Entropy of a dynamical system

In the theory of dynamical systems, the entropy of a dynamical system is a number expressing the degree of randomness of its trajectories. There are metric entropy describing the randomness of dynamics in a system with an invariant measure for a random choice of the initial condition for this measure, and topological entropy describing the randomness of dynamics without assuming the law of choosing the starting point.

Moreover, the variational principle states that for a continuous dynamical system on a compact set, topological entropy is equal to the supremum of metric ones, taken from all possible choices of invariant measures of this system.

Content

Definitions

Topological Entropy

Let a continuous mapping be given $T$ ${\ displaystyle T}$ metric compact $(X,d)$ ${\ displaystyle (X, d)}$ in yourself. Then, the metric $d_{n}$ ${\ displaystyle d_ {n}}$ on $X$ ${\ displaystyle X}$ defined as

d_{n}(x,y)=\max _{0\leq j\leq n}d(T^{j}(x),T^{j}(y)),

{\ displaystyle d_ {n} (x, y) = \ max _ {0 \ leq j \ leq n} d (T ^ {j} (x), T ^ {j} (y)),}

in other words, this is the maximum distance that the orbits $x$ ${\ displaystyle x}$ and $y$ ${\ displaystyle y}$ diverge for $n$ ${\ displaystyle n}$ iterations. Further, for a given $\varepsilon >0,$ ${\ displaystyle \ varepsilon> 0,}$ they say that many $(n,\varepsilon )$ ${\ displaystyle (n, \ varepsilon)}$ -separated if paired $d_{n}$ ${\ displaystyle d_ {n}}$ -distance between its points is not less $\varepsilon$ ${\ displaystyle \ varepsilon}$ , and the power of the largest such set is denoted by $N(n,\varepsilon )$ ${\ displaystyle N (n, \ varepsilon)}$ . Then, the topological entropy of the map $T$ ${\ displaystyle T}$ called double limit

h(T)=\lim _{\varepsilon \to 0}\limsup _{n\to \infty }{\frac {1}{n}}\log N(n,\varepsilon ).

{\ displaystyle h (T) = \ lim _ {\ varepsilon \ to 0} \ limsup _ {n \ to \ infty} {\ frac {1} {n}} \ log N (n, \ varepsilon).}

The same quantity can be determined differently: if denoted by $M(n,\varepsilon )$ ${\ displaystyle M (n, \ varepsilon)}$ least power $\varepsilon$ ${\ displaystyle \ varepsilon}$ networks then

h(T)=\lim _{\varepsilon \to 0}\limsup _{n\to \infty }{\frac {1}{n}}\log M(n,\varepsilon ).

{\ displaystyle h (T) = \ lim _ {\ varepsilon \ to 0} \ limsup _ {n \ to \ infty} {\ frac {1} {n}} \ log M (n, \ varepsilon).}

The equivalence of these definitions is easily derived from the inequalities $N(n,\varepsilon )\leq M(n,\varepsilon )\leq N(n,\varepsilon /2).$ ${\ displaystyle N (n, \ varepsilon) \ leq M (n, \ varepsilon) \ leq N (n, \ varepsilon / 2).}$ It is worth noting that both this and another definition formalize the following non-strict concept: for an unknown starting point, how much information needs to be obtained per iteration in order to predict a large number of iterations with a small fixed error .

Metric Entropy

Let be $(X,T,\mu )$ ${\ displaystyle (X, T, \ mu)}$ - a measure-preserving measurable dynamical system. By definition, the entropy of a partition $X=\bigcup _{j=1}^{n}\xi _{j}$ ${\ displaystyle X = \ bigcup _ {j = 1} ^ {n} \ xi _ {j}}$ called a number

H(\xi ):=\sum _{j=1}^{n}-\mu (\xi _{j})\log \mu (\xi _{j}),

{\ displaystyle H (\ xi): = \ sum _ {j = 1} ^ {n} - \ mu (\ xi _ {j}) \ log \ mu (\ xi _ {j}),}

determining the informational entropy of the definition of a partition element containing $\mu$ ${\ displaystyle \ mu}$ - a random point.

Iterative shredding shredding $\xi$ ${\ displaystyle \ xi}$ ,

\xi ^{(k)}=\left\{\xi _{i_{1}}\cap T^{-1}(\xi _{i_{2}})\cap \dots \cap T^{-k+1}(\xi _{i_{k}})\mid 1\leq i_{1},\dots ,i_{k}\leq n\right\}

{\ displaystyle \ xi ^ {(k)} = \ left \ {\ xi _ {i_ {1}} \ cap T ^ {- 1} (\ xi _ {i_ {2}}) \ cap \ dots \ cap T ^ {- k + 1} (\ xi _ {i_ {k}}) \ mid 1 \ leq i_ {1}, \ dots, i_ {k} \ leq n \ right \}}

determine which elements $\xi$ ${\ displaystyle \ xi}$ turns out to be a point throughout $k$ ${\ displaystyle k}$ iterations, and, accordingly, the quantity

h_{\mu }(T,\xi )=\lim {\frac {1}{k}}H(\xi ^{(k)})

{\ displaystyle h _ {\ mu} (T, \ xi) = \ lim {\ frac {1} {k}} H (\ xi ^ {(k)})}

expresses the informational entropy of such a process. Finally, the metric entropy of the mapping $T$ ${\ displaystyle T}$ as $\mu$ ${\ displaystyle \ mu}$ defined as an exact upper bound $h_{\mu }(\xi )$ ${\ displaystyle h _ {\ mu} (\ xi)}$ on all possible partitions $\xi$ ${\ displaystyle \ xi}$ :

h_{\mu }(T):=\sup _{\xi }h_{\mu }(T,\xi ).

{\ displaystyle h _ {\ mu} (T): = \ sup _ {\ xi} h _ {\ mu} (T, \ xi).}

Literature

Katok A. B. , Hasselblatt B. Introduction to the modern theory of dynamical systems = Introduction to the Modern Theory of Dynamical Systems / trans. from English A. Kononenko with the participation of S. Ferleger. - M .: Factorial, 1999 .-- 768 p. - ISBN 5-88688-042-9 .