$; later the least-period definition also uses $p$.

p

A point that returns to itself after iterating $f$ some number of times.

The video does not specify the ambient set containing $p$.

1. Audio
   Observation

   The speaker says “period m” and later “the first k greater than or equal to 1 for which f to the k of p is p is m.”

2. Formula
   Observation

   Chalkboard uses $m$ in $f^m(p)=p$ and in the least-period statement $f^m(p)=p$.

m

A positive integer period attached to a periodic point.

The video treats $m$ as a period index; it does not explicitly write $m\in\mathbb{Z}_{>0}$, though the spoken definition implies a counting number.

1. Audio
   Observation

   The speaker says “the first k greater than or equal to 1 for which f to the k of p is p is m.”

2. Formula
   Observation

   Chalkboard writes $\forall k, 1 \le k < m$ in the least-period definition.

k

An iterate count used to test whether a smaller positive return time exists.

In the displayed condition, $k$ ranges over integers with $1\le k<m$.

1. Audio
   Observation

   The speaker says “f to the m of p” and explains this as “f applied m times.”

2. Formula
   Observation

   Chalkboard writes $f^m(p)=p$ and later $f^k(p)\neq p$ and $f^m(p)=p$.

$f^m$

The $m$-fold composition (iterate) of $f$ with itself.

Defined implicitly for positive integers $m$ by repeated application of $f$.

1. Audio
   Observation

   The speaker says “f to the k of p is p” while defining least period.

2. Formula
   Observation

   Chalkboard writes $f^k(p)\neq p$ in the universal condition for $1\le k<m$.

$f^k$

The $k$-fold iterate of $f$.

Used here for integers $k$ with $1\le k<m$.

1. Formula
   Observation

   Board text: "Def^n: A FIXED POINT x OF A FUNCTION f SATISFIES THE PROPERTY" followed by the displayed equation $f(x)=x$.

$f(x)=x$

Defining property of a fixed point: applying the function $f$ to the point $x$ returns $x$ itself.

$x$ is a point in the domain of $f$; $f$ is a function.

1. Formula
   Observation

   Board text: "Def^m: A PERIODIC POINT p OF PERIOD m OF A FUNCTION f SATISFIES THE PROPERTY" followed by the displayed equation $f^{m}(p)=p$.

$f^{m}(p)=p$

Defining property of a periodic point of period $m$: the $m$-fold iterate of $f$ at $p$ equals $p$.

$p$ is a point in the domain of $f$; $m$ is a positive integer period.

1. Formula
   Observation

   Board text: "Def^k: A POINT p IS OF PRIME (OR LEAST) PERIOD m IF" followed by $\forall k, 1 \le k < m$ and the condition $f^{k}(p) \ne p$ but $f^{m}(p)=p$. The quantifier line is visible as $\forall k, 1 \le k < m$.

$\forall k$, $1 \le k < m$

Universal quantification over integers $k$ strictly between $1$ and $m$, used to require that no earlier iterate returns to $p$.

$k$ ranges over integers with $1 \le k < m$.

1. Audio
   Observation

   Speaker says this is "Sharkovskii's theorem when n is equal to three."

$n=3$

The specific case of Sharkovskii's theorem being referenced in the spoken introduction.

Integer parameter naming the theorem case.

1. Audio
   Observation

   The speaker gives the formal definition of a fixed point.

2. Formula
   Observation

   Chalkboard writes “A FIXED POINT x OF A FUNCTION f SATISFIES THE PROPERTY” and $f(x)=x$.

A fixed point of a function is a point that remains unchanged after applying the function once. In the video’s notation, if $x$ is a fixed point of $f$, then applying $f$ to $x$ returns $x$ itself.

1. $f$ is a function under consideration.

2. $x$ is a point in the domain where $f(x)$ is defined.

1. Audio
   Observation

   The speaker defines a periodic point of period $m$ by saying $f$ applied $m$ times sends $p$ back to $p$.

2. Formula
   Observation

   Chalkboard writes “A PERIODIC POINT p OF PERIOD m OF A FUNCTION f SATISFIES THE PROPERTY” and $f^m(p)=p$.

A periodic point of period $m$ is a point that returns to itself after $m$ iterations of the function. The video emphasizes that this is recurrence under repeated application of $f$, not the same notion of period used for trigonometric functions.

1. $f$ is a function under consideration.

2. $m$ is a positive integer counting the number of iterations.

3. $p$ is a point for which the iterates are defined.

1. Definition of a fixed point

1. Audio
   Observation

   The speaker says they will focus on points of least or prime period $m$, defined so that the first $k\ge 1$ with $f^k(p)=p$ is $m$.

2. Formula
   Observation

   Chalkboard writes “A POINT p IS OF PRIME (OR LEAST) PERIOD m IF” followed by $\forall k, 1 \le k < m$, $f^k(p)\neq p$ BUT $f^m(p)=p$.

A point has least or prime period $m$ if it returns to itself after $m$ iterations and does not return earlier at any smaller positive iterate. This excludes trivial cases where a point already repeats with a divisor-like smaller period.

1. $m$ is a positive integer.

2. $k$ ranges over integers with $1\le k<m$.

3. $p$ is a periodic point of $f$.

1. Definition of a periodic point of period m

1. Formula
   Observation

   Top-left board definition reads "Def^n: A FIXED POINT x OF A FUNCTION f SATISFIES THE PROPERTY" with $f(x)=x$.

2. Diagram
   Observation

   Nearby label "FIXED POINT THEORY" and small stick-figure doodles accompany the definition.

The board defines a fixed point of a function $f$ as a point $x$ satisfying $f(x)=x$. This is presented as the basic object of "FIXED POINT THEORY".

1. $f$ is a function.

2. $x$ is a point in the domain of $f$.

1. Formula
   Observation

   Middle-right board definition reads "Def^m: A PERIODIC POINT p OF PERIOD m OF A FUNCTION f SATISFIES THE PROPERTY" with $f^{m}(p)=p$.

2. Diagram
   Observation

   A red oval nearby states "WE DON'T WANT TO ALWAYS BE INCLUDING TRIVIALLY PERIODIC POINTS," motivating the next definition.

The board defines a periodic point $p$ of period $m$ for a function $f$ by the iterate equation $f^{m}(p)=p$. The surrounding note indicates that one often wants to exclude points that are only trivially periodic, leading to the notion of prime or least period.

1. $f$ is a function.

2. $p$ is a point in the domain of $f$.

3. $m$ is a positive integer.

1. Definition of a fixed point

1. Formula
   Observation

   Bottom board definition reads "Def^k: A POINT p IS OF PRIME (OR LEAST) PERIOD m IF" followed by $\forall k, 1 \le k < m$ and $f^{k}(p) \ne p$ but $f^{m}(p)=p$.

The board defines prime or least period $m$ by requiring both that the $m$-th iterate returns to $p$ and that no smaller positive iterate below $m$ does so. This sharpens the previous periodic-point definition by excluding earlier repetitions.

1. $m$ is the candidate period.

2. For every integer $k$ with $1 \le k < m$, the iterate $f^{k}(p)$ is not equal to $p$.

3. The iterate $f^{m}(p)$ equals $p$.

1. Definition of a periodic point of period m

1. Audio
   Observation

   Speaker says: "It is a case of Sharkovskii's theorem, or more specifically, Sharkovskii's theorem when n is equal to three."

2. Caption evidence
   Observation

   On-screen caption appears as "SHARKOVSKII'S" around 18.5 seconds.

1. The clip names the theorem case but does not state the full formal hypothesis or conclusion of Sharkovskii's theorem.

In the spoken portion, the presenter identifies the relevant application of fixed point theory as a case of Sharkovskii's theorem, specifically the case $n=3$, and says it concerns the existence of periodic points for a given function.

1. The theorem is invoked as an existence result about periodic points.

2. Only the case name $n=3$ is stated in this clip.

1. Definition of a periodic point of period m
2. Definition of prime (least) period

1. Audio
   Observation

   Speaker says: "In light of the relationship between fixed points and periodic points, one of the more accessible theorems that uses fixed point theory in its proof is a proof about periodic points."

1. The exact mathematical statement of the relationship is not spelled out verbally in this clip; it is supported visually by the board definitions.

Because fixed points and periodic points are related, an accessible theorem proved using fixed point theory can be a theorem about periodic points.

1. The board definitions of fixed points, periodic points, and prime period are available as context.

No explicit quantifiers are spoken in this claim.

1. Audio
   Observation

   Speaker says this is "Sharkovskii's theorem when n is equal to three, which tells us about the existence of periodic points for a given function."

2. Caption evidence
   Observation

   Caption "SHARKOVSKII'S" appears on screen.

1. The precise hypotheses on the function and the exact implication among periods in Sharkovskii's theorem are not stated in this clip.

The case of Sharkovskii's theorem with $n=3$ gives information about the existence of periodic points for a given function.

1. A function under consideration is assumed, though the clip does not specify its domain, regularity, or other conditions.

Existential statement about periodic points is asserted informally; full quantifier structure is not given in the clip.

1. Audio
   Observation

   The speaker contrasts the present notion of period with the period of sine and cosine, saying the trigonometric case references recurrence within the function, whereas here recurrence means after applying a function many times.

2. Diagram
   Observation

   Chalkboard shows “NOT LIKE” with sketches labeled $\sin(x)$ and $\cos(x)$ beside small coordinate axes and waveforms.

1. Expression

   $$$$
   Explanation

   Start from the definition just written on the board: $f^m(p)=p$ means the point $p$ returns after $m$ applications of $f$.

   Justification

   This is the periodic-point definition stated immediately before the contrast.

   Shown in the video
2. Expression

   $$$$
   Explanation

   Then the speaker compares this with familiar periodic functions such as $\sin(x)$ and $\cos(x)$, whose graphs repeat along the input axis.

   Justification

   Supported by the spoken explanation and the chalkboard sketches labeled $\sin(x)$ and $\cos(x)$.

   Shown in the video
3. Expression

   $$$$
   Explanation

   The conclusion drawn is that the word “period” is being used differently: here it refers to recurrence of a point under iteration, not repetition of a function’s graph.

   Justification

   Explicitly stated in the audio: “this is a different notion of period than is used in the context of trigonometric functions.”

   Shown in the video

In this lecture, “period $m$” means that a point returns after $m$ iterations of a function, which is conceptually distinct from the usual periodicity of trigonometric functions.

1. Audio
   Observation

   The speaker says that when a point $p$ has period $m$, what is really being said is that $p$ is fixed by $f^m$, i.e. $f$ applied $m$ times.

2. Formula
   Observation

   The board already contains $f(x)=x$ for fixed points and $f^m(p)=p$ for periodic points.

1. Expression

   $$f^m(p)=p$$
   Explanation

   Take the definition of a periodic point of period $m$.

   Justification

   Written on the chalkboard and spoken aloud.

   Shown in the video
2. Expression

   $$g=f^m$$
   Explanation

   Regard the $m$-fold iterate $f^m$ itself as a new function $g$.

   Justification

   This is a standard reinterpretation of the displayed equation; the video explicitly says the point is fixed by $f^m$.

   Derived from the video
3. Expression

   $$g(p)=p$$
   Explanation

   Then the equation becomes $g(p)=p$, which matches the fixed-point form.

   Justification

   Direct substitution into the fixed-point pattern $f(x)=x$.

   Derived from the video

A point of period $m$ for $f$ can be viewed as a fixed point of the iterate $f^m$; in particular, an ordinary fixed point corresponds to the case $m=1$. This final specialization to $m=1$ is an editorial clarification rather than a separately spoken sentence.

1. Diagram
   Observation

   A hand writes the fixed-point definition across the top of a dark chalkboard in white chalk.

2. Formula
   Observation

   The visible text reads “A FIXED POINT x OF A FUNCTION f SATISFIES THE PROPERTY” with $f(x)=x$ underneath.

1. dark chalkboard

2. hand writing in white chalk

3. text line defining fixed point

4. formula $f(x)=x$

1. The board goes from blank to showing the full fixed-point definition.

2. The formula $f(x)=x$ appears beneath the sentence.

1. The setting remains a single chalkboard view.

2. The notation uses $x$ for the point and $f$ for the function.

The visual establishes the basic object of the lecture: a point unchanged by one application of a function.

1. Diagram
   Observation

   Below the fixed-point definition, the presenter draws two stick figures with a heart and writes a list of fields.

2. Formula
   Observation

   Visible words include “TOPOLOGY”, “CHAOTIC DYNAMICS”, “DIFFERENTIAL EQUATIONS”, and “FIXED POINT THEORY”.

1. stick-figure drawing

2. heart symbol

3. list of application areas

4. label “FIXED POINT THEORY”

1. New colored annotations appear beneath the fixed-point definition.

2. The list of fields is added and linked to the phrase “FIXED POINT THEORY”.

1. The original fixed-point definition remains visible above.

The board visually motivates the topic by connecting fixed-point ideas to topology, chaotic dynamics, and differential equations.

1. Diagram
   Observation

   A second definition is written lower on the board.

2. Formula
   Observation

   The visible text reads “A PERIODIC POINT p OF PERIOD m OF A FUNCTION f SATISFIES THE PROPERTY” with $f^m(p)=p$ underneath.

1. chalkboard text for periodic point

2. formula $f^m(p)=p$

1. A new definition line is added beneath the earlier material.

2. The notation shifts from one application $f(x)$ to an iterate $f^m(p)$.

1. The same function symbol $f$ is reused.

2. The board still concerns recurrence properties of points under maps.

This extends the fixed-point idea from one step to multiple iterations.

1. Diagram
   Observation

   To the right of the periodic-point definition, the presenter writes “NOT LIKE” and sketches small coordinate systems with waveforms labeled $\sin(x)$ and $\cos(x)$.

1. The exact shape of each waveform is schematic rather than a precise graph.

1. text “NOT LIKE”

2. small axes

3. waveform sketches

4. labels $\sin(x)$ and $\cos(x)$

1. A comparison region is added to the right side of the board.

2. The lecture visually separates point recurrence from graph repetition.

1. The periodic-point formula $f^m(p)=p$ remains the main reference point.

The sketch signals that “period” here is not the same as the ordinary periodicity of trigonometric functions.

1. Diagram
   Observation

   A red note appears near the periodic-point section reading “WE DON'T WANT TO ALWAYS BE ANALYZING TRIVIALLY PERIODIC POINTS”, circled in red.

1. red handwritten note

2. circle around the note

3. arrow toward the next definition

1. A warning-style annotation is added to the board.

2. An arrow directs attention downward to the forthcoming least-period definition.

1. The existing periodic-point definition stays in place.

The visual prepares the refinement from any period $m$ to the first, nontrivial return time.

1. Diagram
   Observation

   A green definition is written at the bottom of the board.

2. Formula
   Observation

   The visible text reads “A POINT p IS OF PRIME (OR LEAST) PERIOD m IF” followed by $\forall k, 1 \le k < m$, $f^k(p)\neq p$ BUT $f^m(p)=p$.

1. green chalk text

2. universal quantifier over $k$

3. inequality $1\le k<m$

4. conditions $f^k(p)\neq p$ and $f^m(p)=p$

1. A more restrictive definition replaces the informal idea of merely having some period.

2. The board now distinguishes earlier returns from the first return.

1. The symbols $p$, $m$, and $f$ continue the same roles as before.

This formalizes the notion of the minimal positive return time of a periodic point.

1. Diagram
   Observation

   A chalkboard fills the frame with three definitions in white, a purple list "TOPOLOGY / CHAOTIC DYNAMICS / DIFFERENTIAL EQUATIONS," a pink note "FIXED POINT THEORY," a red oval saying "WE DON'T WANT TO ALWAYS BE INCLUDING TRIVIALLY PERIODIC POINTS," and orange sketches labeled "NOT LIKE" with $\sin(x)$ and $\cos(x)$.

2. Audio
   Observation

   Speaker comments: "I think the math I've written on the board is a bit more straightforward than the English it corresponds to."

1. Three handwritten definitions.

2. Purple subject list.

3. Pink "FIXED POINT THEORY" label.

4. Red oval cautionary note.

5. Orange sine/cosine sketches labeled "NOT LIKE".

1. The shot remains static while the speaker talks over the board.

1. The definitions and annotations stay visible throughout the opening shot.

The visual layout organizes the lesson from fixed points to periodic points to prime period, while the side notes place the topic within topology, chaotic dynamics, and differential equations and warn against counting trivial periodic behavior.

1. Caption evidence
   Observation

   Lower-left caption reads "SHARKOVSKII'S" while the speaker names the theorem.

2. Audio
   Observation

   Speaker says "Sharkovskii's theorem" during the same interval.

1. Presenter speaking to camera.

2. Lower-left text caption.

1. Caption appears briefly and then disappears.

1. The spoken reference and caption align on the theorem name.

The caption reinforces the proper name of the theorem being introduced verbally.

1. Audio
   Observation

   The speaker explicitly warns that this is a different notion of period than is used for trigonometric functions like sine and cosine.

2. Diagram
   Observation

   The board writes “NOT LIKE” beside sketches of $\sin(x)$ and $\cos(x)$.

One may think that saying a point has period $m$ means the same thing as a function such as $\sin x$ repeating its graph every $2\pi$.

Here, period $m$ means that after applying the function $m$ times, the point comes back to itself: $f^m(p)=p$. It is recurrence of a point under iteration, not repetition of a function’s output pattern over its input variable.

1. Audio
   Observation

   The speaker says they want to focus on nontrivial cases and introduces points of least or prime period.

2. Formula
   Observation

   The board adds the condition $\forall k, 1 \le k < m,\; f^k(p)\neq p$ together with $f^m(p)=p$.

If $f^m(p)=p$, one might call $m$ the period without checking whether the point already returned earlier.

The lecture refines this by defining least (prime) period: $m$ is the first positive iterate for which $f^m(p)=p$, and no smaller $k$ with $1\le k<m$ satisfies $f^k(p)=p$.

1. Diagram
   Observation

   Red oval on the board states: "WE DON'T WANT TO ALWAYS BE INCLUDING TRIVIALLY PERIODIC POINTS."

2. Formula
   Observation

   This note sits between the definition of a periodic point and the definition of prime (or least) period.

One might stop at the condition $f^{m}(p)=p$ and count points whose return happens earlier as if they had period $m$.

The board explicitly introduces prime (or least) period to exclude such trivial cases by requiring $f^{k}(p)\ne p$ for every $1\le k<m$.

1. Diagram
   Observation

   Orange annotation "NOT LIKE" points toward sketches labeled $\sin(x)$ and $\cos(x)$.

2. Audio
   Observation

   No accompanying verbal explanation is heard in this clip.

1. The exact intended contrast is not explained in the clip; it may refer to ordinary periodic functions versus discrete dynamical notions of periodic points, but that interpretation is not directly stated.

The presence of $\sin(x)$ and $\cos(x)$ could suggest the lesson is about classical periodic functions in the usual analytic sense.

The board marks these sketches as "NOT LIKE," indicating they are contrastive examples rather than the main objects; the actual definitions concern iterates of a function returning a point to itself.

1. Audio
   Observation

   After defining fixed points, the speaker says fixed points are closely related to periodic points and then defines period via $f^m(p)=p$.

2. Formula
   Observation

   The board juxtaposes $f(x)=x$ with $f^m(p)=p$.

The periodic-point definition extends the fixed-point idea from one application of $f$ to $m$ applications of $f$.

1. Audio
   Observation

   The speaker introduces least or prime period to avoid trivial cases.

2. Formula
   Observation

   The board adds the extra universal condition excluding earlier returns.

Least period is a special, nontrivial version of periodicity: it requires $f^m(p)=p$ and rules out any smaller positive return time.

1. Audio
   Observation

   The speaker says a point of period $m$ is fixed by $f^m$.

2. Formula
   Observation

   Both $f(x)=x$ and $f^m(p)=p$ are visible on the board.

Saying $p$ has period $m$ for $f$ is equivalent to saying $p$ is a fixed point of the iterate $f^m$; this equivalence is made explicit in the spoken reinterpretation.

1. Formula
   Observation

   The board presents $f(x)=x$ first and then $f^{m}(p)=p$ second.

2. Audio
   Observation

   Speaker later refers to "the relationship between fixed points and periodic points."

A fixed point is the one-step case of the iterate-return idea, while a periodic point generalizes this to an $m$-step return under $f^{m}$.

1. Formula
   Observation

   The definition of periodic point is followed by the stricter definition of prime (or least) period with the extra condition on all smaller $k$.

2. Diagram
   Observation

   The red oval about avoiding "TRIVIALLY PERIODIC POINTS" links the two definitions.

Prime period refines the notion of periodic point by adding a minimality requirement, so it distinguishes genuine least period from any larger iterate that also returns to the point.

1. Audio
   Observation

   After defining periodic and prime period on the board, the speaker introduces Sharkovskii's theorem as a result about the existence of periodic points.

1. The clip does not spell out whether Sharkovskii's theorem is stated in terms of prime period explicitly, though the preceding board setup strongly suggests periodic-point existence is the target concept.

The theorem named in the introduction applies the periodic-point framework developed on the board to an existence question for a given function.

1. Audio
   Observation

   Speaker says one accessible theorem "uses fixed point theory in its proof" and is "a proof about periodic points."

1. The actual proof is not shown in this clip.

The introduction states that the proof of the periodic-point theorem uses fixed point theory, linking the fixed-point definition to the method behind Sharkovskii's theorem case $n=3$.

1. Audio
   Observation

   Formal spoken definition of fixed point.

2. Formula
   Observation

   Board writes $f(x)=x$.

1. Definition of a fixed point

1. Audio
   Observation

   The speaker says fixed-point theorems are important to topology, chaotic dynamics, and differential equations.

2. Diagram
   Observation

   Those field names are written on the board.

1. Definition of a fixed point

1. Audio
   Observation

   Spoken definition of periodic point of period $m$.

2. Formula
   Observation

   Board writes $f^m(p)=p$.

1. Definition of a periodic point of period m

1. Audio
   Observation

   The speaker contrasts this use of period with sine and cosine.

2. Diagram
   Observation

   Board writes “NOT LIKE” with $\sin(x)$ and $\cos(x)$ sketches.

1. Definition of a periodic point of period m
2. Confusing iterative period with trigonometric period

1. Audio
   Observation

   The speaker says a point of period $m$ is fixed by $f^m$, meaning $f$ applied $m$ times.

1. Definition of a periodic point of period m
2. s0-cr-fixed-as-iterate-fixed

1. Audio
   Observation

   Spoken definition of least or prime period.

2. Formula
   Observation

   Board writes the universal condition over $k$ with $1\le k<m$.

1. Definition of least (prime) period

1. Audio
   Observation

   The speaker says they focus on nontrivial cases as often as possible.

2. Diagram
   Observation

   Red note says they do not want to always analyze trivially periodic points.

1. Definition of least (prime) period
2. Treating every return time as the period

1. Formula
   Observation

   Board definition $f(x)=x$ is visible.

1. Definition of a fixed point

1. Formula
   Observation

   Board definition $f^{m}(p)=p$ is visible.

1. Definition of a periodic point of period m

1. Diagram
   Observation

   Red oval warns against including "TRIVIALLY PERIODIC POINTS."

2. Formula
   Observation

   Prime-period definition adds $\forall k, 1 \le k < m$, $f^{k}(p)\ne p$.

1. Definition of prime (least) period
2. Not all periodic points should be treated as having genuinely new period

1. Audio
   Observation

   Speaker names "Sharkovskii's theorem when n is equal to three" and says it concerns existence of periodic points.

1. Sharkovskii's theorem, case $n = 3$
2. Sharkovskii's theorem for $n = 3$ concerns existence of periodic points

1. Audio
   Observation

   Speaker says the theorem "uses fixed point theory in its proof."

1. Relationship between fixed points and periodic points motivates a theorem about periodic points
2. s120-cr-fixed-theory-to-sharkovskii-proof