Topological entropy

Topological entropy is an invariant that measures the complexity of dynamic systems . It generalizes the (measure theoretical) concept of Kolmogorow-Sinai entropy to not necessarily measure-preserving dynamic systems .

Entropy measures the chaotic nature of a dynamic system. Dynamic systems are called chaotic if their entropy is positive.

Definitions

Let be a compact topological space and a continuous map . ${\displaystyle X}$ ${\displaystyle f\colon X\to X}$

Definition according to Adler-Konheim-McAndrew

The topological entropy of the dynamic system defined by iteration of is defined as follows. ${\displaystyle h(f)}$ ${\displaystyle f}$

For a finite open cover ${\displaystyle {\mathcal {U}}}$

${\displaystyle X=\cup _{i=1}^{n}U_{i}}$

of by open sets will be denoted by ${\displaystyle X}$ ${\displaystyle U_{i}}$

${\displaystyle H({\mathcal {U}})}$

the logarithm of the minimum number of sets that already completely cover. For two open covers and we denote the cover by open sets of the form ${\displaystyle {\mathcal {U}}}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle {\mathcal {V}}}$ ${\displaystyle {\mathcal {U}}\vee {\mathcal {V}}}$

${\displaystyle U_{i}\cap V_{j}}$with . ${\displaystyle U_{i}\in {\mathcal {U}},V_{j}\in {\mathcal {V}}}$

With these notations, the topological entropy of is defined as ${\displaystyle f}$

${\displaystyle h(f)=sup_{\mathcal {U}}\lim _{n\to \infty }{\frac {1}{n}}H({\mathcal {U}}\vee f^{-1}{\mathcal {U}}\vee \ldots \vee f^{-n+1}{\mathcal {U}})}$,

where the supremum is taken over all open coverages . ${\displaystyle {\mathcal {U}}}$

Metric definition according to Bowen and Dinaburg

Let be a metric space and again a continuous map . ${\displaystyle (X,d)}$ ${\displaystyle f\colon X\to X}$

We are defining a new metric for everyone ${\displaystyle n\in \mathbb {N} }$

${\displaystyle d_{n}(x,y)=max\left\{d(f^{k}(x),f^{k}(y)):0\leq k\leq n-1\right\}}$.

For be ${\displaystyle \epsilon >0}$

${\displaystyle N(d_{n},\epsilon )}$

the maximum cardinality of a set with for all . ${\displaystyle M\subset X}$ ${\displaystyle d_{n}(m,m^{\prime })\geq \epsilon }$ ${\displaystyle m\not =m^{\prime }\in M}$

Then we define the topological entropy of by ${\displaystyle f}$

${\displaystyle h(f)=\lim _{\epsilon \to 0}\limsup _{n\to \infty }{\frac {1}{n}}\log N(d_{n},\epsilon )}$.

If is a compact metric space, then this definition agrees with that of Adler-Konheim-McAndrew. ${\displaystyle X}$

Examples

• For an isometry or more generally a Lipschitz continuous mapping with Lipschitz constant is . ${\displaystyle f}$ ${\displaystyle \leq 1}$ ${\displaystyle h(f)=0}$
• For a logistic picture with is . ${\displaystyle f(x)=\mu x(1-x)}$ ${\displaystyle \mu >2+{\sqrt {5}}}$ ${\displaystyle h(f)=\ln(2)}$
• The topological entropy of the angle doubling map is . ${\displaystyle \ln(2)}$
• The topological entropy of the shift mapping on symbols is . ${\displaystyle k}$ ${\displaystyle \ln(k)}$

properties

• ${\displaystyle h(f^{k})=kh(f)}$.
• For a homeomorphism is . ${\displaystyle h(f^{-1})=h(f)}$
• ${\displaystyle h(gfg^{-1})=h(f)}$for any homeomorphism and any . ${\displaystyle g}$ ${\displaystyle f}$
• The topological entropy only depends on the topology , not on the underlying metric.

literature

• Luis Barreira, Claudia Valls: Dynamical systems. An introduction. Translated from the 2012 Portuguese original. University text. Springer, London 2013, ISBN 978-1-4471-4834-0 .
• RL Adler, AG Konheim, MH McAndrew: Topological entropy. In: Trans. Amer. Math. Soc. 114, 1965, pp. 309-319.
• Rufus Bowen: Entropy for group endomorphisms and homogeneous spaces. In: Trans. Amer. Math. Soc. 153, 1971, pp. 401-414.
• EI Dinaburg: A connection between various entropy characterizations of dynamical systems. In: Izv. Akad. Nauk SSSR Ser. Mat. 35, 1971, pp. 324-366. (Russian)

Individual evidence

1. ↑ Rufus Bowen: Entropy for group endomorphisms and homogeneous spaces. In: Trans. Amer. Math. Soc. 153, 1971, pp. 401-414.
2. ^ Jacques M. Bahi, Christophe Guyeux: Discrete dynamical systems and chaotic machines. Theory and applications. Chapman & Hall / CRC Numerical Analysis and Scientific Computing. CRC Press, Boca Raton, FL 2013, ISBN 978-1-4665-5450-4 .