NestGraph[x{3n+1,2n+2}
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Solve[4x(1-x)xp,x]
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x1-1+
1
2
1-xp
,x1
2
1-xp
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ListPlot[With[{a=2},Catenate[MapIndexed[{#1,-First[#2]}&,NestList[Flatten[{1/a(1-Sqrt[1-#]),1/a(1-Sqrt[1+#])}]&,{.5},5],{2}]]],PlotRangeAll]
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ListPlot[With[{a=1},Catenate[MapIndexed[{#1,-First[#2]}&,NestList[Flatten[{Sin[#],Cos[#]}]&,{.5},5],{2}]]],PlotRangeAll]
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NestGraph[{Sin[#],Cos[#]}&,{.5},3]
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Plot[{Sin[x],Cos[x]},{x,0,1}]
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Contents cannot be rendered at this time; please try again later
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ListPlot[With[{a=1},Catenate[MapIndexed[{#1,-First[#2]}&,NestList[Flatten[{#,1-#}]&,{.3567},5],{2}]]],PlotRangeAll]
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Plot[1-2Abs[1/2-x],{x,0,1}]
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ListPlot[With[{a=1},Catenate[MapIndexed[{#1,-First[#2]}&,NestList[Flatten[{1-2Abs[1/2-#],2Abs[1/2-#]}]&,{.3567},10],{2}]]],PlotRangeAll]
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NestGraph[{1-2Abs[1/2-#],2Abs[1/2-#]}&,1/3,5]
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ListPlot[With[{a=1},Catenate[MapIndexed[{#1,-First[#2]}&,NestList[Flatten[{FractionalPart[2#],FractionalPart[3#]}]&,{.3567},10],{2}]]],PlotRangeAll]
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Contents cannot be rendered at this time; please try again later
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NestList[Union[Flatten[{FractionalPart[2#],FractionalPart[3#]}]]&,{.3567},10]
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{{0.3567},{0.0701,0.7134},{0.1402,0.2103,0.4268},{0.2804,0.4206,0.6309,0.8536},{0.2618,0.5608,0.7072,0.8412,0.8927},{0.1216,0.4144,0.5236,0.6781,0.6824,0.7854},{0.0343,0.0472,0.2432,0.3562,0.3648,0.5708,0.8288},{0.0686,0.0944,0.1029,0.1416,0.4864,0.6576,0.7124,0.7296},{0.1372,0.1888,0.2058,0.2832,0.3087,0.3152,0.4248,0.4592,0.9728},{0.2744,0.3776,0.4116,0.5664,0.6174,0.6304,0.8496,0.9184,0.9261,0.9456},{0.1328,0.2348,0.2608,0.5488,0.6992,0.7552,0.7783,0.8232,0.8368,0.8522,0.8912}}
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Length/@%
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{1,2,3,4,5,6,7,8,9,10,11}
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Length/@NestList[Union[Flatten[{FractionalPart[2#],FractionalPart[3/2#]}]]&,{.3567},10]
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{1,2,3,6,8,12,16,21,32,46,64}
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ListPlot[With[{a=1},Catenate[MapIndexed[{#1,-First[#2]}&,NestList[Flatten[{FractionalPart[2#],FractionalPart[3/2#]}]&,{.3567},10],{2}]]],PlotRangeAll]
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Histogram/@NestList[Flatten[{FractionalPart[2#],FractionalPart[3/2#]}]&,{.3567},20]
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{,,,,,,,,,,,,,,,,,,,,}
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2^24
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16777216
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Histogram[Nest[Flatten[{FractionalPart[2#],FractionalPart[3/2#]}]&,{.3567},24]]
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In[]:=
Histogram[Nest[Flatten[{FractionalPart[2#],FractionalPart[3/2#]}]&,{.3567},24],{.01}]
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Contents cannot be rendered at this time; please try again later
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Histogram[Nest[Flatten[{FractionalPart[2#],FractionalPart[3#]}]&,{.3567},10],{.01}]
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Histogram[Nest[Flatten[{FractionalPart[2#],FractionalPart[3#]}]&,{N[1/3]},10],{.01}]
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Histogram[Nest[Flatten[{FractionalPart[2#],FractionalPart[3#]}]&,{N[1/5]},10],{.01}]
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Histogram[Nest[Flatten[{FractionalPart[2#],FractionalPart[3#]}]&,{N[1/7]},10],{.01}]
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Histogram[Nest[Flatten[{FractionalPart[2#],FractionalPart[3#]}]&,{N[1/11]},10],{.01}]
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NestGraph[{FractionalPart[2#],FractionalPart[3#]}&,{1/11},5,VertexLabelsAutomatic]
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NestGraph[{FractionalPart[2#],FractionalPart[3#]}&,{1/13},15,VertexLabelsAutomatic]
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Graph3D[%]
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NestGraph[{FractionalPart[2#],FractionalPart[3#]}&,{1/17},15,VertexLabelsAutomatic]
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BaseForm[N[1/7,20],2]
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0.001001001001001001001001001001001001001001001001001001001001001001001
2
In[]:=
IntegerPart[2NestList[FractionalPart[2#]&,N[1/7,20],20]]
Mapping characteristics
Mapping characteristics
Normally our x{f[x],g[x]}
The mapping from t=0 to t=∞ for causal invariance it preserves measure
Case 1: consider only nonzero measure initial conditions (i.e. not countable i.c.’s)
Case 2: consider merging “up to ϵ” [this is an analog of a completion]
Case 2: consider merging “up to ϵ” [this is an analog of a completion]
Mappings of measures
Mappings of measures
This is the ensemble average
https://en.wikipedia.org/wiki/Binomial_options_pricing_model
ϵ Bucketing
ϵ Bucketing
Symbolic dynamics binning
Symbolic dynamics binning
“Symbolic dynamics in the middle” just gives a discrete function
This is just filling out the symbolic dynamics of the system
The next case is mapping every value to a kernel
The next case is mapping every value to a kernel
At each step, you are computing Integrate[kern[dx] f[x+dx],{dx,-1,1}] assuming no merging [[ where the pure discrete multiway system would be a tree ]]
First element is red, second is green
At any given stage, every value is a delta function
For a given x, we have p1[x] from one kernel, and p2[x] from the other kernel, combining with Max[ ]
What about continuous time?
What about continuous time?
With no branching, for discrete time this is an iterated convolution;
the behavior is a PDE
the behavior is a PDE
What is the continuous time analog of combining using Max at every infinitesimal step?