Dynamics on Fractals...Dynamics on Fractals plan (1) IFS, attractor A ,points p(s) and addresses s...

Dynamics on Fractals

Michael F. Barnsley (MSI-ANU)

Lecture for Symmetry in Newcastle

1 November 2019

Preliminary Remarks

Thank you

Joint work with Louisa Barnsley, Andrew Vince

Preliminary Remarks

Thank youJoint work with Louisa Barnsley, Andrew Vince

(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

(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

(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

(3) π ×Π : Σ× Σ† → A×T

(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

(3) π ×Π : Σ× Σ† → A×T

(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

(3) π ×Π : Σ× Σ† → A×T

(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

(3) π ×Π : Σ× Σ† → A×T

example of a fractal tiling. see also abstract, monthly pape

IFS, A, points and addresses, golden b

IFS of similitudes

attractors and addresses: definition of π : Σ→ 2A

OSC - components of attractors are non-overlappingalgebraic condition on scaling factors

IFS of similitudesattractors and addresses: definition of π : Σ→ 2A

OSC - components of attractors are non-overlapping

algebraic condition on scaling factors

Sphinx example

(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

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

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

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

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

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

(4) Rigidity, deflation and inflation

the golden b is rigid

deflation and inflation definitions(3) interplay with the big picture

the golden b is rigiddeflation and inflation definitions

(3) interplay with the big picture

the golden b is rigiddeflation and inflation definitions(3) interplay with the big picture

(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

(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

(6) Examples

robinson, golden bsquare, fish-horn, purely fractal

(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

built using 5 similitudesfirst describe the tiling IFS and the graph

then illustrate the construction of the canonical tilings andof a tiling

built using 5 similitudesfirst describe the tiling IFS and the graphthen illustrate the construction of the canonical tilings andof a tiling

(7) Main theorem

key theorem and action of the group of isometries

canonical tilingsbeautiful combinatorial formula explains all

(7) Main theorem

key theorem and action of the group of isometriescanonical tilings

beautiful combinatorial formula explains all

(7) Main theorem

key theorem and action of the group of isometriescanonical tilingsbeautiful combinatorial formula explains all

(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

the big picture including dynamicslocal isomorphism, self-similar tiling theory vs tiling IFStheory

Anderson and Putnam, Solomyakthe importance of boundaries

the big picture including dynamicslocal isomorphism, self-similar tiling theory vs tiling IFStheoryAnderson and Putnam, Solomyak

the importance of boundaries

the big picture including dynamicslocal isomorphism, self-similar tiling theory vs tiling IFStheoryAnderson and Putnam, Solomyakthe importance of boundaries

(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

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

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

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

IFS tiling theory unifies, simplifies and extends s.s. tilingbig picture of shift dynamicskey theorem and its significancesymbolic theory↔ geometryunderstanding canonical tilings is key

Dynamics on Fractals...Dynamics on Fractals plan (1) IFS, attractor A ,points p(s) and addresses s...

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