LECTURES ON SRB MEASURES 1. SRB measures for hyperbolic

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<ul><li><p>LECTURES ON SRB MEASURES</p><p>YAKOV PESIN</p><p>1. SRB measures for hyperbolic attractors</p><p>1.1. Topological attractor. M - a compact smooth Riemannian manifold,f : M M a C2 (or C1+) diffeomorphism.</p><p>U M open and f(U) U - a trapping region.</p><p> =n0 f</p><p>n(U) a topological attractor for f . We allow the case = M .</p><p>Exercise 1. Show that is compact, f -invariant and maximal (i.e., if U is invariant, then ).</p><p>1.2. Natural measures. m is volume, mU =1</p><p>m(U)m|U is the normalizedvolume in U ,</p><p>(1.1) n =1</p><p>n</p><p>n1k=0</p><p>fkmU</p><p>is an evolution of m.</p><p>Exercise 2. Show that the sequence n is compact in the weak topology.</p><p>There is nk a natural measure for f on . If and only if for anyh C1(M): </p><p>h(x)dn =</p><p>1</p><p>n</p><p>n1k=0</p><p>h(fk(x))dmU </p><p>hd.</p><p>Exercise 3. Show that is supported on and is f -invariant.</p><p>1.3. Basin of attraction. is an (ergodic) measure on .</p><p>B ={x U : 1</p><p>n</p><p>n1k=0</p><p>h(fk(x))</p><p>hd for any h C1(M)</p><p>}basin of attraction of . We say that has positive basin of attraction if(B) &gt; 0.</p><p>Date: April 18, 2016.</p><p>1</p></li><li><p>2 YAKOV PESIN</p><p>A (ergodic) natural measure on the attractor is a physical measureif its basin of attraction has positive volume. An attractor with a physicalmeasure is called a Milnor attractor.</p><p>1.4. Hyperbolic measures.</p><p>(x, v) = lim supn</p><p>1</p><p>nlog dfnv, x M.v TxM</p><p>the Lyapunov exponent of v at x.</p><p>(x, ) takes on finitely many values, 1(x) p(x)(x), p(x) dimM and (f(x)) = (x), i.e., the values of the Lyapunov exponent areinvariant functions, also p(f(x)) = p(x).</p><p> is hyperbolic if i(x) 6= 0 and 1(x) &lt; 0 &lt; p(x)(x) that is</p><p>1(x) . . . k(x) &lt; 0 &lt; k+1(x) p(x)(x).</p><p>If is ergodic, then i(x) = i() and p(x) = p() for a.e. x that is</p><p>1() . . . k() &lt; 0 &lt; k+1() p().If is hyperbolic, then for a.e. x </p><p>(1) TxM = Es(x) Eu(x) where</p><p>Es(x) = {v TxM : (x, v) &lt; 0}, Es(x) = {v TxM : (x, v) &lt; 0}stable and unstable subspaces at x and(a) dfEs,u(x) = Es,u(f(x)),(b) (Es(x), Eu(x)) K(x);</p><p>(2) there are V s(x), V u(x) stable and unstable local manifolds at x:(a) we have</p><p>d(fn(x), fn(y)) C(x)n(x)d(x, y), y V s(x), n 0,d(fn(x), fn(y)) C(x)n(x)d(x, y), y V u(x), n 0</p><p>(b) C(x) &gt; 0, K(x) &gt; 0,</p><p>C(f(x)) C(x)e(x), K(f(x)) K(x)e(x),0 &lt; (x) &lt; 1, (x) &gt; 0 and</p><p>(f(x)) = (x), (f(x)) = (x).</p><p>(3) r(x) the size of local manifolds and r(f(x)) r(x)e(x).Exercise 4. Show that V u(x) for every x (for which the localunstable manifold is defined.</p><p>Fix 0 &lt; &lt; 1 and set = {x : 0 &lt; (x) &lt; }. is invariant andthere is s.t. () &gt; 0. We set = .</p><p>` &gt; 1, ` = {x : C(x) `, K(x) 1`} regular set of level `.</p></li><li><p>LECTURES ON SRB MEASURES 3</p><p>(1) ` `+1,`1 ` = ;</p><p>(2) the subspaces Es,u(x) depend continuously on x ; in fact, Holdercontinuously:</p><p>dG(Es,u(x), Es,u(y)) M`d(x, y),</p><p>where dG is the Grasmannian distance in TM ;(3) the local manifolds V s,u(x) depend continuously on x ; in fact,</p><p>Holder continuously:</p><p>dC1(Vs,u(x), V s,u(y)) L`d(x, y);</p><p>(4) r(x) r` for all x `.</p><p>1.5. SRB measures. We can assume that ` are compact and we canchoose ` s.t. (`) &gt; 0. For x ` and small ` &gt; 0 set</p><p>Q`(x) =</p><p>yB(x,`)`</p><p>V u(y).</p><p>Let ` be the partition of Q`(x) by Vu(y), u(y) the conditional measures</p><p>generated by on V u(y) and mV u(y) the leaf volume on Vu(y). Both mea-</p><p>sures are probability (normalized) measures on V u(y).</p><p> is an SRB measure if is hyperbolic and for every ` with (`) &gt; 0,a.e. x ` and a.e. y B(x, `) `, we have u(y) mV u(y).</p><p>For y `, z V u(y) and n &gt; 0 set</p><p>un(y, z) =</p><p>n1k=0</p><p>Jac(df |Eu(fk(z)))Jac(df |Eu(fk(y)))</p><p>.</p><p>Exercise 5. Show that</p><p>(1) for every &gt; 0 there is N &gt; 0 s.t. for every n Nmaxy`</p><p>maxzV u(y)</p><p>|un(y, z) u(y, z)| ;</p><p>in particular,</p><p>u(y, z) = limn</p><p>un(y, z) =k=0</p><p>Jac(df |Eu(fk(z)))Jac(df |Eu(fk(y)))</p><p>;</p><p>(2) u(y, z) depends continuously on y ` and z V u(y);(3) u(y, z)u(z, w) = u(y, w).</p><p>If is an SRB measure, then</p><p>du(y)(z) = u(y)1u(y, z)dmV u(y)(z),</p><p>where</p><p>(1.2) u(y) =</p><p>V u(y)</p><p>u(y, z) dmV u(y)(z)</p></li><li><p>4 YAKOV PESIN</p><p>is the normalizing factor; in particular, u(y)1u(y, z) is the density of theSRB measure.</p><p>1.6. Ergodic properties of SRB measures.</p><p>Theorem 1.1 (Ledrappier). Let be an SRB measure on an attractor for a C1+ diffeomorphism. Then there are An , n = 1, 2, . . . s.t.</p><p>(1) f(An) = An,n0 = , (An) &gt; 0 for n &gt; 0 and (A0) = 0;</p><p>(2) f |An is ergodic;(3) for each n &gt; 0 there are mn 1 and Bn An s.t. the sets f i(Bn)</p><p>are disjoint for i = 0, . . . ,mn 1 and fmn(Bn) = Bn, fmn |Bn is aBernoulli diffeomorphism;</p><p>(4) for each n &gt; 1 there are `n and xn `n s.t.</p><p>An =mZ</p><p>fm(Q`n(xn));</p><p>In addition, one can show that a hyperbolic measure on is an SRBmeasure if and only if</p><p>(1) (B) = 1;(2) the Kolmogorov-Sinai entropy h(f) of is given by the entropy</p><p>formula:</p><p>h(f) =</p><p>i(x)&gt;0</p><p>i(x) d(x);</p><p>For smooth measures (which are a particular case of SRB measures) theupper bound for the entropy was obtained by Margulis (and extension toarbitrary Borel measures by Ruelle) and the lower bound was proved byPesin (thus implying the entropy formula in this case). The extension toSRB measures was given by Ledrappier and Strelcyn. The fact that a hyper-bolic measure satisfying the entropy formula is an SRB measure was provedby Ledrappier and for arbitrary (not necessarily hyperbolic) measures thisresult was obtained by Ledrappier and Young.</p><p>The limit measures for the sequence of measures (1.1) are natural candi-dates for SRB measures.</p><p>Exercise 6. Construct f s.t. n , (B) &gt; 0 and(1) is not hyperbolic;(2) is hyperbolic but is not an SRB measure.</p><p>In this regard we state the following result by Tsujii.</p><p>Theorem 1.2. Let be an attractor for a C1+ diffeomorphism f andsuppose there is a positive Lebesgue measure set S such that for everyx S</p><p>(1) the sequence of measures 1nn1</p><p>k=0 x converges weakly to an ergodicmeasure which we denote by x;</p><p>(2) the Lyapunov exponents at x coincide with the Lyapunov exponentsof the measure x;</p></li><li><p>LECTURES ON SRB MEASURES 5</p><p>(3) the measure x has no zero and at least one positive Lyapunov ex-ponent.</p><p>Then x is an SRB measure for Lebesgue almost every x S.</p><p>It follows from Theorem 1.1 that f admits at most countably many er-godic SRB measures. J. Rodriguez Hertz, F. Rodriguez Hertz, R. Ures andA. Tahzibi have shown that a topologically transitive C1+ surface diffeo-morphism can have at most one SRB measure but the result is not true indimension higher than two, see Section 2.4.</p><p>1.7. Hyperbolic attractors.</p><p>1.7.1. Definition of hyperbolic attractors. a topological attractor for f . Itis (uniformly) hyperbolic if for each x there is a decomposition of thetangent space TxM = E</p><p>s(x)Eu(x) and constants c &gt; 0, (0, 1) s.t. foreach x :</p><p>(1) dxfnv cnv for v Es(x) and n 0;(2) dxfnv cnv for v Eu(x) and n 0.</p><p>Es(x) and Eu(x) are stable and unstable subspaces at x.</p><p>Exercise 7. Show that Es(x) and Eu(x) depend continuously on x.</p><p>In particular, (Es(x), Eu(x)) is uniformly away from zero. In fact, Es(x)and Eu(x) depend Holder continuously on x.</p><p>For each x there are V s(x) and V u(x) stable and unstable local man-ifolds at x. They have uniform size r, depend continuously on x in the C1</p><p>topology and V u(x) for any x .</p><p>1.7.2. An example of hyperbolic attractor. Consider the solid torus P =D2S1. We use coordinates (x, y, ) on P ; x and y give the coordinates onthe disc, and is the angular coordinate on the circle. Fixing parametersa (0, 1) and , (0,min{a, 1 a}), define a map f : P P by</p><p>f(x, y, ) = (x+ a cos , y + a sin , 2).</p><p>The action of f on P may be described as follows:</p><p>(1) Take the torus and slice it along a disc so that it becomes a tube.(2) Squeeze this tube so that its cross-sections are no longer circles of</p><p>radius 1, but ellipses with axes of length and .(3) Stretch the tube along its axis until it is twice its original length.(4) Wrap the resulting longer, skinnier tube twice around the z-axis</p><p>within the original solid torus.(5) Glue the ends of the tube together.</p><p>P is a trapping region and =n0 f</p><p>n(P ) is the attractor for f , known asthe Smale-Williams solenoid.</p></li><li><p>6 YAKOV PESIN</p><p>1.7.3. Existence of SRB measures.</p><p>Theorem 1.3 (Sinai, Ruelle, Bowen). Assume that f is C2 (or C1+). Thefollowing statements hold:</p><p>(1) Every limit measure of the sequence of measures n is an SRBmeasure on .</p><p>(2) There are at most finally many ergodic SRB measures on .(3) If f | is topologically transitive, then the sequence of measures n</p><p>converges to a unique SRB-measure on and B has full measurein U .</p><p>Let be an SRB measure on and A its ergodic component of pos-itive measure. Then there is x s.t. A =</p><p>mZ f</p><p>m(Q(x)), whereQ(x) =</p><p>yV u(x) V</p><p>s(y). Note that Q(x) is open and contains a ball of</p><p>radius &gt; 0. This implies that has only finitely many ergodic compo-nents and they are open (mod 0). It follows that there are at most finallymany ergodic SRB measures on and if f | is topologically transitive, thenthe SRB measure is unique.</p><p>To prove existence consider x and V = V u(x). For y V u(x) let</p><p>c0 = 1 and cn =(n1k=0</p><p>Jac(df |Eu(fk(x))))1</p><p>for n 1</p><p>and consider the sequence of measures given by</p><p>dn(x)(y) = cnu(fn(x), y)dmfn(V u(x))(y).</p><p>Lemma 1.4. n(x) = fn 0(x).</p><p>Proof of the lemma. For a measurable set F V u(x), w F andy = fn(w) fn(F ),n(F ) = n(f</p><p>n(fn(F ))</p><p>=</p><p>fn(F )</p><p>cnu(fn(x), fn(w))</p><p>n1k=0</p><p>Jac(df |Eu(fk(w)) dmV u(x)(w)</p><p>=</p><p>fn(F )</p><p>cnu(fn(x), fn(w))</p><p>n1k=0</p><p>Jac(df |Eu(fk(w)) (x,w)1d0(x)(w)</p><p>=</p><p>fn(F )</p><p>d0(x)(w) = 0(fn(F )).</p><p>Let</p><p>n =1</p><p>n</p><p>n1k=0</p><p>k =1</p><p>n</p><p>n1k=0</p><p>fk 0,</p><p>where we view k and n as measures on . We shall show that every limitmeasure for this sequence of measures is an SRB-measure. In fact, every</p></li><li><p>LECTURES ON SRB MEASURES 7</p><p>SRB-measure can be constructed in this way, i.e., it can be obtained as thelimit measure for a subsequence of measures n.</p><p>We have that</p><p>dn(x)(y) = cnu(fn(x), y)mun(x)(y),</p><p>where</p><p>mun(x) =1</p><p>n</p><p>n1k=0</p><p>mfk(V u(x)).</p><p>To prove that is an SRB measure. Let z be a Lebesgue point of , so that(B(z, r)) &gt; 0 for every r &gt; 0. Consider the set Q(z) and its partition into unstable local manifolds V u(y), y B(z, r). We identify the factorspace Q(z)/ with W = V s(z) and we denote by Vn = fn(V ). Set</p><p>An = {y W : V u(y) Vn 6= },Bn = {y W : V u(y) Vn 6= },</p><p>Cn = An \Bn, Dn =yBn</p><p>V u(y),</p><p>Fn = {y Vn : du(y, Vn) 2r},where du is the distance in Vn induced by the Riemannian metric. Notethat Bn An, Dn Fn and that An, Bn and Cn are finite set. If h is acontinuous function on with support in Q(z), then</p><p>h dn =</p><p>Q(z)</p><p>h dn</p><p>=yAn</p><p>V u(z)Vn</p><p>h dn</p><p>=yCn</p><p>V u(z)Vn</p><p>h dn +yBn</p><p>V u(z)Vn</p><p>h dn</p><p>= I(1)n + I(2)n .</p><p>We have that</p><p>n(Fn) = cn</p><p>Fn</p><p>u(fn(x), y) dmfn(V u(x))(y)</p><p>= cn</p><p>fn(Fn)</p><p>u(fn(x), fn(w))n1k=0</p><p>Jac(df |Eu(fk(w))))mV u(x)(w)</p><p>=</p><p>fn(Fn)</p><p>u(x,w) dmV u(x)(w) C0(fn(Fn)) C(+ )n2r,</p><p>where C &gt; 0 is a constant and is sufficiently small so that + &lt; 1. Itfollows that</p><p>I(2)n Cn(Dn) Cn(Fn) C</p><p>n.</p></li><li><p>8 YAKOV PESIN</p><p>Denote by n the measure on W , which is the uniformly distributed pointmass on Cn. We have that</p><p>h dn = cn</p><p>yCn</p><p>u(fn(x), y)</p><p>V u(y)</p><p>h(w)u(y, w) dmV u(y)(w)</p><p>=</p><p>Wcn</p><p>u(fn(x), y)u(y) dn(y)</p><p>V u(y)</p><p>h(w)u(y, w)</p><p>u(y)dmV u(y)(w),</p><p>where u(y) is given by (1.2). Hence,</p><p>I(2)n =1</p><p>n</p><p>n1k=0</p><p>Wck</p><p>u(fk(x), y)u(y) dk(y)</p><p>V u(y)</p><p>h(w)u(y, w)</p><p>u(y)dmV u(y)(w),</p><p>The desired result is now a corollary of the following statement.</p><p>Lemma 1.5. Let n be a sequence of Borel probability measures on Q(z)such that</p><p>(1) if (n, un(y)) is the systems of conditional measures for n with re-</p><p>spect to the partition , so that n is a measure on the factor spaceW = Q(z)/ and un(y)) is a measure on V</p><p>u(y), then</p><p>dun(y)(w) = Pn(y, w)dmV u(y)(w),</p><p>where Pn(y, w) is a continuous function on Q(z);(2) there is a sequence of numbers n` s.t. the sequence of measures n`</p><p>converges in the weak topology to a measure on Q(z);(3) the sequence of functions Pn`(y, w) converges uniformly in Q(z) to</p><p>a continuous function P (y, w).</p><p>Then the system of conditional measures for with respect to the partition has the form (, u(y)) where is the measure on the factor space W thatis the limit of measures n` and </p><p>u(y) is a measure on V u(y) for which</p><p>du(y)(w) = P (y, w)dmV u(y)(w).</p><p>We stress that in the definition of the sequence of measures n one canreplace the local unstable manifold V u(x) with any admissible manifold, i.e.,a local manifold passing through x and sufficiently close to V u(x) in the C1</p><p>topology.</p><p>If f | is topologically transitive, then the SRB measure is unique andhence, the sequence of measures converges to .</p><p>Exercise 9. Show that the sequence of measures n converges to .</p><p>In the particular, case when = M that is f is a C2 Anosov diffeomor-phism, the above theorem guaranties existence and uniqueness of the SRBmeasure for f (provided f is topologically transitive). Reversing the timewe obtain the unique SRB measure for f1. One can show that = ifand only if is a smooth measure.</p></li><li><p>LECTURES ON SRB MEASURES 9</p><p>2. SRB measures for partially hyperbolic attractors</p><p>2.1. Definition of partially hyperbolic attractors. a topological at-tractor for f . It is (uniformly) partially hyperbolic if for each x thereis a decomposition of the tangent space TxM = E</p><p>s(x)Ec(x)Eu(x) andfor n 0:</p><p>(1) dxfnv cnv for v Es(x);(2) c1n1dxfnv cn2v;(3) dxfnv cnv for v Eu(x).</p><p>Es(x), Ec(x) and Eu(x) are strongly stable, central and strongly unstablesubspaces at x. They depend (Holder) continuously on x. In particular, theangle between any two of them is uniformly away from zero.</p><p>For each x there are V s(x) and V u(x) strongly stable and stronglyunstable local manifolds at x. They have uniform size r, depend continu-ously on x in the C1 topology and V u(x) for any x .</p><p>An example of a hyperbolic attractor is a map which the direct productof a map f with a hyperbolic attractor and Id map of any manifold.</p><p>2.2. u-measures. is a u-measure if for every x and y B(x, ) ,we have u(y) mV u(y).</p><p>Theorem 2.1 (Pesin, Sinai). Any limit measure of the sequence of measuresn is a u-measure and so is any limit measure of the sequence of measuresn.</p><p>Properties of u-measures:</p><p>(1) Any measure whose basin has positive volume is a u-measure;(2) If there is a unique u-measure for f , then its basin has full volume</p><p>in the topological basin of attraction;(3) Every ergodic component of a u-measure is again a u-measure.</p><p>Exercise 10. Give an example of a map f with a partially hyperbolicattractor and a u-measure whose basin has zero volume.</p><p>2.3. u-measures with negative central exponents. We say that f hasnegative (positive) central exponents (with respect to ) if there exists an in-variant subset A with (A) &gt; 0 s.t. the Lyapunov exponents (x, v) &lt; 0(respectively, (x, v) &gt; 0) for every x A and every vector v Ec(x).</p><p>If f has negative central exponents on a set A of full measure with respectto a u-measure , then is an SRB measure for f .</p><p>Theorem 2.2. Assume that f has negative central exponents on an invari-ant set A of positive measure with respect to a u-measure for f . Then thefollowing statements hold:</p></li><li><p>10 YAKOV PESIN</p><p>(1) Every ergodic component of f |A of positive -measure is open (mod 0);in particular, the set A is open (mod 0) (that is there exists an openset U s.t. (A4U) = 0).</p><p>(2) If for -almost every x the trajectory {fn(x)} is dense in supp(),then f is ergodic with respect to .</p><p>Proof. f |A has non-zero Lyapunov exponents with TxM = E(x) Eu(x) for every z A where E(x) =...</p></li></ul>

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