Volume entropy

<h2 id="definition">Definition</h2>
Let (M, g) be a compact Riemannian manifold, with <a href="/facts/Universal_cover/bDRGasl7">universal cover</a> 
 
 
 
 
 
 
 M
 ~
 
 
 
 .
 
 
 {\displaystyle {\tilde {M}}.}
 
 Choose a point 
 
 
 
 
 
 
 
 x
 ~
 
 
 
 
 0
 
 
 ∈
 
 
 
 M
 ~
 
 
 
 
 
 {\displaystyle {\tilde {x}}_{0}\in {\tilde {M}}}
 
.
The volume entropy (or asymptotic volume growth) 
 
 
 
 h
 =
 h
 (
 M
 ,
 g
 )
 
 
 {\displaystyle h=h(M,g)}
 
 is defined as the limit

h
        (
        M
        ,
        g
        )
        =
        
          lim
          
            R
            →
            +
            ∞
          
        
        
          
            
              log
              ⁡
              
                (
                
                  vol
                  ⁡
                  B
                  (
                  R
                  )
                
                )
              
            
            R
          
        
        ,
      
    
    {\displaystyle h(M,g)=\lim _{R\to +\infty }{\frac {\log \left(\operatorname {vol} B(R)\right)}{R}},}

where B(R) is the ball of radius R in 
 
 
 
 
 
 
 M
 ~
 
 
 
 
 
 {\displaystyle {\tilde {M}}}
 
 centered at 
 
 
 
 
 
 
 
 x
 ~
 
 
 
 
 0
 
 
 
 
 {\displaystyle {\tilde {x}}_{0}}
 
 and vol is the Riemannian <a href="/facts/Volume_(mathematics)/aeAXCBUd">volume</a> in the universal cover with the natural Riemannian metric.
A. Manning proved that the limit exists and does not depend on the choice of the base point. This asymptotic invariant describes the exponential growth rate of the volume of balls in the universal cover as a function of the radius.

<h2 id="properties">Properties</h2>
<ul><li>Volume entropy h is always bounded above by the topological entropy htop of the geodesic flow on M. Moreover, if M has nonpositive sectional curvature then h = htop. These results are due to Manning.</li>
<li>More generally, volume entropy equals topological entropy under a weaker assumption that M is a closed Riemannian manifold without <a href="/facts/Conjugate_points/7GbGNFxF">conjugate points</a> (Freire and Mañé).</li>
<li><a href="/facts/Locally_symmetric_space/AWYuHfNp">Locally symmetric spaces</a> minimize entropy when the volume is prescribed. This is a corollary of a very general result due to Besson, Courtois, and Gallot (which also implies <a href="/facts/Mostow_rigidity/KXtuIhnr">Mostow rigidity</a> and its various generalizations due to Corlette, Siu, and <a href="/facts/William_Thurston/K1Qe8801">Thurston</a>):
Let X and Y be compact oriented connected n-dimensional smooth manifolds and f: Y → X a continuous map of non-zero <a href="/facts/Degree_of_a_map/yWxq9e4B">degree</a>. If g0 is a negatively curved locally symmetric Riemannian metric on X and g is any Riemannian metric on Y then

h
          
            n
          
        
        (
        Y
        ,
        g
        )
        vol
        ⁡
        (
        Y
        ,
        g
        )
        ≥
        
          |
          
            deg
            ⁡
            (
            f
            )
          
          |
        
        
          h
          
            n
          
        
        (
        X
        ,
        
          g
          
            0
          
        
        )
        vol
        ⁡
        (
        X
        ,
        
          g
          
            0
          
        
        )
        ,
      
    
    {\displaystyle h^{n}(Y,g)\operatorname {vol} (Y,g)\geq \left|\deg(f)\right|h^{n}(X,g_{0})\operatorname {vol} (X,g_{0}),}

and for n ≥ 3, the equality occurs if and only if (Y,g) is locally symmetric of the same type as (X,g0) and f is homotopic to a homothetic covering (Y,g) → (X,g0).</li></ul>
<h2 id="application-in-differential-geometry-of-surfaces">Application in differential geometry of surfaces</h2>
Katok's entropy inequality was recently exploited to obtain a tight asymptotic bound for the <a href="/facts/Systolic_geometry/exKrWG9e">systolic</a> ratio of surfaces of large genus, see <a href="/facts/Systoles_of_surfaces/dIVJqvsL">systoles of surfaces</a>.

<ul><li>Besson, G., Courtois, G., Gallot, S. Entropies et rigidités des espaces localement symétriques de courbure strictement négative. (French) [Entropy and rigidity of locally symmetric spaces with strictly negative curvature] Geom. Funct. Anal. 5 (1995), no. 5, 731–799</li>
<li>Katok, A.: Entropy and closed geodesics, Erg. Th. Dyn. Sys. 2 (1983), 339–365</li>
<li>Katok, A.; Hasselblatt, B.: Introduction to the modern theory of dynamical systems. With a supplementary chapter by Katok and L. Mendoza. Encyclopedia of Mathematics and its Applications, 54. Cambridge University Press, Cambridge, 1995</li>
<li>Katz, M.; Sabourau, S.: Entropy of systolically extremal surfaces and asymptotic bounds. Erg. Th. Dyn. Sys. 25 (2005), 1209-1220</li>
<li>Manning, A.: Topological entropy for geodesic flows. Ann. of Math. (2) 110 (1979), no. 3, 567–573</li></ul>

Volume entropy open-in-new

Volume entropy