Dynamics on Fractals...Dynamics on Fractals plan (1) IFS, attractor A ,points p(s) and addresses s...
Transcript of Dynamics on Fractals...Dynamics on Fractals plan (1) IFS, attractor A ,points p(s) and addresses s...
Dynamics on Fractals
Dynamics on Fractals
Michael F. Barnsley (MSI-ANU)
Lecture for Symmetry in Newcastle
1 November 2019
Dynamics on Fractals
Preliminary Remarks
Thank you
Joint work with Louisa Barnsley, Andrew Vince
Dynamics on Fractals
Preliminary Remarks
Thank youJoint work with Louisa Barnsley, Andrew Vince
Dynamics on Fractals
plan
(1) IFS, attractor A ,points π(σ) and addresses σ ∈ Σ,golden b
(2) T, tilings Π(θ), and addresses θ ∈ Σ†
(3) π ×Π : Σ× Σ† → A×T
(4) rigidity, recognizability, deflation α and inflation α−1
(5) subshifts of finite type, graph IFS, and Markovmeasures(6) three examples, pedal triangle, golden bsquare,fish-horn(7) key theorem and action of the group of isometries(8) shift invariant measures and their relation todisjunctive points(9) the big picture
Dynamics on Fractals
plan
(1) IFS, attractor A ,points π(σ) and addresses σ ∈ Σ,golden b(2) T, tilings Π(θ), and addresses θ ∈ Σ†
(3) π ×Π : Σ× Σ† → A×T
(4) rigidity, recognizability, deflation α and inflation α−1
(5) subshifts of finite type, graph IFS, and Markovmeasures(6) three examples, pedal triangle, golden bsquare,fish-horn(7) key theorem and action of the group of isometries(8) shift invariant measures and their relation todisjunctive points(9) the big picture
Dynamics on Fractals
plan
(1) IFS, attractor A ,points π(σ) and addresses σ ∈ Σ,golden b(2) T, tilings Π(θ), and addresses θ ∈ Σ†
(3) π ×Π : Σ× Σ† → A×T
(4) rigidity, recognizability, deflation α and inflation α−1
(5) subshifts of finite type, graph IFS, and Markovmeasures(6) three examples, pedal triangle, golden bsquare,fish-horn(7) key theorem and action of the group of isometries(8) shift invariant measures and their relation todisjunctive points(9) the big picture
Dynamics on Fractals
plan
(1) IFS, attractor A ,points π(σ) and addresses σ ∈ Σ,golden b(2) T, tilings Π(θ), and addresses θ ∈ Σ†
(3) π ×Π : Σ× Σ† → A×T
(4) rigidity, recognizability, deflation α and inflation α−1
(5) subshifts of finite type, graph IFS, and Markovmeasures(6) three examples, pedal triangle, golden bsquare,fish-horn(7) key theorem and action of the group of isometries(8) shift invariant measures and their relation todisjunctive points(9) the big picture
Dynamics on Fractals
plan
(1) IFS, attractor A ,points π(σ) and addresses σ ∈ Σ,golden b(2) T, tilings Π(θ), and addresses θ ∈ Σ†
(3) π ×Π : Σ× Σ† → A×T
(4) rigidity, recognizability, deflation α and inflation α−1
(5) subshifts of finite type, graph IFS, and Markovmeasures
(6) three examples, pedal triangle, golden bsquare,fish-horn(7) key theorem and action of the group of isometries(8) shift invariant measures and their relation todisjunctive points(9) the big picture
Dynamics on Fractals
plan
(1) IFS, attractor A ,points π(σ) and addresses σ ∈ Σ,golden b(2) T, tilings Π(θ), and addresses θ ∈ Σ†
(3) π ×Π : Σ× Σ† → A×T
(4) rigidity, recognizability, deflation α and inflation α−1
(5) subshifts of finite type, graph IFS, and Markovmeasures(6) three examples, pedal triangle, golden bsquare,fish-horn
(7) key theorem and action of the group of isometries(8) shift invariant measures and their relation todisjunctive points(9) the big picture
Dynamics on Fractals
plan
(1) IFS, attractor A ,points π(σ) and addresses σ ∈ Σ,golden b(2) T, tilings Π(θ), and addresses θ ∈ Σ†
(3) π ×Π : Σ× Σ† → A×T
(4) rigidity, recognizability, deflation α and inflation α−1
(5) subshifts of finite type, graph IFS, and Markovmeasures(6) three examples, pedal triangle, golden bsquare,fish-horn(7) key theorem and action of the group of isometries
(8) shift invariant measures and their relation todisjunctive points(9) the big picture
Dynamics on Fractals
plan
(1) IFS, attractor A ,points π(σ) and addresses σ ∈ Σ,golden b(2) T, tilings Π(θ), and addresses θ ∈ Σ†
(3) π ×Π : Σ× Σ† → A×T
(4) rigidity, recognizability, deflation α and inflation α−1
(5) subshifts of finite type, graph IFS, and Markovmeasures(6) three examples, pedal triangle, golden bsquare,fish-horn(7) key theorem and action of the group of isometries(8) shift invariant measures and their relation todisjunctive points
(9) the big picture
Dynamics on Fractals
plan
(1) IFS, attractor A ,points π(σ) and addresses σ ∈ Σ,golden b(2) T, tilings Π(θ), and addresses θ ∈ Σ†
(3) π ×Π : Σ× Σ† → A×T
(4) rigidity, recognizability, deflation α and inflation α−1
(5) subshifts of finite type, graph IFS, and Markovmeasures(6) three examples, pedal triangle, golden bsquare,fish-horn(7) key theorem and action of the group of isometries(8) shift invariant measures and their relation todisjunctive points(9) the big picture
Dynamics on Fractals
example of a fractal tiling. see also abstract, monthly pape
Dynamics on Fractals
IFS, A, points and addresses, golden b
(1)
IFS of similitudes
attractors and addresses: definition of π : Σ→ 2A
OSC - components of attractors are non-overlappingalgebraic condition on scaling factors
Dynamics on Fractals
IFS, A, points and addresses, golden b
(1)
IFS of similitudesattractors and addresses: definition of π : Σ→ 2A
OSC - components of attractors are non-overlappingalgebraic condition on scaling factors
Dynamics on Fractals
IFS, A, points and addresses, golden b
(1)
IFS of similitudesattractors and addresses: definition of π : Σ→ 2A
OSC - components of attractors are non-overlapping
algebraic condition on scaling factors
Dynamics on Fractals
IFS, A, points and addresses, golden b
(1)
IFS of similitudesattractors and addresses: definition of π : Σ→ 2A
OSC - components of attractors are non-overlappingalgebraic condition on scaling factors
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Sphinx example
Dynamics on Fractals
(2) How to construct tilings
Π(θ) for θ ∈ Σ
given θ ∈ Σ†, how to construct Π(θ)
Π(θ|k) = f−θ|k {π(σ) : ξ− (σ) ≤ ξ (θ) < ξ (σ)}Theorem: Π(θ|0) ⊂ Π(θ|1) ⊂ Π(θ|2)...tiling metricillustrate for the golden bapplies to any OSC IFS
Dynamics on Fractals
(2) How to construct tilings
Π(θ) for θ ∈ Σ
given θ ∈ Σ†, how to construct Π(θ)Π(θ|k) = f−θ|k {π(σ) : ξ− (σ) ≤ ξ (θ) < ξ (σ)}
Theorem: Π(θ|0) ⊂ Π(θ|1) ⊂ Π(θ|2)...tiling metricillustrate for the golden bapplies to any OSC IFS
Dynamics on Fractals
(2) How to construct tilings
Π(θ) for θ ∈ Σ
given θ ∈ Σ†, how to construct Π(θ)Π(θ|k) = f−θ|k {π(σ) : ξ− (σ) ≤ ξ (θ) < ξ (σ)}Theorem: Π(θ|0) ⊂ Π(θ|1) ⊂ Π(θ|2)...
tiling metricillustrate for the golden bapplies to any OSC IFS
Dynamics on Fractals
(2) How to construct tilings
Π(θ) for θ ∈ Σ
given θ ∈ Σ†, how to construct Π(θ)Π(θ|k) = f−θ|k {π(σ) : ξ− (σ) ≤ ξ (θ) < ξ (σ)}Theorem: Π(θ|0) ⊂ Π(θ|1) ⊂ Π(θ|2)...tiling metric
illustrate for the golden bapplies to any OSC IFS
Dynamics on Fractals
(2) How to construct tilings
Π(θ) for θ ∈ Σ
given θ ∈ Σ†, how to construct Π(θ)Π(θ|k) = f−θ|k {π(σ) : ξ− (σ) ≤ ξ (θ) < ξ (σ)}Theorem: Π(θ|0) ⊂ Π(θ|1) ⊂ Π(θ|2)...tiling metricillustrate for the golden b
applies to any OSC IFS
Dynamics on Fractals
(2) How to construct tilings
Π(θ) for θ ∈ Σ
given θ ∈ Σ†, how to construct Π(θ)Π(θ|k) = f−θ|k {π(σ) : ξ− (σ) ≤ ξ (θ) < ξ (σ)}Theorem: Π(θ|0) ⊂ Π(θ|1) ⊂ Π(θ|2)...tiling metricillustrate for the golden bapplies to any OSC IFS
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
(4) Rigidity, deflation and inflation
the golden b is rigid
deflation and inflation definitions(3) interplay with the big picture
Dynamics on Fractals
(4) Rigidity, deflation and inflation
the golden b is rigiddeflation and inflation definitions
(3) interplay with the big picture
Dynamics on Fractals
(4) Rigidity, deflation and inflation
the golden b is rigiddeflation and inflation definitions(3) interplay with the big picture
Dynamics on Fractals
(5) Subshifts and definition of tiling IFS
subshifts of finite type, graph IFS, and Markov measures
disjunctive points have measure zero and correspond toboundaries of tiles
Dynamics on Fractals
(5) Subshifts and definition of tiling IFS
subshifts of finite type, graph IFS, and Markov measuresdisjunctive points have measure zero and correspond toboundaries of tiles
Dynamics on Fractals
(6) Examples
robinson, golden bsquare, fish-horn, purely fractal
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
(6) golden rectangle with golden b
built using 5 similitudes
first describe the tiling IFS and the graphthen illustrate the construction of the canonical tilings andof a tiling
Dynamics on Fractals
(6) golden rectangle with golden b
built using 5 similitudesfirst describe the tiling IFS and the graph
then illustrate the construction of the canonical tilings andof a tiling
Dynamics on Fractals
(6) golden rectangle with golden b
built using 5 similitudesfirst describe the tiling IFS and the graphthen illustrate the construction of the canonical tilings andof a tiling
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
Dynamics on Fractals
(7) Main theorem
key theorem and action of the group of isometries
canonical tilingsbeautiful combinatorial formula explains all
Dynamics on Fractals
(7) Main theorem
key theorem and action of the group of isometriescanonical tilings
beautiful combinatorial formula explains all
Dynamics on Fractals
(7) Main theorem
key theorem and action of the group of isometriescanonical tilingsbeautiful combinatorial formula explains all
Dynamics on Fractals
(8 and 9) Big picture
the big picture including dynamics
local isomorphism, self-similar tiling theory vs tiling IFStheoryAnderson and Putnam, Solomyakthe importance of boundaries
Dynamics on Fractals
(8 and 9) Big picture
the big picture including dynamicslocal isomorphism, self-similar tiling theory vs tiling IFStheory
Anderson and Putnam, Solomyakthe importance of boundaries
Dynamics on Fractals
(8 and 9) Big picture
the big picture including dynamicslocal isomorphism, self-similar tiling theory vs tiling IFStheoryAnderson and Putnam, Solomyak
the importance of boundaries
Dynamics on Fractals
(8 and 9) Big picture
the big picture including dynamicslocal isomorphism, self-similar tiling theory vs tiling IFStheoryAnderson and Putnam, Solomyakthe importance of boundaries
Dynamics on Fractals
(10) main points I care about
IFS tiling theory unifies, simplifies and extends s.s. tiling
big picture of shift dynamicskey theorem and its significancesymbolic theory↔ geometryunderstanding canonical tilings is key
Dynamics on Fractals
(10) main points I care about
IFS tiling theory unifies, simplifies and extends s.s. tilingbig picture of shift dynamics
key theorem and its significancesymbolic theory↔ geometryunderstanding canonical tilings is key
Dynamics on Fractals
(10) main points I care about
IFS tiling theory unifies, simplifies and extends s.s. tilingbig picture of shift dynamicskey theorem and its significance
symbolic theory↔ geometryunderstanding canonical tilings is key
Dynamics on Fractals
(10) main points I care about
IFS tiling theory unifies, simplifies and extends s.s. tilingbig picture of shift dynamicskey theorem and its significancesymbolic theory↔ geometry
understanding canonical tilings is key
Dynamics on Fractals
(10) main points I care about
IFS tiling theory unifies, simplifies and extends s.s. tilingbig picture of shift dynamicskey theorem and its significancesymbolic theory↔ geometryunderstanding canonical tilings is key