A count of maximal small copies in Multibrot sets

Bridy, Andrew; Pérez, Rodrigo A.

doi:10.1088/0951-7715/18/5/004

Cited by 6 publications

(4 citation statements)

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“…in agreement with the time-evolution count (27); all itineraries are allowed, except that the periodicity of r n = 1 1 accounts for 0 and s − 1 fixed points (see figure 3) being a single periodic point.…”

Section: Hill's Formula For a 1st Order Difference Equationsupporting

confidence: 64%

“…Poincaré [149] was the first to recognize the fundamental role periodic orbits play in shaping ergodic dynamics. The first step in this program is a census of periodic orbits, addressed in [186,22,27,30,33,34,34,47,122,123,187], starting with 1950's Myrberg investigations of periodic orbits of quadratic maps, in what was arguably the first application of computers to dynamics [136,137,138,139,140]. Such orbit counts are most elegantly encoded by topological zeta functions of section 11.…”

Section: Discussionmentioning

confidence: 99%

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A chaotic lattice field theory in one dimension*

Liang¹,

Cvitanović²

2022

J. Phys. A: Math. Theor.

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Motivated by Gutzwiller's semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB `skeleton' for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no `Lyapunov horizon'; there is only an enumeration of lattice states that contribute to the theory's partition sum, each a global spatiotemporal solution of system's deterministic Euler-Lagrange equations. The reformulation aligns `chaos theory' with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the time-periodic orbits of dynamical systems theory are replaced by periodic d-dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells. The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already 1-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, `time'-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions.

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Section: Hill's Formula For a 1st Order Difference Equationsupporting

confidence: 64%

Section: Discussionmentioning

confidence: 99%

A chaotic lattice field theory in one dimension*

Liang¹,

Cvitanović²

2022

J. Phys. A: Math. Theor.

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“…where (• • •) denotes a transpose. The Bernoulli equation (22), rewritten as a first-order difference equation…”

Section: Temporal Bernoullimentioning

confidence: 99%

“…Poincaré [141] was the first to recognize the fundamental role periodic orbits play in shaping ergodic dynamics. The first step in this program is the count of periodic orbits, addressed in [8,19,22,25,27,28,40,119,120,170], starting with 1950's Myrberg investigations of periodic orbits of quadratic maps, in what was arguably the first application of computers to dynamics [129][130][131][132][133]. Such orbit counts are most elegantly encoded by topological zeta functions.…”

Section: Lind Zeta Functionmentioning

confidence: 99%

A chaotic lattice field theory in one dimension

Liang,

Cvitanović

2022

Preprint

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Motivated by Gutzwiller's semiclassical quantization, in which unstable periodic orbits of low-dimensional deterministic dynamics serve as a WKB 'skeleton' for chaotic quantum mechanics, we construct the corresponding deterministic skeleton for infinite-dimensional lattice-discretized scalar field theories. In the field-theoretical formulation, there is no evolution in time, and there is no 'Lyapunov horizon'; there is only an enumeration of lattice states that contribute to the theory's partition sum, each a global spatiotemporal solution of system's deterministic Euler-Lagrange equations.The reformulation aligns 'chaos theory' with the standard solid state, field theory, and statistical mechanics. In a spatiotemporal, crystallographer formulation, the timeperiodic orbits of dynamical systems theory are replaced by periodic d-dimensional Bravais cell tilings of spacetime, each weighted by the inverse of its instability, its Hill determinant. Hyperbolic shadowing of large cells by smaller ones ensures that the predictions of the theory are dominated by the smallest Bravais cells.The form of the partition function of a given field theory is determined by the group of its spatiotemporal symmetries, that is, by the space group of its lattice discretization, best studied on its reciprocal lattice. Already 1-dimensional lattice discretization is of sufficient interest to be the focus of this paper. In particular, from a spatiotemporal field theory perspective, 'time'-reversal is a purely crystallographic notion, a reflection point group, leading to a novel, symmetry quotienting perspective of time-reversible theories and associated topological zeta functions.

show abstract