page_title: Horseshoe map
section_title: Invariant set
hierarchical_section_title: Horseshoe map / Dynamics of the map / Invariant set
caption_attribution_description: English: This tartan-like graph shows the Ising model probability density P ( &#x03C3; ) {\displaystyle P(\sigma )} for the two-sided lattice using the dyadic mapping. That is, a lattice configuration of length 2 N {\displaystyle 2N} &#x03C3; = ( &#x03C3; &#x2212; N &#x2212; 1 , &#x22EF; , &#x03C3; &#x2212; 2 , &#x03C3; &#x2212; 1 , &#x03C3; 0 , &#x03C3; 1 , &#x03C3; 2 , &#x22EF; , &#x03C3; N ) {\displaystyle \sigma =(\sigma _{-N-1},\cdots ,\sigma _{-2},\sigma _{-1},\sigma _{0},\sigma _{1},\sigma _{2},\cdots ,\sigma _{N})} is understood to consist of a sequence of "spins" &#x03C3; k = &#x00B1; 1 {\displaystyle \sigma _{k}=\pm 1} . This sequence may be represented by two real numbers 0 &#x2264; x , y &#x2264; 1 {\displaystyle 0\leq x,y\leq 1} with x ( &#x03C3; ) = &#x2211; k = 0 N ( &#x03C3; k + 1 2 ) 2 &#x2212; ( k + 1 ) {\displaystyle x(\sigma )=\sum _{k=0}^{N}\left({\frac {\sigma _{k}+1}{2}}\right)2^{-(k+1)}} and y ( &#x03C3; ) = &#x2211; k = 0 N ( &#x03C3; &#x2212; k &#x2212; 1 + 1 2 ) 2 &#x2212; ( k + 1 ) {\displaystyle y(\sigma )=\sum _{k=0}^{N}\left({\frac {\sigma _{-k-1}+1}{2}}\right)2^{-(k+1)}} The energy of a given configuration &#x03C3; {\displaystyle \sigma } is computed using the classical Hamiltonian, H ( &#x03C3; ) = &#x2211; k = &#x2212; N N V ( &#x03C4; k &#x03C3; ) {\displaystyle H(\sigma )=\sum _{k=-N}^{N}V(\tau ^{k}\sigma )} Here, &#x03C4; {\displaystyle \tau } is the shift operator, acting on the lattice by shifting all spins over by one position: &#x03C4; ( &#x22EF; , &#x03C3; 0 , &#x03C3; 1 , &#x03C3; 2 , &#x22EF; ) = ( &#x22EF; , &#x03C3; &#x2212; 1 , &#x03C3; 0 , &#x03C3; 1 , &#x22EF; ) {\displaystyle \tau (\cdots ,\sigma _{0},\sigma _{1},\sigma _{2},\cdots )=(\cdots ,\sigma _{-1},\sigma _{0},\sigma _{1},\cdots )} The interaction potential V {\displaystyle V} is given by the Ising model interaction V ( &#x03C3; ) = J &#x03C3; 0 &#x03C3; 1 + B &#x03C3; 0 {\displaystyle V(\sigma )=J\sigma _{0}\sigma _{1}+B\sigma _{0}} Here, the constant J {\displaystyle J} is the interaction strength between two neighboring spins &#x03C3; 0 {\displaystyle \sigma _{0}} and &#x03C3; 1 {\displaystyle \sigma _{1}} , while the constant B {\displaystyle B} may be interpreted as the strength of the interaction between the magnetic field and the magnetic moment of the spin. The set of all possible configurations &#x03A9; = { &#x03C3; } {\displaystyle \Omega =\{\sigma \}} form a canonical ensemble, with each different configuration occurring with a probability P ( &#x03C3; ) {\displaystyle P(\sigma )} given by the Boltzmann distribution P ( &#x03C3; ) = 1 Z ( T ) e &#x2212; H ( &#x03C3; ) / k B T {\displaystyle P(\sigma )={\frac {1}{Z(T)}}e^{-H(\sigma )/k_{B}T}} where k B {\displaystyle k_{B}} is Boltzmann's constant, T {\displaystyle T} is the temperature, and Z ( T ) {\displaystyle Z(T)} is the partition function. The partition function is defined to be such that the sum over all probabilities adds up to one; that is, so that Z ( T ) = &#x2211; &#x03C3; &#x2208; &#x03A9; e &#x2212; H ( &#x03C3; ) / k B T {\displaystyle Z(T)=\sum _{\sigma \in \Omega }e^{-H(\sigma )/k_{B}T}} Contents 1 Image details 2 As an invariant measure 3 Licensing 4 Original upload log Image details The image here shows P ( &#x03C3; ) = P ( x , y ) {\displaystyle P(\sigma )=P(x,y)} for the Ising model, with J = 0.3 {\displaystyle J=0.3} , B = 0 {\displaystyle B=0} and temperature T = 1 / k B {\displaystyle T=1/k_{B}} . The lattice is finite sized, with N = 10 {\displaystyle N=10} , so that all 1024 &#x00D7; 1024 = 2 N &#x00D7; 2 N {\displaystyle 1024\times 1024=2^{N}\times 2^{N}} lattice configurations are represented, each configuration denoted by one pixel. The color choices here are such that black represents values where P ( &#x03C3; ) = P ( x , y ) {\displaystyle P(\sigma )=P(x,y)} are zero, blue are small values, with yellow and red being progressively larger values. As an invariant measure This fractal tartan is invariant under the Baker's map. The shift operator &#x03C4; {\displaystyle \tau } on the lattice has an action on the unit square with the following representation: &#x03C4; ( x , y ) = ( x + &#x230A; 2 y &#x230B; 2 , 2 y &#x2212; &#x230A; 2 y &#x230B; ) {\displaystyle \tau (x,y)=\left({\frac {x+\left\lfloor 2y\right\rfloor }{2}}\,,\,2y-\left\lfloor 2y\right\rfloor \right)} This map (up to a reflection/rotation around the 45-degree axis) is essentially the Baker's map or equivalently the Horseshoe map. As the article on the Horseshoe map explains, the invariant sets have such a tartan pattern (an appropriately deformed Sierpinski carpet). In this case, the invariance arises from the translation invariance of the Gibbs states of the Ising model: that is, the energy H ( &#x03C3; ) {\displaystyle H(\sigma )} associated with the state &#x03C3; {\displaystyle \sigma } is invariant under the action of &#x03C4; {\displaystyle \tau } : H ( &#x03C4; k &#x03C3; ) = H ( &#x03C3; ) {\displaystyle H\left(\tau ^{k}\sigma \right)=H(\sigma )} for all integers k {\displaystyle k} . Similarly, the probability density is invariant as well: P ( &#x03C4; k &#x03C3; ) = P ( &#x03C3; ) {\displaystyle P\left(\tau ^{k}\sigma \right)=P(\sigma )} The naive classical treatment given here suffers from conceptual difficulties in the N &#x2192; &#x221E; {\displaystyle N\to \infty } limit. These problems can be remedied by using a more appropriate topology on the set of states that make up the configuration space. This topology is the cylinder set topology, and using it allows one to construct a sigma algebra and thus a measure on the set of states. With this topology, the probability density can be understood to be a translation-invariant measure on the topology. Indeed, there is a certain sense in which the seemingly fractal patterns generated by the iterated Baker's map or horseshoe map can be understood with a conventional and well-behaved topology on a lattice model. Created by Linas Vepstas User:Linas on 24 September 2006
context_page_description: In the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator. The action of the map is defined geometrically by squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a horseshoe.
  Most points eventually leave the square under the action of the map. They go to the side caps where they will, under iteration, converge to a fixed point in one of the caps. The points that remain in the square under repeated iteration form a fractal set and are part of the invariant set of the map.
  The squishing, stretching and folding of the horseshoe map are typical of chaotic systems, but not necessary or even sufficient.
  In the horseshoe map, the squeezing and stretching are uniform. They compensate each other so that the area of the square does not change. The folding is done neatly, so that the orbits that remain forever in the square can be simply described.
  For a horseshoe map:
context_section_description: If a point is to remain indefinitely in the square, then it must belong to a set Λ that maps to itself. Whether this set is empty or not has to be determined. The vertical strips V₁ map into the horizontal strips H₁, but not all points of V₁ map back into V₁. Only the points in the intersection of V₁ and H₁ may belong to Λ, as can be checked by following points outside the intersection for one more iteration.
  The intersection of the horizontal and vertical stripes, Hₙ ∩ Vₙ, are squares that in the limit n → ∞ converge to the invariant set Λ (this set is an intersection of a Cantor set of vertical lines with a Cantor set of horizontal lines). The structure of this set can be better understood by introducing a system of labels for all the intersections—a symbolic dynamics.
caption: Example of an invariant measure
exif: {}