1.1 Measure rigidity

We start with the classification of stationary measures. Recall that given a measure $\mu $ on H, we denote by $\Gamma _{\mu }$ the closed subgroup generated by the support of $\mu $ .

Using the properties of H-expanding measures, the above theorem is deduced by an iterative application of the recent measure classification results of Eskin–Lindenstrauss [Reference Eskin and Lindenstrauss30]; see §4.1. The proof is similar to the argument Eskin–Lindenstrauss use to show that their result implies Benoist–Quint’s measure classification.

In certain cases, the last conclusion of Theorem 1.1 allows us to show that $\nu $ is actually H-invariant; see Proposition 8.2 and also the corollary below. For its statement, recall that a discrete subgroup $\Lambda $ is said to be a lattice in G if $X=G/\Lambda $ admits a G-invariant probability measure $m_X$ . In this case, we refer to $m_X$ as the Haar measure on X. A lattice $\Lambda $ in a connected semisimple Lie group G without compact factors is said to be irreducible if $\Lambda \cap S$ is not a lattice in S for every nontrivial proper connected normal subgroup S of G. Equivalently, $S\Lambda $ is dense in G for every such S.

We note that finite $\Gamma _{\mu }$ -orbits do only occur when $\Gamma _{\mu }$ is virtually contained in a conjugate of $\Lambda $ . The proof of part (i) of the corollary above relies on Margulis’ arithmeticity theorem and a careful analysis of stationary measures charging an orbit of the centralizer of $\Gamma _{\mu }$ , which is carried out in §4.2. The last statement in part (ii) additionally requires countability of finite $\Gamma _{\mu }$ -orbits, which follows from a general countability result for homogeneous subspaces in §5.

1.2 Recurrence and Lyapunov functions

Now, we assume in addition that $\Lambda $ is a lattice and that $\mu $ has finite exponential moments. Under certain assumptions including semisimplicity of the noncompact part of the Zariski closure of $\Gamma _{\mu }$ , Eskin–Margulis [Reference Eskin and Margulis31] and later Benoist–Quint [Reference Benoist and Quint8] have shown that the random walk on $X=G/\Lambda $ given by $\mu $ satisfies strong recurrence properties. If $\delta _x$ denotes the Dirac measure at $x\in X$ and $\mu ^{* n} $ is the n-fold convolution power of $\mu $ , these recurrence statements take the general form that $\mu ^{*n}*\delta _x(M)$ is close to $1$ for large n, where $M\subset X$ is a certain compact set. We obtain analogous results for H-expanding measures.

Theorem 1.4. Let $\Lambda $ be a lattice in a real Lie group G. Let $H\leqslant G$ be a connected semisimple subgroup without compact factors and with finite center. Let $\mu $ be an H-expanding probability measure with finite exponential moments on H. Let Y be a $\Gamma _{\mu }$ -ergodic homogeneous subspace of $X=G/\Lambda $ or the empty set. Finally, let $K_L$ be any compact subset of the centralizer L of $\Gamma _{\mu }$ in G, and set $\mathcal {N}=K_LY$ . Then for any compact subset $Z\subset X\setminus \mathcal {N}$ and $\delta>0$ there exists a compact subset $M_{Z,\delta }$ of $X\setminus \mathcal {N}$ such that

$$ \begin{align*} \mu^{*n}*\delta_x(M_{Z,\delta})\ge 1-\delta \end{align*} $$

for every $n\ge 0$ and $x\in Z$ .

Loosely speaking, the basic case (with $Y=\emptyset $ ) implies that the random walk does not spend too much time in the cusp. The general case ensures that the random walk also does not accumulate near lower-dimensional homogeneous subspaces.

This result will be proved in §7.1 using height functions on $X=G/\Lambda $ satisfying a contraction property with respect to the averaging operator $A_{\mu }$ defined by

$$ \begin{align*} A_{\mu}(f)(x)=\int_Gf(gx)\mathop{}\!\mathrm{d}\mu(g) \end{align*} $$

for nonnegative Borel functions f on X. Heuristically, if $\beta $ is a function on X with values in $[0,\infty ]$ such that

(1.1) $$ \begin{align} A_{\mu}(\beta)\le a\beta +b \end{align} $$

for constants $a\in (0,1)$ and $b\ge 0$ , then, with high probability, the dynamics of the random walk are driven towards the part of the space where $\beta $ takes values below a certain threshold, and $X_{\infty }=\beta ^{-1}(\{\infty \})$ acts as a repeller. Putting this heuristic into quantitative terms yields strong recurrence properties of the random walk away from $X_{\infty }$ , which play a key role not only in the proof of Theorem 1.4, but also for orbit closure and equidistribution results to be described in what follows.

Ideas of this kind have a rich history in the theory of stochastic processes and dynamical systems and trace back to the work of Foster [Reference Foster35] and Lyapunov [Reference Liapounoff63] (see also [Reference Meyn and Tweedie68, §15]). In homogeneous dynamics, they first appear in Eskin–Margulis–Mozes’ work on a quantitative version of the Oppenheim conjecture [Reference Eskin, Margulis and Mozes32]. In the study of random walks on homogeneous spaces, height functions were first systematically used by Eskin–Margulis [Reference Eskin and Margulis31] to establish recurrence properties. Functions satisfying the contraction property (1.1) are therefore often referred to either as Lyapunov functions or Margulis functions.

To obtain our results, we will need to construct two types of Lyapunov functions.

• ○ Height functions with respect to the cusps (§6.1): First, corresponding to the case $Y=\emptyset $ in Theorem 1.4, we require a Lyapunov function $\beta _{\infty }$ that stays bounded on a prescribed compact subset Z of X and tends to infinity when leaving compact parts of the space into the cusps of X. Its role is to rule out escape of mass, that is, ensure that the random walk does not escape to infinity. For this case, we will show that we can use the height function constructed by Benoist–Quint [Reference Benoist and Quint8]. Indeed, as it turns out, the algebraic condition that is imposed in their paper on the Zariski closure of $\Gamma _{\mu }$ is only crucially used to ensure an expansion property in representations of H, so that the proof also goes through under our H-expansion assumption.

• ○ Height functions with respect to singular subspaces (§6.2): Secondly, corresponding to the case of a lower-dimensional homogeneous subspace Y in Theorem 1.4, we also need Lyapunov functions which blow up near the singular subspace Y. These are used to ensure that random walk trajectories do not accumulate near Y when starting outside of it. Here, we give a construction inspired by the work of Eskin–Mirzakhani–Mohammadi [Reference Eskin, Mirzakhani and Mohammadi34] for random walks on moduli space. This will allow us to avoid the use of the first return cocycles and operators appearing in [Reference Benoist and Quint9, Reference Benoist and Quint10], and to obtain a height function $\beta _{\mathcal {N}}$ which satisfies the contraction property (1.1) with respect to $A_{\mu }$ itself.

1.3 Orbit closures and equidistribution

Measure classification and recurrence properties at hand, the next step is the question of equidistribution of random walks with respect to a homogeneous probability measure, which, once established, yields orbit closure descriptions analogous to Ratner’s theorems in unipotent dynamics.

Let $\Gamma ^+_{\mu }$ be the closed semigroup generated by the support of $\mu $ . If $\Gamma _{\mu }$ has Zariski dense image in $\operatorname {\mathrm {Ad}} (H)$ , then it is proved in [Reference Benoist and Quint10] that the orbit closure $\overline {\Gamma _{\mu }^+x}$ is a homogeneous subspace of X inside which the random walk equidistributes. Our next result is a generalization of this and other rigidity results for the random trajectory of points proved in [Reference Benoist and Quint10, Reference Prohaska and Sert79, Reference Simmons and Weiss93].

In statements (ii) and (iii) of the theorem above, convergence is understood with respect to the weak* topology, where weak* convergence of a sequence of probability measures $\nu _n$ on X to a finite measure $\nu $ on X is defined to mean that

(1.2) $$ \begin{align} \lim_{n\to\infty}\int_Xf\mathop{}\!\mathrm{d}\nu_n=\int_Xf\mathop{}\!\mathrm{d}\nu \end{align} $$

for every compactly supported continuous test function f on X. In case the limit measure $\nu $ is a probability measure, weak* convergence $\nu _n\to \nu $ implies that equation (1.2) holds for any bounded continuous function f on X.

Theorem 1.6 will be proved in §7.2. It has the nontrivial topological consequence that any infinite $\Gamma _{\mu }^+$ -orbit in X is dense in a homogeneous subspace of positive dimension. In the G-expanding case with an irreducible lattice $\Lambda <G$ , this means that every infinite $\Gamma _{\mu }^+$ -orbit in $X=G/\Lambda $ is dense.

1.4 The space of homogeneous measures

Given a closed subgroup $\Gamma $ of the Lie group G, we consider

$$ \begin{align*} \mathcal{S}(\Gamma)=\{\Gamma\text{-invariant }\Gamma\text{-ergodic homogeneous subspaces }Y\subset X\}, \end{align*} $$

where, as before, $X=G/\Lambda $ is the quotient of G by a lattice $\Lambda $ . By definition, associated to each $Y\in \mathcal {S}(\Gamma )$ is a $\Gamma $ -invariant and ergodic probability measure $\nu _Y$ with support Y. This defines an embedding of $\mathcal {S}(\Gamma )$ into the space of probability measures on X, which we use to endow $\mathcal {S}(\Gamma )$ with the weak* topology. In the unipotent case, Mozes–Shah [Reference Mozes and Shah73] proved that convergence of homogeneous subspaces in this topology behaves in a very rigid way. Benoist–Quint [Reference Benoist and Quint10, §1.3] later obtained a version of this result for a subgroup $\Gamma $ that is Zariski dense in a semisimple group. Following their strategy, we obtain similar results in our setup.

Given a subset Z of X, let us write $\mathcal {S}_Z(\Gamma )=\{Y\in \mathcal {S}(\Gamma )\mid Y\cap Z\neq \emptyset \}$ and denote by $\delta _{\infty }$ the Dirac measure at $\infty $ in the one-point compactification $\overline {X}=X\cup \{\infty \}$ of X.

This proposition is a manifestation of strong rigidity of the $\Gamma _{\mu }$ -invariant and ergodic homogeneous subspaces. For example, given a compact subset $Z \subset X$ and $Y_{\infty } \in \mathcal {S}(\Gamma _{\mu })$ with $Z^\circ \cap Y_{\infty } \neq \emptyset $ , if for a sequence $Y_n \in \mathcal {S}(\Gamma _{\mu })$ we have $Y_n \cap Z \to Y_{\infty } \cap Z$ in the Hausdorff metric, then one can conclude that $Y_n \to Y_{\infty }$ in $\mathcal {S}(\Gamma _{\mu })$ . In particular, the weak* topology on $\mathcal {S}(\Gamma _{\mu })$ coincides with the restriction to $\mathcal {S}(\Gamma _{\mu })$ of the Fell topology on closed subsets of X.

Another consequence of Proposition 1.8 is the following equidistribution result for sequences of homogeneous subspaces in the case that $\Gamma _{\mu }$ has discrete centralizer in G.

Corollary 1.9. Retain the notation and assumptions of Theorem 1.6, and assume in addition that the centralizer $C_G(\Gamma _{\mu })$ of $\Gamma _{\mu }$ in G is discrete. Let $Y_{\infty }\in \mathcal {S}(\Gamma _{\mu })$ , and consider the set

$$ \begin{align*} \mathcal{S}(\Gamma_{\mu},Y_{\infty})=\{Y\in\mathcal{S}(\Gamma_{\mu})\mid Y\subset Y_{\infty}\} \end{align*} $$

of ergodic homogeneous subspaces of $Y_{\infty }$ . Suppose that $(Y_n)_n$ is a sequence in $\mathcal {S}(\Gamma _{\mu },Y_{\infty })$ such that for every fixed $Y\in \mathcal {S}(\Gamma _{\mu },Y_{\infty })\setminus \{Y_{\infty }\}$ one has $Y_n\not \subset Y$ for all but finitely many n, and such that no subsequence of $(Y_n)_n$ escapes to infinity. Then $Y_n\to Y_{\infty }$ in $\mathcal {S}(\Gamma _{\mu })$ .

Here, by ‘escape to infinity’ we mean weak* convergence towards the Dirac measure $\delta _{\infty }$ at infinity.

The proofs of both statements above will be given in §7.3.

1.5 Birkhoff genericity

We still assume that $\Lambda $ is a lattice in the Lie group G. Let $(a(t))_{t\in \mathbb {R} }$ be a one-parameter $\operatorname {\mathrm {Ad}}$ -diagonalizable subgroup of H and $\nu $ a probability measure on $X=G/\Lambda $ invariant under $a(t)$ for every $t\in \mathbb {R}$ . We say that a Radon measure $\eta $ on H is $a(t)$ -Birkhoff generic at $x\in X$ with respect to $\nu $ if

$$ \begin{align*} \frac1T\int_0^T\delta_{a(t)hx}\mathop{}\!\mathrm{d} t\to\nu \end{align*} $$

in the weak* topology as $T\to \infty $ for $\eta $ -almost every $h\in H$ . It was first noticed by Simmons–Weiss [Reference Simmons and Weiss93] that, in certain situations, pathwise equidistribution of random walks as in Theorem 1.6(iii) can be used to deduce Birkhoff genericity of fractal measures $\eta $ on unipotent subgroups of H with respect to the Haar measure on X, which has consequences in Diophantine approximation thanks to the Dani correspondence principle. Recently, more results were obtained in this direction in [Reference Prohaska and Sert79]. Both of these papers only deal with cases corresponding to $H=G$ in our setup. We are going to extend the existing results by removing this restriction. Even in the case where $H=G$ , we obtain Birkhoff genericity for more general one-parameter subgroups and fractal measures, which will also give new results on Diophantine approximation (see §1.6).

The one-parameter subgroups to which our results apply are required to satisfy certain expansion condition with respect to a unipotent subgroup of H. To phrase it, we use the concept of an a-expanding subgroup of H introduced in [Reference Shi91]. Namely, given an $\operatorname {\mathrm {Ad}}$ -diagonalizable element $a\in H$ , a connected $\operatorname {\mathrm {Ad}}$ -unipotent subgroup U of H normalized by a is said to be a-expanding if for any nontrivial irreducible representation of H on a finite-dimensional real vector space V, the subspace $V^U$ of U-fixed vectors is expanded by a, that is, $\lim _{n\to \infty }a^{-n} v=0$ for any $v\in V^U$ . If the projection of a to each simple factor of H is nontrivial, then certain horospherical subgroups of H are a-expanding. For example, this holds for the unstable horospherical subgroup

of a; see §3.2.

Now, let U be an $a(1)$ -expanding subgroup contained in the unstable horospherical subgroup $H^+_{a(1)}$ of $a(1)$ . We wish to introduce a family of measures on U which are generated by random walks, in a sense to be made precise in what follows. Let $A'=\{a(t)\mid t\in \mathbb {R}\}$ , K be a maximal compact subgroup of H, and $K'=C_K(A')\cap N_H(U)$ . Here and hereafter, $C_K(A')$ denotes the centralizer of $A'$ in K and $N_H(U)$ the normalizer of U in H. We set and denote by $\lambda $ the function which associates to $g\in P$ the real parameter of its $A'$ component in its $K'A'U$ factorization; that is, $\lambda (g)=t \in \mathbb {R}$ for $g=k a(t) u\in K'A'U$ . Finally, given $\omega =(g_1,g_2, \dots )\in P^{\mathbb {N}}$ and $n\in \mathbb {N}$ , let $k_{\omega , n} \in K'$ , $a_{\omega , n}\in A'$ and $u_{\omega , n}\in U$ be such that

$$ \begin{align*} g_n \dotsm g_1 = k_{\omega, n} a_{\omega, n} u_{\omega, n}. \end{align*} $$

With this notation, we are ready to define the class of measures on U we shall be interested in.

The existence of the limit in condition (3) above will be proved in Lemma 8.1. Moreover, we will see as part of our discussion in §8 that conditions (1) and (2) imply that $\mu $ is H-expanding which will allow us to employ our main measure classification and equidistribution results discussed above.

For the statement of our result on Birkhoff genericity, recall that by Ratner’s theorems the orbit closure $\overline {Hx}$ is homogeneous for any $x\in X$ . We denote the homogeneous probability measure corresponding to $\overline {Hx}$ by $\nu _{ \overline {Hx}}$ .

Theorem 1.11 extends the main results of [Reference Shi91], which used the method of Chaika–Eskin [Reference Chaika and Eskin22] developed for the Teichmüller geodesic flow to prove Birkhoff genericity for the Haar measure on U. The same method was employed in [Reference Fraczek, Shi and Ulcigrai37] to obtain Birkhoff genericity for volume measures on curves. The proof of Theorem 1.11 will be given in §8, using the connection to random walks observed in [Reference Simmons and Weiss93].

Probability measures generated by expanding random walks include a piece of Haar measure on U and, under irreducibility conditions, self-similar measures on $\mathbb {R}^m$ as well as natural self-affine measures on Bedford–McMullen carpets. The latter example is crucial for our application to Diophantine approximation problems on fractals described next. In §9.2, we will also discuss a more general class of fractal measures covered by Definition 1.10.

1.6 Diophantine approximation

By virtue of a correspondence principle going back to the work of Dani [Reference Dani25] and Kleinbock [Reference Kleinbock59], Theorem 1.11 on Birkhoff genericity has consequences for problems in Diophantine approximation, which we shall now describe.

Let $m \in \mathbb {N}$ be a positive integer, $\mathbf {v}=(v_1,\dots ,v_m)^t$ a (column) vector in $\mathbb {R}^m$ , and $\mathbf {r}=(r_1, \dots ,r_m) \in (0,1]^m$ such that $\sum _{i=1}^m r_i=1$ . The vector $\mathbf {v}$ is called $\mathbf {r}$ -badly approximable if there exists a constant $C>0$ such that

(1.3) $$ \begin{align} \max_{1\le i\le m} \left\lvert {v_iq-p_i} \right\rvert ^{1/r_i}\cdot \left\lvert {q} \right\rvert \ge C \end{align} $$

for every $\mathbf {p}=(p_1, \dots , p_m) \in \mathbb {Z}^m$ and $q \in \mathbb {Z}\setminus \{0\}$ . When $r_i=1/m$ for every $i=1,\dots , m$ , such a vector is simply called badly approximable. In the case $m=1$ , the latter corresponds to the classical definition of a badly approximable number. It is easily seen by Dirichlet’s principle that for any vector $\mathbf {v} \in \mathbb {R}^m$ , the left-hand side of equation (1.3) is $\le 1$ for infinitely many pairs $(\mathbf {p}, q) \in \mathbb {Z}^m \times (\mathbb {Z}\setminus \{0\})$ .

The existence of badly approximable vectors was observed by Perron [Reference Perron76] a century ago. It follows from Schmidt’s results [Reference Schmidt88] that such vectors constitute a subset of $\mathbb {R}^m$ of everywhere-full Hausdorff dimension. This was strengthened in more recent works [Reference Kleinbock and Weiss55, Reference Kristensen, Thorn and Velani62] to the statement that badly approximable vectors contained in a sufficiently regular fractal $\mathcal {K}$ form a subset of full Hausdorff dimension in $\mathcal {K}$ . For a general weight $\mathbf {r}$ , the results of [Reference Kleinbock and Weiss57, Reference Kristensen, Thorn and Velani62, Reference Pollington and Velani77] imply that $\mathbf {r}$ -badly approximable vectors have everywhere-full Hausdorff dimension in $\mathbb {R}^m$ . For $\mathbf {r}$ -badly approximable vectors on a fractal set $\mathcal {K}$ , the full-dimension statement is known to hold when $\mathcal {K}$ has a certain product structure (see [Reference Kleinbock and Weiss55, Theorem 8.4], [Reference Kristensen, Thorn and Velani62, Theorems 11,13]).

The results outlined above can be summarized by saying that ( $\mathbf {r}$ -)badly approximable vectors are abundant from the viewpoint of Hausdorff dimension. On the Lebesgue measure side, however, Khintchine’s theorem [Reference Khintchine54] implies that badly approximable vectors have zero Lebesgue measure. Using a generalization of Khintchine’s theorem [Reference Schmidt87], the same is seen to be true for $\mathbf {r}$ -badly approximable vectors. The question whether badly approximable vectors on a given fractal $\mathcal {K}$ also form a null set with respect to a natural measure on the fractal proved to be rather more delicate. The first results in this direction are due to Einsiedler–Fishman–Shapira [Reference Einsiedler, Fishman and Shapira29], who proved that badly approximable vectors have zero Hausdorff measure on certain fractals invariant under toral endomorphisms (in case the dimension is $m=1$ ) or toral automorphisms (in case $m=2$ ). For example, their results apply to the middle-third Cantor set. This was vastly generalized by Simmons–Weiss [Reference Simmons and Weiss93], who established the same statement for general self-similar fractals satisfying a separation condition. To the best of our knowledge, for general weights $\mathbf {r}$ or on fractals which are not strictly self-similar, the question of the measure of badly approximable vectors is open. Our methods allow us to make an initial contribution in this direction. For simplicity, here in the introduction we will describe only the special case of Bedford–McMullen carpets; see §9 for the discussion in full generality.

Bedford–McMullen carpets are two-dimensional self-affine fractals, introduced independently by Bedford [Reference Bedford2] and McMullen [Reference McMullen67], which admit a particularly simple construction. Let $a,b\ge 2$ be distinct integers and divide the unit square $[0,1]^2$ into an $a\times b$ -grid parallel to the coordinate axes. Choose an arbitrary subcollection S of the $ab$ rectangles created and discard the remaining ones. Iteratively proceed in the same way for each of the rectangles that remain, using the same pattern S. The points remaining after infinite iteration form a Bedford–McMullen carpet $\mathcal {K}$ . If $(c_i,d_i)_{i=1}^k$ denote the coordinates of the bottom-left corners of the rectangles kept in the first construction step and we define the affine maps $\phi _i\colon \mathbb {R}^2\to \mathbb {R}^2$ by

$$ \begin{align*} \phi_i(x,y)=\begin{pmatrix}\frac1a & \\ &\frac1b\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}+\begin{pmatrix}c_i\\ d_i\end{pmatrix}, \end{align*} $$

then $\mathcal {K}$ is the unique nonempty compact subset of $\mathbb {R}^2$ satisfying $\bigcup _{i=1}^k\phi _i(\mathcal {K})=\mathcal {K}$ . The Hausdorff dimension of fractals of this type was explicity calculated by Bedford and McMullen. Except for special cases, it turns out that their Hausdorff measure in the correct dimension is infinite [Reference Peres75]. However, there exists another natural measure $\nu _{\mathcal {K}}$ on $\mathcal {K}$ , known as the McMullen measure: It is the unique T-invariant ergodic probability measure on $\mathcal {K}$ of full Hausdorff dimension, where T is the toral endomorphism corresponding to $(\begin {smallmatrix}a&\\ & b\end {smallmatrix})$ [Reference Kenyon and Peres52, Reference McMullen67]. For further background on the fractal geometry of Bedford–McMullen carpets, we refer to the survey article [Reference Fraser, Pollicott and Vaienti36].

The following is a specialization of our Theorem 9.3 to the case of weighted badly approximable vectors on Bedford–McMullen carpets (see Corollary 9.5).

Theorem 1.12. Let $a,b$ be positive integers satisfying $\min \{a^2,b^2\}> \max \{a,b\}$ and let $\mathcal {K}\subset \mathbb {R}^2$ be a Bedford–McMullen carpet invariant under the toral endomorphism $T=(\begin {smallmatrix}a&\\&b\end {smallmatrix})$ . Suppose that $\mathcal {K}$ is not contained in any straight line. Then for the choice of weights

$$ \begin{align*} \mathbf{r}=\biggl(\frac{2\log a-\log b}{\log a+\log b},\frac{2\log b-\log a}{\log a+\log b}\biggr), \end{align*} $$

the set of $\mathbf {r}$ -badly approximable vectors on $\mathcal {K}$ has measure zero with respect to the McMullen measure $\nu _{\mathcal {K}}$ on $\mathcal {K}$ .

The requirement above that $\mathcal {K}$ is not contained in any straight line plays the role of an irreducibility condition. It is satisfied when, in the construction of the Bedford–McMullen carpet described above, the kept rectangles in the pattern S do not all belong to a single line or column in the $a\times b$ -grid.

As mentioned before its statement, the above theorem will follow from a much more general result about Diophantine properties of ‘ $(\mathbf {r},\mathbf {s})$ -matrix sponges’ (Theorem 9.3)—a class of fractals that we will introduce in §9.2.3. In fact, the latter result will imply a version of Theorem 1.12 for higher-dimensional analogues of Bedford–McMullen carpets, which are called ‘self-affine Sierpiński sponges’ in [Reference Kenyon and Peres52]; see Corollary 9.5.